Now that we seem to have survived Y2K, the show must go on. 4 new papers this time. Mark Hovey New papers uploaded to hopf between 12/14/99 and 1/2/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/charact Title: An algebraic geometric realization of the Chern character Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph---math.stanford.edu and plfilho---math.tamu.edu Text of abstract Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map $ch : K^{-i}_{hol}(X) \to \prod_* L^*H^{2*-i}(X)\otimes Q$ from the "holomorphic $K$-theory of $X$ to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic $K$-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from $X$ into $BU$. It has been further studied by the authors in a companion paper. Holomorphic $K$-theory sits between algebraic $K$-theory and topological $K$-theory in the same way that morphic cohomology sits between motivic cohomology and ordinary cohomology. Our constructions provide a bridge between these two worlds. We also realize Chern classes in the case where $X$ is smooth, and establish a universal relation between the Chern character and the Chern classes. For this we use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/holo-k-th Title: Holomorphic $K$-theory, algebraic co-cycles, and loop groups Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph---math.stanford.edu and plfilho---math.tamu.edu Text of abstract In this paper we study the ``holomorphic $K$-theory" of a projective variety. This theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we show that there is a rational isomorphism between holomorphic $K$-theory and the appropriate "morphic cohomology", defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the ``symmetrized loop group" $\om U(n)/\Sigma_n$ where the symmetric group $\Sigma_n$ acts on $U(n)$ via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians $BU(k)$ by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic $K$-theory by inverting the Bott class, then it is rationally isomorphic to topological $K$-theory. Finally we produce explicit obstructions to periodicity in holomorphic $K$ - theory, and show that these obstructions vanish for generalized flag manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/ab Classifying subcategories of modules Mark Hovey mhovey---wesleyan.edu In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We claim that these wide subcategories are analogous to thick subcategories of the derived category D(R). Indeed, let C_0 denote the wide subcategory generated by R; C_0 is the collection of all finitely presented modules precisely when R is coherent. When R is a quotient of a regular commutative coherent ring by a finitely generated ideal, we classify wide subcategories of C_0. In fact, they are on 1-1 correspondence with thick subcategories of small objects of D(R). The proof relies heavily on Thomason's thick subcategory theorem for D(R). We also classify wide subcategories closed under arbitrary coproducts; these are analogous to localizing subcategories of D(R). In this case, we must assume that R is Noetherian, where we use Neeman's classification of localizing subcategories of D(R). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lawson-Lima-Filho-Michelsohn/alg-cycles1 Title: Algebraic Cycles and the Classical Groups - Part I, Real Cycles Authors : H. Blaine Lawson, Jr. and Paulo Lima-Filho and Marie-Louise Michelsohn Email addresses: blaine---math.sunysb.edu, plfilho---math.tamu.edu, mmichelsohn---math.sunysb.edu The groups of algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real structure, it is natural to ask for the properties of the groups of real algebraic cycles on P(V). Similarly, if V carries a quaternionic structure, one can define quaternionic algebraic cycles and ask the same question. In this paper and its sequel the homotopy structure of these cycle groups is completely determined. It turns out to be quite simple and to bear a direct relationship to characteristic classes for the classical groups. It is shown, moreover, that certain functors in K-theory extend directly to these groups. It is also shown that, after taking colimits over dimension and codimension, the groups of real and quaternionic cycles carry E_{\infty}-ring structures, and that the maps extending the K-theory functors are E_{\infty}-ring maps. In fact this stabilized space is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed at the representations R^{n,n} for n \geq 0. This gives a wide generalization of the results in [BLLMM] on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a simple quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. These calculations yield an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory. --------------- 7 new papers this time. Mark Hovey New papers uploaded to hopf between 1/2/00 and 1/25/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Anderson-DavisJ/MacPherson Title: Mod 2 Cohomology of Combinatorial Grassmannians Authors: Laura Anderson and James F. Davis Abstract: Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes. The homotopy groups of this poset are related to the image of the J-homomorphism from stable homotopy theory. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dwyer-Wilkerson/center-calc/center-calc Centers and Coxeter elements by W. G. Dwyer and C. W. Wilkerson dwyer.1---nd.edu wilker---math.purdue.edu Abstract: Suppose that $G$ is a connected compact Lie group. We show that simple numerical information about the Weyl group of $G$ can be used to obtain bounds, often sharp, on the size of the center of $G$. These bounds are obtained with the help of certain Coxeter elements in the Weyl group. Variants of the method use generalized Coxeter elements and apply to $p$-compact groups; in this case a splitting theorem emerges. The Lie group results are mostly known, but our arguments have a conceptual appeal. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey-Palmieri/quillen Stably thick subcategories of modules over Hopf algebras by Mark Hovey and John Palmieri hovey---member.ams.org and palmieri---member.ams.org We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional cocommutative Hopf algebra B. Our method is based on that of Benson-Carlson-Rickard, who classify such subcategories when B=kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of Z/2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/bdi4 ON THE 2-COMPACT GROUP DI(4) Author: D. Notbohm Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group $DI(4)$. The mod-2 cohomology of the associated classifying space $BDI(4)$ realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space $BDI(4)$ and that the maximal torus normalizer determines the isomorphism type of $DI(4)$ as 2-compact group. We also calculate the set of homotopy classes of self maps of $BDI(4)$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/orthogonal A UNIQUENESS RESULT FOR ORTHOGONAL GROUPS AS 2-COMPACT GROUPS D. Notbohm Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizer are isomorphic. It is conjectured that this result generalizes to p-compact groups. Here, we prove the generalization for orthogonal groups $O(n)$, the special orthogonal groups $SO(2k+1)$ and the spinor groups $Spin(2k+1)$ considered as 2-compact groups. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SchwartzL/FK Author: Lionel Schwartz Title: La filtration de Krull de la categorie U et la cohomologie des espaces Jan. 6, 2000 The present paper gives a proof of a conjecture of N. Kuhn : if the mod 2 cohomology of a space has finite Krull filtration in the category of unstable modules, it has to be a locally finite unstable module. Some technical assumptions are required. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WuJ/newwgroup Title of Paper: A braided simplicial group Author(s): Jie Wu Email address of Authors: matwuj---nus.edu.sg Text of Abstract: By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general homotopy group of the 3-sphere is the fixed set of the pure braid group action on a certain combinatorially described group. We also give a certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere. Comments are welcome. --------------- This seems a good time to remind you that if you have submitted a paper to Hopf and it does not appear on this list, it is NOT because Clarence has rejected it. Hopf is not an automated archive, so sometimes it takes a while for papers to be moved into the appropriate spot. On the other hand, it is always possible that Clarence or I have made a mistake, so it doesn't hurt to send e-mail reminding us. 5 new papers this time. Mark Hovey New papers uploaded to hopf between 1/25/00 and 1/29/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/homotopy-ext On spaces of self homotopy equivalences of p-completed classifying spaces of finite groups and homotopy group extensions By Carles Broto and Ran Levi Fix a prime p. A mod-p homotopy group extension of a group $\pi$ by a group G is a fibration with base space $B\pi^\wedge_p$ and fibre $BG^\wedge_p$. In this paper we study homotopy group extensions for finite groups. We observe that there is a strong analogy between homotopy group extensions and ordinary group extensions. The study involves investigating the space of self homotopy equivalences of a p-completed classifying space. In particular we show that under the appropriate assumption on $G$, the identity component of this space is homotopy equivalent to $BZ(G)$, the classifying space of the centre of $G$. We proceed by studying the group of components. We show that this group maps into a group of natural equivalences of a certain functor with kernel and cokernel, which are computable in terms of the first and second derived functors of the inverse limit for a certain diagram of abelian groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/simp-models Combinatorial models for iterated loop spaces By: Fred Cohen and Ran Levi The objective of this paper is to provide free simplicial group models for the functors $\Omega^n X$ and $\Omega^n\Sigma^{n+k}X$. The models are based on classical constructions in simplicial homotopy theory. Specifically, Milnor's functor F, Kan's loops group functor G and the Moore loop space construction $\Omega$ are used to produce these models. The models are given in terms of free groups with specific generators and the formulas defining the simplicial operators are given. The utility of these models is that in them certain maps can be written explicitly in a relatively easy way. To illustrate this a null homotopy of the commutator map on a double loop space is given. Similar ideas are used to give a model for pointed mapping spaces out of a Riemann surface. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/stunted-proj ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES By: Fred Cohen and Ran Levi Let $X_n$ denote the infinite stunted projective space ${\Bbb R}P^\infty/{\Bbb R}P^{n-1}$. In this note we study the homotopy type of this family of spaces. In particular we show that for $n=2 $ and 4, the space $X_n$ splits after looping once and for $n=3$ after looping four times and passing to connected covers. In each case the factors are loop spaces on naturally occuring finite complexes. These result generalise to higher values of $n$, but in those cases without a splitting result. The splittings enable us to carry out a calculation of low dimensional homotopy and loop space homology for these spaces, which complements a computer calculation of Sergeraert and Smirnov. A number of interesting related facts and questions is also discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McGibbon-Strom/numphant Numerical invariants of phantom maps C. A. McGibbon and Jeffrey Strom Wayne State University and Dartmouth College Two numerical homotopy invariants of phantom maps, the Gray index G(f) and the essential category weight E(f), are studied. The possible values of these invariants are determined. In certain cases bounds on these values are given in terms of rational homotopy data. Examples are provided showing that the Gray index can take any positive finite value. For certain cases it is shown that every essential phantom f: X --> Y has finite Gray index. However it is also shown that there exist spaces, e. g. CP^\infty, which are the domains of essential phantoms with infinite index. The same type of analysis is carried out on the essential category weight of a phantom map. If the loop space of X is homotopy equivalent to a finite complex, then every phantom f: X --> Y has E(f) = \infty. However, in certain other cases it is shown that E(f) is strictly less than the rational Lusternik-Schnirelmann category of the domain. A homotopy classification of phantoms f: K(Z, n)--> S^m is given along with the values of E(f). The invariants G and E provide decreasing filtrations on the set of homotopy classes of phantoms from X to Y. A third filtration on this set is introduced for certain special targets. When the rational cohomology of the domain X is finitely generated, this filtration enables one to reduce the search for essential phantoms (into finite type targets) to a finite list of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dickson Dickson Invariants hit by the Steenrod Squares BY K. F. Tan and Kai Xu Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$ by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in attacking the question in the general case. (This paper contains some tedious computations which will be dropped in the simplified version that will be written later.) ---------------- Sorry for the delay; I seem to be getting old and tired. 6 new papers this time. Mark Hovey New papers uploaded to hopf between 1/29/00 and 3/4/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Morava/amrrrfls A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space Authors: Matthew Ando mando---math.uiuc.edu Jack Morava jack---math.jhu.edu We show that the fixed-point formula in an equivariant complex-oriented cohomology theory $E$, applied to the free loop space of a manifold $X$, may be viewed as a (renormalized) Riemann-Roch formula for the quotient of the group law of $E$ by a free cyclic subgroup. If $E$ is $K$-theory, this explains how the elliptic genus associated to the Tate elliptic curve emerges from Witten's analysis of the fixed-point formula in $K$-theory. In general this quotient is not representable, but we show that its torsion subgroup is. In the case that $E$ is the Borel theory associated to the Lubin-Tate theory $E_n$, this leads to a description of the functor represented by $E_n[[q]], analogous to the relationship between the Tate curve and $K$-theory. For a more general equivariant $E$, we show that the formal products which arise in this discussion may be naturally viewed as Thom classes for Thom prospectra as considered by Cohen-Jones-Segal. These prospectra seem to define interesting models for the physicists' space of `small' loops on $X$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Crespo-Saumell/aqfh Title: Non-simply connected $H$-spaces with finiteness conditions Authors: Carlos Broto, Juan A. Crespo and Laia Saumell e-mail addresses: broto---mat.uab.es, chiqui---crm.es, and laia---mat.uab.es This article is concerned with homotopy properties of $H$-spaces $X$ that are reflected in the module of indecomposables $QH^*(X;\F_p)$. It is shown that mod $p$ $H$-spaces $X$ of finite type with finite transcendence degree mod $p$ cohomology and locally finite $QH^*(X;\F_p)$ are $B\Z/p$-null spaces, Eilenberg-MacLane spaces $K(\padic,2)$, $K(\Z/p^r,1)$, and extensions of those. If we restrict attention to $H$-spaces with noetherian mod $p$ cohomology algebra, then we are left with finite mod $p$ $H$-spaces and Eilenberg-MacLane spaces. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fisher/bous Title: A Proof of an Exponent Conjecture of Bousfield Author: Michael J. Fisher Email: mjf7---lehigh.edu Abstract: Let p be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the p-exponent of the spectrum Phi SU(n) is (n-1) + nu_p((n-1)!) for n >= 2. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grodal/limsub Title: Higher limits via subgroup complexes Author: Jesper Grodal Email: jg---math.mit.edu Abstract: We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with values in a G-local coefficient system. We examine when smaller complexes can be used e.g., taking only p-radical subgroups, p-centric subgroups, elementary abelian p-subgroups or various subcollections thereof. Since the subgroup complexes are finite complexes, and often rather small, this provides concrete, computable formulas for these higher limits, generalizing earlier work of especially Jackowski-McClure- Oliver. It also gives a conceptual explanation of high dimensional vanishing results previously established in more indirect ways. As an application we look at the special case where all the higher limits vanish, as for example is the case for group cohomology. If F is a functor on the orbit category our formulas for the higher limits in this case yield five different expressions of F(G) in terms of values of F on proper subgroups. Two of these are `classical' namely Webb's exact sequence of Mackey functors and a formula for calculating stable elements, previously obtained using Alperin's fusion theorem. Examining this case also leads to improvements of sharpness results of homology decompositions due to Dwyer and others. Central to many of the proofs are properties of the Steinberg chain complex of a finite group G, as well as other concepts from the emerging Lie theory for arbitrary finite groups of Alperin, Webb, and others. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhang-Woo/forgetnew1 Title: Phantom maps and Forgetful maps Authors: Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing China and Department of Mathematics Education , Korea University , Seoul , Korea email: pjz62---hotmail.com Moo Ha Woo Department of Mathematics Education , Korea University , Seoul , Korea ABSTRACT: In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so- called Forgetful map. We show that we can insert the Forgetful map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. Remark:This is an upgraded version of a preprint which has been on the archive. A problem in Theorem2.8 has been corrected following a suggestion from K.Iriye. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Karoubi/A_descent_theorem Max KAROUBI A descent theorem in topological K-theory karoubi---math.jussieu.fr Let A be a Banach algebra and A' its complexification. In this paper we show that the homotopy fixed point set of K(A'), the topological K-theory space of A', under complex conjugation is just K(A), the topological K-theory space of A. This result generalizes the well known fact that BO is BU^hZ/2. The proof uses in an essential way Atiyah's KR theory and the Clifford algebra definition of higher K-groups. ---------------- 6 new papers this time. Mark Hovey New papers uploaded to hopf between 3/4/00 and 4/9/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/comp1 Compositions in the v1-periodic homotopy groups of spheres Martin Bendersky and Donald M. Davis mbenders---shiva.hunter.cuny.edu, dmd1---lehigh.edu 21 pages, completed March 7, 2000, submitted to Forum Mathematicum Abstract Let p_i in pi_{n+8i-1}(S^n) denote an element which suspends to a generator of the image of the stable 2-primary J-homomorphism. We determine the image of the composite p_j o p_k in v1-periodic homotopy v_1^{-1} pi_{n+8i+8j-2}(S^n). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson. Odd-primary analogues are also obtained. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org April, 2000 This is a revised version. The basic idea is to automate the passage from unstable to stable homotopy theory, so that it applies in particular to the A^1 category of Voevodsky. So if we start with a model category C and a left Quillen endofunctor G of C, we want to make a new model category, the stabilization of C, where G becomes a Quillen equivalence. The simplest way to do this is with ordinary spectra. Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms. In particular, this holds for the A^1 category. If C has a tensor product, and G is given by tensoring with a cofibrant object K, then we also can construct symmetric spectra. The localization techniques apply here as well, so we get a stable model structure of symmetric spectra without having to assume anything like the Freudenthal suspension theorem. In particular, this is a new construction of the stable model structure on simplicial symmetric spectra. Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general. Also new to this revision is a much more careful comparison between symmetric spectra and ordinary spectra when both are defined. Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity. Under some additional technical hypotheses (which again are satisfied in the A^1 category), we construct a zigzag of Quillen equivalences between symmetric spectra and ordinary spectra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Oliver-Segev/2dim Fixed point free actions on $Z$-acyclic 2-complexes by Bob Oliver and Yoav Segev E-mail: bob---math.univ-paris13.fr, yoavs---math.bgu.ac.il We show that a finite group has an "essential" fixed point free action on an acyclic 2-complex if and only if it is one of the simple groups in the following list: - $PSL_2(2^k)$ for $k\ge2$, - $PSL_2(q)$ for $q\equiv3,5$ (mod 8) and $q\ge5$, - $Sz(2^k)$ for odd $k\ge3$. More precisely, for any finite group $G$, and any 2-dimensional acyclic $G$-CW complex $X$ without fixed points, there is a normal subgroup $H$ in $G$ such that $G/H$ is in the above list, and such that the $G$-action on $X$ looks "essentially" like the $G/H$-action which we construct. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-simpl-alg-proper Title: Every homotopy theory of simplicial algebras admits a proper model Author: Charles Rezk rezk---math.nwu.edu Abstract: We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory, which is allowed to be multi-sorted. The results have applications to the construction of localization model category structures. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Scheerer-Stanley-Tanre/Qcat Fibrewise construction applied to Lusternik-Schnirelmann category by Hans Scheerer, Donald Stanley and Daniel Tanr\'e scheerer---math.fu-berlin.de Don.Stanley---agat.univ-lille1.fr Daniel.Tanre---agat.univ-lille1.fr Abstract: In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted by Qcat(X). It is obtained by applying a base-point free version of Q = Omega-infinity Sigma-infinity fibrewise to the Ganea fibrations. We prove cat(X) >= Qcat(X) >= scat(X), where scat(X) denotes Y. Rudyak's strict category weight. However, Qcat(X) approximates cat(X) better, because e.g. in the case of a rational space Qcat(X)=cat(X) and scat(X) equals the Toomer invariant. We show that Qcat(X x Y) <= Qcat(X)+Qcat(Y). The invariant Qcat is designed to measure the failure of the formula cat(X x S^r)=cat(X)+1. In fact for 2-cell complexes Qcat(X)< cat(X) if and only if cat(X x S^r) <= cat(X) for some r >= 1. We note that the paper is written in the more general context of a functor L from the category of spaces to itself satisfying certain conditions; L= Q, Omega^n Sigma^n, Sp^infinity or L_f are just particular cases. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dicknew (This is a revised version) Dickson Invariants hit by the Steenrod Squares BY K. F. Tan and Kai Xu Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$ by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in attacking the question in the general case. (This paper contains some tedious computations which will be dropped in the simplified version that will be written later.) --------------- 9 new papers this time, including the Mahowald-Ravenel-Shick paper returning the telescope conjecture to the community. Mark Hovey New papers uploaded to hopf between 4/9/00 and 6/4/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Carlson-Karagueuzian-Milgram/hs The Cohomology of the Sylow 2-Subgroup of the Higman-Sims Group A. Adem Mathematics Department University of Wisconsin Madison WI 53706 J. F. Carlson Mathematics Department University of Georgia Athens GA 30602 D. B. Karagueuzian Mathematics Department University of Wisconsin Madison WI 53706 R. James Milgram Mathematics Department Stanford University Stanford CA 94305 Abstract In this paper we compute the mod 2 cohomology of the Sylow 2-subgroup of the Higman--Sims group HS, one of the 26 sporadic simple groups. We obtain its Poincare series as well as an explicit description of it as a ring with 17 generators and 79 relations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Pakianathan/adpak On the Cohomology of Central Frattini Extensions Alejandro Adem and Jonathan Pakianathan Mathematics Department University of Wisconsin Madison, Wisconsin, 53706 adem---math.wisc.edu, pakianat---math.wisc.edu Abstract In this paper we provide calculations for the mod p cohomology of certain p-groups, using topological methods. More precisely, we look at p-groups G defined as central extensions 1-> V -> G ->W ->1 of elementary abelian groups such that the mod p reduction of G/[G,G] is W and the defining k-invariants span the entire image of the Bockstein. We show that if p>dim V-dim W+1, then the mod p cohomology of G can be explicitly computed as an algebra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ausoni-Rognes/tcl_us Title: Algebraic K-theory of topological K-theory Author: Christian Ausoni Author2: John Rognes Email: ausoni---math.ethz.ch Email2: rognes---math.uio.no Abstract: Let l_p = BP<1>_p be the p-complete connective Adams summand of topological K-theory, and let V(1) be the Smith-Toda complex. For p>3 we explicitly compute the V(1)-homotopy of the algebraic K-theory spectrum of l_p. In particular we find that it is a free finitely generated module over the polynomial algebra P(v_2), except for a sporadic class in degree 2p-3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is actually made in the V(1)-homotopy of the topological cyclic homology of l_p. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dwyer-Greenlees/CompleteTorsion Complete modules and torsion modules by W. G. Dwyer and J. P. C. Greenlees Suppose that $R$ is a ring and that $A$ is a chain complex over $R$. Inside the derived category of differential graded $R$-modules there are naturally defined subcategories of $A$-torsion objects and of $A$-complete objects. Under a finiteness condition on $A$, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case $R=Z$ and $A=Z/p$. Addresses: University of Notre Dame, Notre Dame, IN 46556, USA dwyer.1---nd.edu School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK j.greenlees---sheffield.ac.uk 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kuhn/kuhnsplit Stable Splittings and the Diagonal Nicholas J. Kuhn Department of Mathematics, University of Virginia, Charlottesville, VA 22903 njk4x---virginia.edu AMS classification numbers: Primary 55P35; Secondary 55P42 Many approximations to function spaces admit natural stable splittings, with a typical example being the stable splitting of a space C_d(X) approximating Omega^d Sigma^d X. With an eye towards understanding cup products in the cohomology of such function spaces, we describe how the diagonal interacts with the stable splitting. The description involves group theoretic transfers. In an appendix independent of the rest of the paper, we use ideas from Goodwillie calculus to show that such natural stable splittings are unique, and discuss three different constructions showing their existence. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mahowald-Ravenel-Shick/telconj Title: The triple loop space approach to the telescope conjecture Authors: Mark Mahowald, Doug Ravenel, Paul Shick Addresses: Northwestern University, University of Rochester, John Carroll University email: mark---math.mwu.edu, drav---harpo.cc.rochester.edu, shick---jcu.edu AMS Classification: 55 Abstract: The purpose of this paper is to describe an unsuccessful attempt to prove that the telescope conjecture is false for all $n \ge 2$ and all primes $p$. At the time it was originally proposed over 20 years ago, the telescope conjecture appeared to be the simplest and most plausible statement about the relationship between two different localization functors. We hope that the present paper will show that this is no longer the case. We will set up a spectral sequence converging to the homotopy of one of the two localizations (the geometrically defined telescope) of a certain spectrum, and it will be apparent that only a bizarre pattern of differentials would lead to the known homotopy of the localization defined in terms of $BP$-theory. While we cannot exclude such a pattern, it is certainly not favored by Occam's razor. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mahowald-Ravenel-Shick/temss Title: The Thomified Eilenberg-Moore spectral sequence Authors: Mark Mahowald, Doug Ravenel, Paul Shick Addresses: Northwestern University, University of Rochester, John Carroll University email: mark---math.mwu.edu, drav---harpo.cc.rochester.edu, shick---jcu.edu AMS Classification: 55 Abstract: We construct a generalization of the Eilenberg-Moore spectal sequence, which in some interesting cases turns out to be a form the Adams spectral sequence. We apply the spectral sequence to give a new construction of the $Z /p$-equivariant Adams spectral sequence of Greenlees. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert This one has an abstract in .dvi form, so I do not include it. The title is A Proof of the Hilbert-Smith Conjecture by Louis F. McAuley (The Hilbert-Smith conjecture is the one about a topological group having to be a Lie group under certain conditions). ---------------- 7 new papers this time. Sometimes there is a considerable delay between the time the author puts a paper on Hopf and the time it is announced. This delay is sometimes at my end, and sometimes at Clarence's end. I believe the delay on Clarence's end is longer when the author e-mails him the paper, as Clarence then has to do more work. I believe this is the reason that some of the papers announced this time were actually submitted sooner than some of the papers announced last time. Mark Hovey New papers appearing on hopf between 6/4/00 and 6/16/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenD-CohenF-Xicotencatl/CCX Title: Lie algebras associated to fiber-type arrangements Authors: Daniel C. Cohen, Frederick R. Cohen, Miguel Xicotencatl math.AT/0005091 Addresses of Authors D. Cohen, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 F. Cohen, Department of Mathematics, University of Rochester, Rochester, NY 14627 M. Xicotencatl, Depto. de Mathematicas, Cinvestav del IPN, Mexico City Max-Plank-Institut fur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany Email address of Authors cohen---math.lsu.edu cohf---math.rochester.edu xico------mpim-bonn.mpg.de Abstract: Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive elements in the homology of the loop space for the complement of the associated subspace arrangement. Furthermore, this last Lie algebra is given by the homotopy groups modulo torsion of the loop space of the complement of the subspace arrangement. Looping further yields an associated Poisson algebra, and generalizations of the "universal infinitesimal Poisson braid relations." 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fausk-Lewis-May/FLMApril20 The Picard Group of Equivariant Stable Homotopy Theory by H. Fausk, L.G. Lewis, Jr, and J.P. May The University of Chicago (Fausk and May) Syracuse University (Lewis) fausk---math.uchicago.edu, lglewis---mailbox.syr.edu, may---math.uchicago.edu April 20, 2000 Let G be a compact Lie group. We describe the Picard group Pic(HoGS) of invertible objects in the stable homotopy category of G-spectra in terms of a suitable class of homotopy representations of G. Combining this with results of tom Dieck and Petrie, which we reprove, we deduce an exact sequence that gives an essentially algebraic description of Pic(HoGS) in terms of the Picard group of the Burnside ring of G. The deduction is based on an embedding of the Picard group of the endomorphism ring of the unit object of any stable homotopy category C in the Picard group of C. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org June, 2000 (This is an updated version; following an idea of Voevodsky, we strengthen our proof that stable homotopy isomorphisms agree with stable equivalences of ordinary spectra so that it applies to one version of motivic homotopy theory. ) The basic idea is to automate the passage from unstable to stable homotopy theory, so that it applies in particular to the A^1 category of Voevodsky. So if we start with a model category C and a left Quillen endofunctor G of C, we want to make a new model category, the stabilization of C, where G becomes a Quillen equivalence. The simplest way to do this is with ordinary spectra. Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms. If C has a tensor product, and G is given by tensoring with a cofibrant object K, then we also can construct symmetric spectra. The localization techniques apply here as well, so we get a stable model structure of symmetric spectra without having to assume anything like the Freudenthal suspension theorem. In particular, this is a new construction of the stable model structure on simplicial symmetric spectra. Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general. Also new to this revision is a much more careful comparison between symmetric spectra and ordinary spectra when both are defined. Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity. Under some additional technical hypotheses (which again are satisfied in the A^1 category), we construct a zigzag of Quillen equivalences between symmetric spectra and ordinary spectra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May/MMM Equivariant orthogonal spectra and S-modules by M.A. Mandell and J.P. May The University of Chicago mandell---math.uchicago.edu may---math.uchicago.edu April 20, 2000 The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of S-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of S-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to S-modules. We then develop the equivariant theory. For a compact Lie group G, we construct a symmetric monoidal model category of orthogonal G-spectra whose homotopy category is equivalent to the classical stable homotopy category of G-spectra. We also complete the theory of S_G-modules and compare the categories of orthogonal G-spectra and S_G-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/PicApril20 Picard groups, Grothendieck rings, and Burnside rings of categories J.P. May The University of Chicago may---math.uchicago.edu For Saunders Mac Lane, on his 90th birthday April 20, 2000 We discuss the Picard group, the Grothendieck ring, and the Burnside ring of a symmetric monoidal category, and we consider examples from algebra, homological algebra, topology, and algebraic geometry. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May-Neumann/MNApril20 On the cohomology of generalized homogeneous spaces by J.P. May and F. Neumann The University of Chicago Georg-August-Universit\"at, G\"ottingen, Germany may---math.uchicago.edu neumann---cfgauss.uni-math.gwdg.de April 20, 2000 We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces G/H, where G is a finite loop space or a p-compact group and H is a ``subgroup'' in the homotopical sense. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Santos/equivariant-D-T A note on the equivariant Dold-Thom theorem by Pedro F. dos Santos Addresses of Author: Department of Mathematics, Texas A&M University, College Station TX-77840 Department of Mathematics, Instituto Superior Tecnico, 1049 Lisboa, Portugal Email: pedfs---math.ist.utl.pt In this note we prove a version of the classical Dold-Thom theorem for the RO(G)-graded equivariant homology functors H^G_*(-;RM), where G is a finite group, M is a discrete Z[G]-module, and RM is the Mackey functor associated to M. In the case where M=Z with the trivial G-action, our result says that, for a G-CW-complex X, and for a finite dimensional G-representation V, there is a natural isomorphism [S^V,Z_0(X)]_G \cong H^G_V(X;RM); where Z_0(X) denotes the free abelian group on X. ---------------- 13 new papers this time. Mark Hovey New papers appearing on hopf between 6/16/00 and 7/16/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/AlAgl-Brown-Steiner/multiplecat Multiple categories: the equivalence of a globular and a cubical approach Fahd A. A. Al-Agl, Ronald Brown, Richard Steiner math.CT/0007009 Fahd A. A. Al-Agl\\Um-Alqura University,\\ Makkah\\Saudi Arabia Ronald Brown, \\ School of Informatics, \\ Mathematics Division, \\ University of Wales,\\ Bangor, Gwynedd LL57 1UT, \\ United Kingdom. Richard Steiner, \\ Department of Mathematics, \\ University of Glasgow, \\University Gardens, \\ Glasgow G12 8QW \\ United Kingdom r.brown---bangor.ac.uk r.steiner---maths.gla.ac.uk We show the equivalence of two kinds of strict multiple category, namely the well known globular omega-categories, and the cubical omega-categories with connections. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Strom/TrivModF Homotopy Classes that are Trivial Mod F Martin Arkowitz (Martin.Arkowitz---Dartmouth.edu) Jeffrey Strom (Jeffrey.Strom---Dartmouth.edu) Dartmouth College If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha _* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_{F}(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_{\Sigma}(X,Y) \subseteq Z_{\M}(X,Y) \subseteq Z_{\S}(X,Y), and we find examples of {\it finite complexes} X and Y for which these inclusions are strict. We are also interested in Z_{F}(X) = Z_{F}(X,X) which under composition has the structure of a semi-group with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semi-group Z_{F}(X) is nilpotent. More generally, the nilpotency of Z_{F}(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_{F}(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arlettaz/Arlettaz-survey Title: Algebraic K-theory of rings from a topological viewpoint Author: Dominique Arlettaz Dominique Arlettaz, Institut de math\'ematiques, Universit\'e de Lausanne, CH-1015 Lausanne, Switzerland dominique.arlettaz---ima.unil.ch Abstract: This paper is a long survey providing the basic definitions of the algebraic K-theory of rings and an overview of the main classical theorems which have been obtained by arguments from algebraic topology (in particular by using methods from stable homotopy theory, group cohomology and Postnikov theory). It will appear in Publicacions Matem\`atiques. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arlettaz-Ausoni-Mimura-Yagita/Arlettaz-A-M-Y Title: Integral cohomology and Chern classes of the special linear group over the ring of integers Author1: Dominique Arlettaz Author2: Christian Ausoni Author3: Mamoru Mimura Author4: Nobuaki Yagita Author1: Dominique Arlettaz, Institut de math\'ematiques, Universit\'e de Lausanne, CH-1015 Lausanne, Switzerland Author2: Christian Ausoni, Departement Mathematik, HG, ETH-Zentrum, 8092 Z\"urich, Switzerland Author3: Mamoru Mimura, Department of Mathematics, Faculty of Science, Okayama University, Okayama, Japan 700 Author4: Nobuaki Yagita, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan E-mail1: dominique.arlettaz---ima.unil.ch E-mail2: ausoni---math.ethz.ch E-mail3: mimura---math.okayama-u.ac.jp E-mail4: yagita---mito.ipc.ibaraki.ac.jp Abstract: This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(Z) over the ring of integers Z. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Casacuberta-Scherer/casasche Homological localizations preserve 1-connectivity by Carles Casacuberta and Jerome Scherer Universitat Autonoma de Barcelona Universite de Lausanne casac---mat.uab.es jerome.scherer---ima.unil.ch To appear in Contemporary Mathematics, Proceedings of the 1999 Arolla Conference on Algebraic Topology. Every generalized homology theory $E$ yields a localization functor $L$ that sends the $E$-equivalences to homotopy equivalences. We prove that if $X$ is any $1$-connected space, then $LX$ is also $1$-connected, for every generalized homology theory $E$. This is deduced from a result by Hopkins and Smith stating that if $K(\Z,2)$ is $E$-acyclic then $E$ is trivial. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dugger/ddpres Title: Combinatorial Model Categories Have Presentations Author: Daniel Dugger Purdue University West Lafayette, IN 47906 Email: ddugger---math.purdue.edu We show that every combinatorial model category can be obtained---up to Quillen equivalence---by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of `generators' and a set of `relations' ---i.e., any combinatorial model category has a presentation. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dugger/dduniv Title: Universal Homotopy Theories Author: Daniel Dugger Address: Purdue University West Lafayette, IN 47906 Email: ddugger---math.purdue.edu Abstract: Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and also discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Goebel-Rodriguez-Shelah/locsimple TITLE: Large localizations of finite simple groups AUTHORS: Ruediger Goebel, Jose L. Rodriguez, and Saharon Shelah R.Goebel---uni-essen.de, jlrodri---mat.uab.es, shelah---math.huji.ac.il ABSTRACT: A group homomorphism $\eta: H\to G$ is called a localization of $H$ if every homomorphism $\varphi : H\to G$ can be `extended uniquely' to a homomorphism $\Phi :G\to G$ in the sense that $\Phi \eta = \varphi$. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation $A_n\to SO_{n-1}(\R)$ of the alternating group $A_n$, which turns out to be a localization for $n$ even and $n\geq 10$. Dror Farjoun asked if there is any upper bound in cardinality for localizations of $A_n$. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group $H$, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space $K(H,1)$ for any non abelian finite simple group $H$. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/finite Equivariant p-adic Homotopy Theory Michael A. Mandell mandell---math.uchicago.edu Let G be a finite group. We show that the cochain functor with coefficients in \FPbar is an equivalence between the p-adic G-equivariant homotopy category of finite type nilpotent G-spaces and a full subcategory of the homotopy category of diagrams of \einf \FPbar-algebras indexed on the orbit category of G. This turns out to be an easy consequence of Elmendorf's Theorem and Kan's work on diagrams in closed model categories plus the equivalence in the nonequivariant context. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity Title: Pretty Good Gravity Author: Jack Morava (not yet on xxx, but will be soon) Address: Dept. of Mathematics, the Johns Hopkins Uniperversity e-mail address: jack---math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. [This is a writeup of a talk at the RIMS Symposium on algebraic geometry and integrable systems related to string theory, June 12-16, 2000.] 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ravenel/first Title of paper: The first Adams-Novikov differential for the spectrum T(m) Author: Douglas C. Ravenel Address of Author: University of Rochester, Rochester, NY 14627 Email address of author: drav---math.rochester.edu Abstract: There are p-local spectra T(m) with $BP_{*}(T(m))=BP_{*}[t_{1},\dots ,t_{m}]$. In this paper we determine the first nontrivial differential in the Adams--Novikov spectral sequence for each of them for p odd. For m=0 (the sphere spectrum) this is the Toda differential, whose source has filtration 2 and whose target is the first nontrivial element in filtration 2p+1. The same goes for m=1, and for larger m the target is $v_2$ times the first such element. The proof uses the Thomified Eilenberg-Moore spectral sequence. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ravenel/micro Title of paper: The microstable Adams-Novikov spectral sequence Author: Douglas C. Ravenel Address of Author: University of Rochester, Rochester, NY 14627 Email address of author: drav---math.rochester.edu Abstract: In the Adams--Novikov spectral sequence one considers Ext groups over the Hopf algebroid $\Gamma =BP_{*}(BP)$. There are spectra $T(m)$ with $BP_{*} (T (m))=BP_{*}[t_{1},...,t_{m}]$, which leads one to replace $\Gamma $ by $\Gamma (m+1)=\Gamma / (t_{1},... ,t_{m})$. The corresponding Ext groups have certain structural features that are independent of $m$. In this paper we set up an algebraic framework for studying the limit as $m \to \infty $. In particular there is an analog of the chromatic spectral sequence in which the Morava stabilizer group gets replaced by an infinitesimal analog, hence the title. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/koszulii I can't read the abstract of this file, but I think this is not Larry's fault. Clarence is out of town though, and I am about to be, so I wanted to announce it now. It has to do with invariant theory of Z/p acting on a polynomial ring F[V]. The detailed abstract will appear next time. --------------- Sorry for the long delay since the last such announcement. One big factor contributing to the delay is e-mail attachments. Clarence has trouble dealing with these, and it also messes up my system. So it would be a big help to us if you could follow the old ftp method, or the newer web browser method, of uploading papers to Hopf. 14 new papers this time, including the abstract of Larry Smith's paper that was announced last time. Mark Hovey New papers appearing on hopf between 7/16/00 and 9/14/00. 0. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/koszulii (This paper was announced last time without abstract. Here is the abstract.) Title of Paper: Invariant Theory and the Koszul Complex Representations of Z/p in Characteristic p Applications Author: Larry Smith AMS Code: 13A50 Invariant Theory Address: Mathematisches Institut Bunsenstrasse 3--5 D 37073 Goettingen Federal republic of germany e-mail: larry---sunrise.uni-math.gwdg.de THIS IS a POstScript file. Summary: We study the ring of invariants $\F[V]^{\Z/p}$\/, and its derived functors $H^i(\Z/p\semicolon \F[V])$\/, of the cyclic group $\Z/p$ of prime order $p$ over a field $\F$ of characteristic $p$\/. We verify a formula of Ellingsrud and Skjelbred \cite{norway} for the homological codimension, show the quotient algebra $\F[V]^{\Z/p}/\Im(\Tr^{\Z/p})$ is Cohen-Macaulay, and that the ideal generated by the elements in the image of the transfer homomorphism, $\Im(\Tr^{\Z/p}) \subset \F[V]^{\Z/p}$\/, is primary of height $n-1$ when $V$ is an $n$-dimensional irreducible representation of $\Z/p$\/. Using our cohomological computations and a previous result \cite{vectors} about permutation representations we are able to obtain an upper bound for the degree of homogeneous forms in a minimal algebra generating set for $\F[V]^{\Z/p}$\/. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Basterra/abwgeec Title: The Witten genus and equivariant elliptic cohomology Authors: Matthew Ando mando---math.uiuc.edu Maria Basterra basterra---math.uiuc.edu Department of Mathematics, The University of Illinois at Urbana-Champaign Abstract: We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of the Witten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Hopkins-Strickland/eswgtc-2/ Elliptic spectra, the Witten genus, and the theorem of the cube. (revised version) M. Ando, M. J. Hopkins, and N. P. Strickland University of Illinois at Urbana-Champaign mando---math.uiuc.edu MIT mjh---math.mit.edu University of Sheffield N.P.Strickland---sheffield.ac.uk This is a revised version of an earlier paper (1998) with the same title. We show that every elliptic spectrum receives a natural MU<6>-orientation. For the elliptic spectrum defined by the Tate curve, this orientation specializes to the Witten genus. The naturality of the orientation implies that the modularity of the Witten genus for MU<6>-manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi-Oliver/blo1 Homotopy equivalences of p-completed classifying spaces of finite groups by Carles Broto, Ran Levi, and Bob Oliver We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category with the following properties. Two p-completed classifying spaces BG_p^\wedge and BG'_p^\wedge have the same homotopy type if and only if the associated categories are equivalent. And the topological group Aut(BG_p^\wedge) of self equivalences is determined by the self equivalences of the associated category. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Davis-Mahowald/eo2/ Nonimmersions of real projective spaces implied by eo2 Robert R. Bruner Wayne State University, Detroit, MI 48202 rrb---math.wayne.edu Donald M. Davis Lehigh University, Bethlehem, PA 18018 dmd1---lehigh.edu Mark Mahowald Northwestern University, Evanston, IL 60201 mark---math.nwu.edu AMS Classifications: 57R42, 55N20 Abstract: Recently Hopkins and Mahowald constructed a new 2-primary ring spectrum eo2, satisfying H^*(eo2)=A//A2. We use eo2 to obtain new results regarding nonimmersions of real projective spaces in Euclidean space. The method is to say enough about eo2-cohomology of a product of real projective spaces to obtain nonexistence of certain axial maps. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Chacholski-Dwyer-Intermont/complication The A-complication of a space W. Chacholski, W. G. Dwyer, and M. Intermont Suppose that A is a pointed CW-complex. We look at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complication of B. Studying the complication leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CW_A(X) of an arbitrary space X. Yale University, New Haven, CT 06520 USA University of Notre Dame, Notre Dame IN 46556 USA Kalamazoo College, Kalamazoo MI, 49006 USA MSC2000: 55P60, 55P99 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fors/AugmHom Title: Augmental Homology Theory and the Künneth Formula for Topological Joins. Author: Göran Fors. AMS Classification numbers: 55N10. Address: Department of Mathematics, University of Stockholm, SE-106 91 Stockholm, Sweden E-mail address: goranf---matematik.su.se We prove topological join versions of the relative Eilenberg-Zilber Theorem and the relative Künneth Formula. We also express the local homology groups for topological joins and products in terms the local homology groups for the factors. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbunov-Malikov-Schechtman/group-all-fedin1 On chiral differential operators over homogeneous spaces Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA;\ vgorb\---ms.uky.edu F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA;\ fmalikov\---mathj.usc.edu V.S.: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France;\ vadik\---ihes.fr The notion of an algebra of chiral differential operators (cdo for short) over a smooth algebraic variety X has been studied by the authors previously. We give a classification of cdo over X in the following cases: X=G is an affine algebraic group; X=G/N or G/P where N is a unipotent subgroup and P is a parabolic subgroup and G is simple (the extension to the case of a semisimple G being straightforward). The above sheaves are constructed using the BRST (or quantum Hamiltonian) reduction of the corresponding cdo's on G. The classification of cdo over homogeneous spaces is exactly reflected in the BRST world: namely the square of the corresponding BRST charge is zero at all levels for G/N, only at the critical level for G/B and is never zero for G/P. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ishiguro/G2 "Classifying spaces and a subgroup of the exceptional Lie group G_2" Kenshi Ishiguro Mathematics subject classification: 55R35 Abstract: We consider a problem on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group, as well as some related problems. A previously obtained necessary condition is shown to be not sufficient. Our counterexample is given by a quotient group \Gamma_2 of a subgroup of the exceptional Lie group G_2 at p=3. The 3-adic K-theory of B\Gamma_2 and BG_2 are isomorphic , though the loop space of the 3-completion of B\Gamma_2 is not a 3-compact group. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/quinn Title: The dualizing spectrum of a topological group Author: John R. Klein AMS subjclass: Primary: 55P91, 55N91, 55P42, 57P10. Secondary: 55P25, 20J05,18G15. Address: Dept. Of Mathematics, Wayne State University, Detroit, MI 48202 e-mail: klein---math.wayne.edu Abstract: To a topological group G, we assign a naive G-spectrum D_G, called the "dualizing spectrum" of G. When the classifying space BG is finitely dominated, we show that D_G detects Poincare duality in the sense that BG is a Poincare duality space if and only if D_G is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a "norm map" which is defined for any G and for any naive G-spectrum E. Applications include: (1) a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of finitely dominated spaces, the total space satisfies Poincare duality if and only if the base and fiber do. (2) An entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincare duality space. (3) A new proof of Browder's theorem that every finite H-space satisfies Poincare duality. (4) We show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. We prove a vanishing result for this theory. In an appendix, we identify the homotopy type of D_G for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. (This paper has already been accepted for publication in Math. Annalen.) 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/chains Cochain Multiplications Michael A. Mandell mandell---math.uchicago.edu Abstract We describe a refinement of the Eilenberg--Steenrod axioms that provides a necessary and sufficient condition for functors from spaces to algebras or E-infty algebras to be naturally quasi-isomorphic to the singular cochain functor. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert (This is a revised version of the author's paper proving the Hilbert-Smith conjecture about certain topological groups being forced to be Lie. The abstract has appeared at least twice before here, so I omit it). MH 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pakianathan-YalcinE/nc Title: On Commuting and Non-Commuting Complexes Authors: Jonathan Pakianathan and Erg\"un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 06A09, 05E25. Addresses: Department of Mathematics University of Rochester N.Y., U.S.A. Department of Mathematics Bilkent University Ankara, Turkey Abstract: In this paper we study various simplicial complexes associated to the commutative structure of a finite group $G$. We define $NC(G)$ (resp. $C(G)$) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets of nontrivial elements in $G$. We observe that $NC(G)$ has only one positive dimensional connected component, which we call $BNC(G)$, and we prove that $BNC(G)$ is simply connected. Our main result is a simplicial decomposition formula for $BNC(G)$ which follows from a result of A. Bj\"orner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary we obtain that if $G$ has a nontrivial center or if $G$ has odd order, then the homology group $H_{n-1}(BNC(G))$ is nontrivial for every $n$ such that $G$ has a maximal noncommuting set of order $n$. We discuss the duality between $NC(G)$ and $C(G)$, and between their $p$-local versions $NC_p(G)$ and $C_p(G)$. We observe that $C_p(G)$ is homotopy equivalent to the Quillen complexes $A_p(G)$, and obtain some interesting results for $NC_p(G)$ using this duality. Finally, we study the family of groups where the commutative relation is transitive, and show that in this case, $BNC(G)$ is shellable. As a consequence we derive some group theoretical formulas for the orders of maximal non-commuting sets. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YalcinE/clpg4 Title: Set Covering and Serre's Theorem on the Cohomology Algebra of a $p$-Group Author: Erg\" un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J06; Secondary: 20D15, 20D60, 51E20. Address: Department of Mathematics Bilkent University Ankara, Turkey Email: yalcine---math.mcmaster.ca Abstract: We define a group theoretical invariant, denoted by $s(G)$, as a solution of a certain set covering problem, and show that it is closely related to $chl(G)$, the cohomology length of a $p$-group $G$. By studying $s(G)$, we improve the known upper bounds for the cohomology length of a $p$-group, and determine $chl(G)$ completely for extra-special $2$-groups of real type. ---------------- Two new papers this time. Mark Hovey New papers appearing on hopf between 9/14/00 and 9/28/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lundell/stircenid1 Title: Stirling and Central Factorial Number Identies Author: Albert T. Lundell Address: Department of Mathematics, Box 395 University of Colorado Boulder, Colorado 80309 E-mail: lundell---euclid.colorado.edu This paper contains many identities related to Stirling numbers and central factorial numbers, with an emphasis toward divisibility properties. The paper is self-contained and contains proofs of the identities. There is a short section relating these numbers to the James numbers U(n,r), i.e., the index of p_*(\pi_{2n-1}(W_{n,r})\subset\pi_{2n-1}(S^{2n-1}), where p:W_{n,r}\arrow S^{2n-1} is the fibration of complex Stiefel manifolds. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Menichi/Free_Loop Title: The cohomology ring of free loop spaces Author: Luc Menichi AMS classification numbers: 55P35, 16E40, 55P62, 57T30, 55U10. address: Universite d'Angers Faculte des Sciences Departement de Mathematiques 2 Boulevard Lavoisier 49045 ANGERS Cedex 01 - FRANCE Luc.Menichi---univ-angers.fr Abstract: Let $X$ be a simply connected space and $\Bbbk$ a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space $HH^{*}S_*(\Omega X)$ is isomorphic to the free loop space cohomology $H^{*}(X^{S^{1}})$. We proved that this isomorphism is compatible with both the cup product on $HH^{*}S_*(\Omega X)$ and on $H^{*}(X^{S^{1}})$. In particular, we explicit the algebra $H^{*}(X^{S^{1}})$ when $X$ is a suspended space, a complex projective space or a finite CW-complex of dimension $p$ such that $\frac {1}{(p-1)!}\in {\Bbbk}$. --------------- Four new papers this time, all from some energetic guy named Greenlees. He maintains a bibliography on Hopf as well, under http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/greenleesbiblio Mark Hovey New papers appearing on hopf between 9/28/00 and 10/2/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/axiomatic Title: ``Tate cohomology in axiomatic stable homotopy theory.'' Author: J.P.C.Greenlees AMS classification numbers: 55U35, 55T99, 55P42, 55P91, 55N91 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: Any smashing localization in an axiomatic stable homotopy theory in the sense of Hovey-Palmieri-Strickland gives rise to a Tate theory. Various known versions of Tate cohomology (for example in commutative algebra, in the cohomology of groups, in equivariant homotopy theory and in chromatic stable homotopy theory) are considered from this point of view. Status: Submitted for publication. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/guanajuato Title: Local cohomology in equivariant topology Author: J.P.C.Greenlees AMS classification numbers: 13D45, 19L41, 20Jxx, 55N91, 55N22, 55P43 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: The article (based on talks at the Guanajuato Workshop on Local Cohomology, December 1999) describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather different families of examples. Both topology and commutative algebra benefit from the connection, and many interesting questions remain open. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/so3q Title: Rational SO(3)-equivariant cohomology theories Author: J.P.C.Greenlees AMS classification numbers: 55N91, 55P42, 55P62, 55P91 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: The results of previous work for the circle and O(2) are used to give an explicit algebraic model of the category of rational SO(3)-spectra. This gives a complete classification of rational SO(3)-equivariant cohomology theories. A number of new features appear for the first time for this group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees-Hopkins-Rosu/ellT Title: Rational S^1-equivariant elliptic cohomology Authors:J.P.C.Greenlees, M.J.Hopkins and I.Rosu AMS Class numbers: 55N34, 55N91, 55P42, 55P62 \address{JPCG: Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} \address{MJH: Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.} \email{mjh---math.mit.edu} \address{IR: Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.} \email{ioanid---math.mit.edu} Abstract: We give a functorial construction of a rational $S^1$-equivariant cohomology theory from an elliptic curve equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an $S^1$-representation is given by the sheaf cohomology of a suitable line bundle on the curve. The construction is easy: by considering functions on the elliptic curve with specified poles one may write down the representing $S^1$-spectrum in the first author's algebraic model of rational $S^1$-spectra. ---------------- Ten new papers this time. Mark Hovey New papers appearing on hopf between 10/2/00 and 11/8/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Luptin-Murillo/SubgpsofHEs Title: Subgroups of the Group of Self-Homotopy Equivalences Authors: Martin Arkowitz, Gregory Lupton and Aniceto Murillo. Classification Nos. (1991): Primary 55P10; Secondary 55P62, 55Q05. Addresses: Department of Mathematics, Dartmouth College, Hanover NH 03755 U.S.A. Department of Mathematics, Cleveland State University, Cleveland OH 44115 U.S.A. Departmento de Algebra, Geometria y Topologia, Universidad de Malaga, Ap. 59, 29080 Malaga, Spain e-mail Addresses: Martin.Arkowitz---Dartmouth.edu Lupton---math.csuohio.edu Aniceto---agt.cie.uma.es Abstract: Denote by $\mathcal{E}(Y)$ the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex $Y$. We give a selection of results about certain subgroups of $\mathcal{E}(Y)$. We establish a connection between the Gottlieb groups of $Y$ and the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of $Y$, denoted by $\mathcal{E}_{\#}(Y)$. We give an upper bound for the solvability class of $\mathcal{E}_{\#}(Y)$ in terms of a cone decomposition of $Y$. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org October, 2000 This is the final version, to appear in JPAA. There are several significant notational changes, and many minor corrections in this version. (Rest of abstract elided, since it has appeared twice already.) 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/torsext2 Cotorsion theories, model category structures, and representation theory by Mark Hovey mhovey---wesleyan.edu AMS Classification: 20C05,20J05,18E30,18G35, 55U35 We make a general study of Quillen model structures on abelian categories. Given a proper class P of short exact sequences on an abelian cateory A, we define what it means for a model structure to be compatible with P. We then give a complete characterization of model structures compatible with P. This characterization is in terms of cotorsion theories, which were introduced by Salce and have been much studied recently by Enochs and coauthors. We apply the general method to construct a stable category of $K[G]$-modules where $K$ is a principal ideal domain and $G$ is a finite group. This is a compactly generated triangulated category that generalizes the well-known stable category of $k[G]$-modules, where $k$ is a field. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/brown-II A Brown representability theorm via coherent functors Author: Henning Krause Address of Author: University of Bielefeld, Germany Email address of Author: henning---mathematik.uni-bielefeld.de Abstract: We discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-Shipley/telescope Title: A telescope comparison lemma for THH (to appear in Topology and its Applications) Authors: Mike Mandell and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Mike Mandell 5734 University Ave. Chicago, IL 60637 USA Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: mandell---math.uchicago.edu bshipley---math.purdue.edu Abstract: The usual telescope or sequential homotopy colimit construction of the underlying infinite loop space must be replaced for symmetric spectra by a homotopy colimit over the category of finite sets and injections. Here we show that for convergent symmetric spectra this modified homotopy colimit agrees with the usual telescope construction. This sharpens B\"okstedt's original lemma because no connectivity conditions are necessary here. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/2local `The stable homotopy category has a unique model at the prime 2' Stefan Schwede Fakultaet fuer Mathematik Universitaet Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de ABSTRACT: In a closed model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences. But passage to the homotopy category loses information and in general the `homotopy theory' can not be recovered from the homotopy category. We show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory, at least 2-locally. We prove a uniqueness theorem which says that there is only one model structure (up to so called Quillen equivalence) underlying the stable homotopy category of 2-local spectra. This theorem is a 2-local strenghtening of a result with B. Shipley, given in `A uniqueness theorem for stable homotopy theory', in that we use only the triangulated structure of the stable homotopy catgory. The earlier result with Shipley works integrally, but needs additional structure, namely the action of the ring of stable homotopy groups of spheres. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede-Shipley/unique Title: A uniqueness theorem for stable homotopy theory Authors: Stefan Schwede and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Stefan Schwede Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: schwede---mathematik.uni-bielefeld.de bshipley---math.purdue.edu Abstract: In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres, $\pi^s$. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of $\pi^s$. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space $QS^0$ to the endomorphism space is a weak equivalence. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/monoid.unique Title: Monoidal uniqueness of stable homotopy Author: Brooke Shipley AMS Classification numbers: 55U35 55P42 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley---math.purdue.edu Abstract: We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences produced here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, $\mathcal{W}$-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/rational.circle Title: An algebraic model for rational $S^1$-equivariant stable homotopy theory Author: Brooke Shipley AMS Classification numbers: 55P62 55P91 55P42 55N91 18E30 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley---math.purdue.edu Greenlees defined an abelian category $A$ whose derived category is equivalent to the rational $S^1$-equivariant stable homotopy category whose objects represent rational $S^1$-equivariant cohomology theories. We show that in fact the model category of differential graded objects in $A$ ($dgA$) models the whole rational $S^1$-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between $dgA$ and the model category of rational $S^1$-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The new ingredients here are certain Massey product calculations and the work on rational stable model categories from "Classification of stable model categories" and "Equivalences of monoidal model categories" with Schwede; see http://www.math.purdue.edu/~bshipley/ 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Tamanoi/orbifold Title: Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory Author: Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz Santa Cruz, CA 95064 Email: tamanoi---math.ucsc.edu Abstract: We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mutually commuting elements (of order powers of $p$) in the wreath product G~S_n in terms of corresponding numbers of G. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K-theories of symmetric products of a G-manifold M. AMS Classification Numbers: 55N20, 55N91, 57S17, 57D15, 20E22 --------------- Twelve new papers this time. Mark Hovey New papers appearing on hopf between 11/8/00 and 11/26/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dorabiala/transfer Abstract: The goal of this paper is to show that if a smooth fiber bundle has a compact Lie group as structure group, then the transfer map for the algebraic K-theory of spaces satisfies analogs of the Mackey Double coset formula and Feshbach's sum formula. We also prove a "cut and paste" formula for parametrized Reidemeister torsion. Wojtek Dorabiala 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/local Author: Stephen A. Mitchell Title: The algebraic K-theory spectrum of a 2-adic local field e-mail: mitchell---math.washington.edu We explicitly determine the homotopy type of the 2-completed algebraic K-theory spectrum KF, where F is an arbitrary finite extension of the 2-adic rational numbers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localhom Author: Stephen A. Mitchell Title: The mod 2 homology of the general linear group of a 2-adic local field e-mail: mitchell---math.washington.edu Let F be a finite extension of the 2-adic rational numbers. We compute the mod 2 homology of the general linear group GL(F) as a Hopf algebra over the Steenrod algebra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morton-Strickland/cost-hrk The Hopf Rings for KO and KU Dena S. Cowen Morton and Neil P. Strickland 55N15; 55P43 math.AT/0011125 Department of Mathematics Xavier University Cincinnati OH 45207 USA Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We compute the mod two homology Hopf rings of the spectra KO and KU. The spaces in these spectra are the infinite classical groups and their coset spaces, and their homology was first calculated in the Cartan seminars, but the Hopf ring structure was first determined in the second author's unpublished PhD thesis. The presentation given here serves as an introduction to the first author's much more intricate work on the connective spectrum bo. The Hopf ring viewpoint turns out to be very convenient for understanding the homological effect of various maps between classical groups and fibrations of their connective covers. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Saneblidze-Umble/SUpaper Title: A Diagonal on the Associahedra Authors: Samson Saneblidze and Ronald Umble MSC-class: 57T30; 55U10; 55N20; 55N10 xxx.LANL.gov: math.AT/0011065 Author's Addresses: A. Razmadze Mathematical Institute, M. Aleksidze St., 1, 380093 Tbilisi, Georgia Department of Mathemaitcs, Millersville Univ. of PA, Millersville, PA 17551 Author's e-mail addresses: sane---rmi.acnet.ge ron.umble---millersville.edu ABSTRACT: An associahedral set is a combinatorial object generated by Stasheff associahedra K_n and equipped with appropriate face and degeneracy operators. Associahedral sets are similar in many ways to simplicial or cubical sets. In this paper we give a formal definition of an associahedral set, discuss some naturally occurring examples and construct an explicit geometric diagonal \Delta :C_*(K_n) --> C_*(K_n) \otimes C_*(K_n) on the cellular chains C_*(K_n). The diagonal \Delta, which is analogous to the Alexander-Whitney diagonal on the simplices, gives rise to a diagonal on any associahedral set and leads immediately to an explicit diagonal on the A_\infty operad. As an application of this, we use the diagonal \Delta to define a tensor product in the A_\infty category. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-bce The BP cohomology of elementary Abelian groups Neil P. Strickland 20J06; 55N20; 14L05 math.AT/0011120 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk In this paper we study E^*BV_k, where E=BP is a cohomology theory with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n] (if m=0). We use ideas from the theory of multiple level structures, developed in earlier work of the author with John Greenlees. Our results apply when k is less than or equal to w=n+1-m. If k E^0BG is an isomorphism. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-csi Common subbundles and intersections of divisors Neil P. Strickland 55N20; 14L05; 14M15 math.AT/0011123 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that V_0\cap V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-fsfg Formal schemes and formal groups Neil P. Strickland 14L05; 55N22 math.AT/0011121 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We set up a framework for using algebraic geometry to study the generalised cohomology rings that occur in algebraic topology. This idea was probably first introduced by Quillen and it underlies much of our understanding of complex oriented cohomology theories, exemplified by the work of Morava. Most of the results have close and well-known analogues in the algebro-geometric literature, but with different definitions or technical assumptions that are often inconvenient for topological applications. We merely put everything together in a systematic and convenient way. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-ghd Gross-Hopkins duality Neil P. Strickland 55N20; 55P42; 20E18 math.AT/0011108 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We give a new and simpler proof of a result of Hopkins and Gross relating Brown-Comenetz duality to Spanier-Whitehead duality in the K(n)-local stable homotopy category. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-kld K(n) local duality for finite groups and groupoids Neil P. Strickland 55P42; 55P60; 55R40 math.AT/0011109 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Included postscript file: st-kld.eps We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the usual inner product on the representation ring when n=1, and to the Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure. In many places we find it convenient to use groupoids rather than groups, and to assist with this we include a detailed treatment of the homotopy theory of groupoids. We also explain some striking formal similarities between our duality and Atiyah-Poincare duality for manifolds. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-pmm Products on MU-modules Neil P. Strickland 55T25 math.AT/0011122 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Included postscript file: st-pmm.eps We use the new categories of spectra and MU-modules constructed by Elmendorf, Kriz, Mandell and May to get improved results about multiplicative structures on spectra such as P(n) and E(n), particularly in the case p=2. ---------------- There are so many new papers this time that I am breaking this post into at least 2 posts. 8 new papers have modification dates in December, and those are announced here. The January ones will be in the next message. Mark Hovey New papers appearing on hopf between 11/26/00 and 12/31/00 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/f4 A stable approach to an unstable homotopy spectral sequence Martin Bendersky Hunter College, CUNY, NY 10021 mbenders---shiva.hunter.cuny.edu Donald M. Davis Lehigh University, Bethlehem, PA 18015 dmd1---lehigh.edu AMS classification: 55T15, 55Q52 Abstract: Recently Bendersky and Thompson introduced a spectral sequence which, for many spaces X, converges to the v1-periodic homotopy groups of X. It is proved that the E2-term of this spectral sequence is given by Ext in the category of stable p-adic Adams modules of PK^1(X;Zphat)/im(psi^p). We compute this spectral sequence when p=2 and X is the exceptional Lie group F4, yielding as a new result the 2-primary v1-periodic homotopy groups of F4. Some new general results about convergence of this spectral sequence are also proved. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc---julian.uwo.ca hovey---member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category of chain complexes of objects of A has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Intermont-JohnsonM/ijxspace Model Structures on the Category of Ex-spaces Michele Intermont Mark W. Johnson Primary: 55R70, 55U35; Secondary: 55P91, 55U40 Department of Mathematics Kalamazoo College Kalamazoo, MI 49006 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 intermon---kzoo.edu johnson.295---nd.edu Abstract: This paper describes several model structures on the categories of ex-spaces and ex-$G$-spaces when $G$ is a compact Lie group. Two of these are of particular interest in that they have expected applications to the study of transfer maps and to parametrized spectra. These two structures are shown to coincide on the collection of Hurewicz fibrations, and an indication is also given, mainly via examples, of how they differ. The last two sections of this paper are mostly expository; they set forth the model category techniques needed to prove the main theorems. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kashiwabara-Wilson/kash-wil The Morava K-theory and Brown-Peterson cohomology of spaces related to BP Takuji Kashiwabara Institut Fourier, Universit\'{e} de Grenoble I, U.M.R. au C.N.R.S., B. P. 74, 38402 Saint-Martin-d'H\`{e}res CEDEX France Takuji.Kashiwabara---ujf-grenoble.fr W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu This is the "final" version of the paper. We calculate the Morava K-theory of the spaces in the Omega spectra for BP. They fit into an exotic array of short and long exact sequences of Hopf algebras. We apply this to calculate the p-adically completed Brown-Peterson cohomology, as well as all of the intermediary cohomology theories, E, of these spaces. We give two descriptions of the answer, both of which turn out to be surprisingly nice. One part of our first description is just the image in the E cohomology of the corresponding space in the Omega spectrum for BP, which is as big as it could possibly be and which we show how to calculate. The other part is just the E cohomology of several copies of Eilenberg-MacLane spaces, something which is already known. Our second description is inductive and gives us a new way of looking at the Brown-Peterson cohomology of Eilenberg-MacLane spaces. The Brown-Comenetz dual of BP shows up in our calculations and so we take up the study of this spectrum as well. It was already known that the Morava K-theory of the spaces in the Omega spectrum for the Brown-Comenetz dual of BP made it look like a product of Eilenberg-MacLane spaces and we find, somewhat to our surprise, that the same is true for the BP cohomology. In order to state our answers we set up the foundations for the category of completed Hopf algebras. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/mandell-taq Topological Andre-Quillen Cohomology and E-infty Andre-Quillen Cohomology Michael A. Mandell mandell---math.uchicago.edu Abstract This paper compares Andre-Quillen cohomology in various categories of E-infty rings. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Piriou-Schwartz/schwartz La filtration du degre sur la cohomologie modulo 2 des 2-groupes abeliens elementaires Laurent Piriou Université de Nantes, Département de mathématiques 2 rue de la Houssiničre BP 92208 Nantes Cedex 03 France laurent.piriou---math.univ-nantes.fr Lionel Schwartz Université Paris 13 Institut Galilée LAGA UMR 7539 du CNRS Av. J. B. Clément 93430 Villetaneuse France schwartz---math.univ-paris13.fr Code AMS 55S10 This article considers two filtrations on the mod-$2$ cohomology $H^*E$ of an abelian $2$-groups $E$. The first one is the primitive fitration, recall that $H^*E$ is a Hopf algebra. The second one is a kind of socle or Loewy filtration of $H^*E$ as unstable module. If dimension of $E$ is $1$ the two filtrations are the same, if the dimension is larger than $2$ it is shown that the filtration are, in some sense compatible. There is an analogous statement in ${\cal F}$, the category of functors from the category of finite dimensional ${\bf F}_2$-vector spaces to the category of all ${\bf F}_2$-vector spaces, for the functor $V \mapsto {\rm map}({\rm Hom}(V,E),{\bf F}_2)$. However, it is better to work with unstable modules because the Steenrod algebra allows computation on certain classes, that are central in the proof, given by the representation theory of symmetric groups that are central in the proof. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scherer-Thevenaz/simplegroups Finite simple groups and localization Jose L. Rodriguez, Jerome Scherer and Jacques Thevenaz 20D06, 20D08, 55P60 Departamento de Geometria, Topologia y Quimica Organica Universidad de Almeria E--04120 Almeria Spain Institut de Mathematiques Universite de Lausanne CH--1015 Lausanne Switzerland jlrodri---ual.es, jerome.scherer---ima.unil.ch, jacques.thevenaz---ima.unil.ch The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Weibel/Homotopyends-R TITLE: Homotopy Ends and Thomason Model Categories AUTHOR: Chuck Weibel weibel---math.rutgers.edu AUTHOR ADDRESS: Math. Dept. Rutgers University New Brunswick, NJ 08903 USA AMS CLASSIFICATION: Primary 55U35; Secondary 18F20, 55P05, 55Q05 ABSTRACT: In the last year of his life, Bob Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. In this paper we explain and prove Thomason's results, based on his private notebooks. The first half presents Thomason's ideas about homotopy ends and its generalizations. This material may be of independent interest. Then we define Thomason model categories and give some examples. The usual proof shows that the homotopy category exists. In the last two sections we prove the main theorem: functor categories inherit a Thomason model structure, at least when the original category is enriched over simplicial sets and fibrations are preserved by limits. These are the January papers, of which there are 13. Mark Hovey New papers appearing on hopf between 1/1/01 and 2/3/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker/regquotients On the homology of regular quotients Andrew Baker Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker---maths.gla.ac.uk We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. We easily deduce that for $s>1$, the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker-Lazarev/Rmod-ASS On the Adams Spectral Sequence for $R$-modules Andrew Baker \& Andrej Lazarev Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker---maths.gla.ac.uk Department of Mathematics, University of Bristol, Bristol BS8 1TW, England. A.Lazarev---bris.ac.uk We consider the Adams Spectral Sequence for $R$-modules based on commutative localized regular quotient ring spectra of a commutative $S$-algebra $R$ in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case, and we reduce to algebra involving the cohomology of certain `brave new Hopf algebroids' $E^R_*E$. In order to work out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an $R$ ring spectrum. We show that the Adams Spectral Sequence for $S_R$ based on $E=R/I[X^{-1}]$ converges to the homotopy of the $E$-nilpotent completion which has homotopy \[ \pi_*\hat{\mathrm{L}}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}. \] We also show that $\hat{\mathrm{L}}^R_ES_R$ is equivalent to $\L^R_ES_R$, the Bousfield localization of $S_R$ with respect to $E$-theory. This seems surprising since the spectral sequence collapses at $\E_2$, but $\E_r$ does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree, thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve a construction of the internal $I$-adic tower \[ R/I\la R/I^2\la\cdots\la R/I^s\la R/I^{s+1}\la\cdots \] whose homotopy limit is $\hat{\mathrm{L}}^R_ES_R$. Finally, we describe some examples for the case $R=MU$. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Greenlees/kubg The Connective K-theory of Finite Groups Robert Bruner and John Greenlees MSC2000: Primary 19L41, 19L47, 19L64, 55N15. Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30. Department of Mathematics, School of Mathematics and Statistics, Wayne State University, Hicks Building, Detroit MI 48202-3489, Sheffield S3 7RH, USA. UK. rrb---math.wayne.edu, j.greenlees---sheffield.ac.uk Included graphics files: AdamsA4.eps AdamsBip.eps AdamsC2.eps AdamsC4.eps AdamsC5.eps AdamsD8.eps AdamsQ8.eps AdamsSl23.eps AdamsV2.eps AdamsX.eps ExtIE.eps Extku.eps Extl.eps L.eps Qrank4.eps Qrank4lc.eps T3rank6.eps T3rank6lc.eps Xku.eps rank8.eps string.eps tku2.eps Abstract: This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen, we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. The variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, the interest lying in the way these parts fit together. The main technical obstacle is that the Kunneth spectral sequence does not collapse, so we have to show that it collapses up to isomorphism of varieties. In Chapter 2 we give several families of new complete and explicit calculations of the ring ku^*(BG). In Chapter 3 we consider the associated homology ku_*(BG), as a module over ku^*(BG) by using the local cohomology spectral sequence. This gives new specific calculations, but also illuminating structural information, including remarkable duality properties. Finally, in Chapter 4 we make a particular study of elementary abelian groups V. Despite the group-theoretic simplicity of V, the detailed calculation of ku^*(BV) and ku_*(BV) exposes a very intricate structure, and gives a striking illustration of our methods. Unlike earlier work, our description is natural for the action of GL(V). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/JohnsonM/shfloop Loop Spaces as Sheaves: A Sheaf-Theoretic View of Loop Spaces Mark W. Johnson \address {Department of Mathematics\\ University of Notre Dame\\ Notre Dame, IN 46556} \email{johnson.295---nd.edu} The context of enriched sheaf theory introduced in \cite{thesis} provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current document is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as $n$-fold loop spaces and pointed $G$-spaces. This language also includes the diagram categories of spectra from \cite{MMSS} as well as spectra in the sense of \cite{Lewis}, which will be discussed in some detail. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision Title: Excision for simplicial sheaves on the Stein site and Gromov's Oka Principle Author: Finnur Larusson AMS classification numbers: Primary: 32Q28; secondary: 18F10, 18F20, 18G30, 18G55, 32E10, 32H02, 55U35 arXiv:math.CV/0101103 Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada larusson---uwo.ca ABSTRACT: A complex manifold $X$ satisfies the Oka-Grauert property if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak equivalence for every Stein manifold $S$, where the spaces of holomorphic and continuous maps from $S$ to $X$ are given the compact-open topology. Gromov's Oka principle states that if $X$ has a spray, then it has the Oka-Grauert property. The purpose of this paper is to investigate the Oka-Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka-Grauert property is equivalent to $X$ representing a finite homotopy sheaf on the Stein site. This expresses the Oka-Grauert property in purely holomorphic terms, without reference to continuous maps. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert This is another revised version of the proof of the Hilbert-Smith conjecture. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Looptan Title: The equivariant tangent bundle of a free smooth loopspace Author: Jack Morava AMS classification: 58Dxx; 53C29, 55P91 Address: The Johns Hopkins Uniperversity e-mail: jack---math.jhu.edu ABSTRACT: The space of free loops on a manifold X inherits an action of the circle group \T. A Riemannian metric on X defines an equivariant isomorphism of the complexified tangent bundle of the loopspace with \bT X \otimes (\oplus \C(n)), where \C(n) is the standard one-dimensional representation of \T, and \bT X \otimes \C is an equivariant bundle on the loopspace, nonequivariantly isomorphic to the pullback of the complexified tangent bundle of X along evaluation at the basepoint. On a flat manifold, this analogue of Fourier analysis is quite familiar. [Perhaps this is all nonsense; if so, please let me know.] 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity5 Title: Pretty Good Gravity Author: Jack Morava AMS Classification: 19Dxx, 57Rxx, 83Cxx (not yet on xxx, but will be soon) Address: Dept. of Mathematics, the Johns Hopkins Uniperversity e-mail address: jack---math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. [This is a writeup of a talk at the RIMS Symposium on algebraic geometry and integrable systems related to string theory, June 12-16, 2000.] 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Tate2MU Title: Duality of Tate cohomology of framed circle actions Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: Abstract: The complex Mahowald pro-spectrum \CP^{\infty}_{-\infty} is not, as might seem at first sight, Spanier-Whitehead self-dual; rather, its S-dual is its own double suspension. This assertion makes better sense as a claim about the Tate cohomology spectrum t_{\T}S^0 defined by circle actions on framed manifolds. A subtle twist in some duality properties of infinite-dimensional projective space results, which has consequences [via work of Madsen and Tillmann] for the Virasoro symmetries [discovered by Witten and Kontsevich] of the stable cohomology of the Riemann moduli space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moreno/moreno Author: Guillermo Moreno Title: Alternative elements in the Cayley--Dickson algebras We describe the alternative elements in the Cayley-Dickson algebras for n>3. Also we ``measure'' the failure of these algebras of being a normed algebra in terms of the alternative elements. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk-Schwede-Shipley/simplicial Title: Simplicial structures on model categories and functors Authors: Charles Rezk, Stefan Schwede, Brooke Shipley To appear in American Journal of Mathematics Institute for Advanced Study School of Mathematics Olden Lane Princeton, NJ 08540, USA rezk---ias.edu Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de Department of Mathematics Purdue University West Lafayette, IN 47907, USA bshipley---math.purdue.edu We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/ks-hgr The homotopy groups $\pi_*(L_nT(m)\wedge V(n-2))$ Katsumi Shimomura Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520 Japan katsumi---math.kochi-u.ac.jp Let $V_{T(m)}(n)$ denote the spectrum such that $BP_*(V_{T(m)}(n))=BP_*/I_{n+1}[t_1,\dots, t_m]$ for the ideal $I_{n+1}=(p,v_1,\dots, v_{n})$. In the title, we write $T(m)\wedge V(n-2)$ as $V_{T(m)}(n-2)$. Ravenel determined the structure of the Adams-Novikov $E_2$-term for the homotopy groups $\pi_*(L_nV_{T(m)}(n-1))$ for $n\le m+2$ and $n3$. Here are the February papers on Hopf, of which there are 9. So far this "monster snowstorm" hasn't amounted to much, but the real action is supposed to be tonight and tomorrow. Mark Hovey New papers appearing on hopf between 2/3/01 and 3/5/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Clarke-Crossley-Whitehouse/KKbases Bases for cooperations in $K$-theory Francis Clarke, M. D. Crossley and Sarah Whitehouse Primary: 55S25; % K-theory operations and generalized cohomology operations Secondary: 19L64, % Computations, geometric applications 11B65. % Binomial coefficients; factorials; q-identities Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, Wales Laboratoire de G\'eom\'etrie-Alg\`ebre, Universit\'{e} d'Artois, 62307 Lens, France F.Clarke---Swansea.ac.uk M.D.Crossley---Swansea.ac.uk whitehouse---euler.univ-artois.fr Gaussian polynomials are used to define bases with good multiplicative properties for the algebra $K_{*}(K)$ of cooperations in $K$-theory and for the invariants under conjugation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Devoto/elbg-disc Title of Paper: On the elliptic cohomology of the classifying space of discrete groups Author: Jorge A. Devoto AMS Classification: 20J06, 55N34 Addresses of authors: Dept.\ de Matem\'aticas, ITBA, Av. E. Madero 399, Buenos Aires, Argentina and Dept.\ de Matem\'aticas, FCEN, Ciudad Univ. (1428) Buenos Aires, Argentina e-mail: jdevoto---itba.edu.ar We study, for $\Gamma$ a discrete group of finite virtual cohomological dimension, the elliptic cohomology of the classifying space $B\Gamma$. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision Title: Excision for simplicial sheaves on the Stein site and Gromov's Oka Principle Author: Finnur Larusson This is an updated version of a paper announced last month, with the same abstract, so the abstract is omitted. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McClure-SmithJH/deligne-conj (This is also an updated version, but the previous version was announced in 10/99, so I include the abstract). A solution of Deligne's Hochschild cohomology conjecture. James E. McClure and Jeffrey H. Smith ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action by an operad that is equivalent to the little 2-cubes operad. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/bertin AUTHOR: Mara D. Neusel TITLE: The Transfer in the Invariant Theory of Modular Permutation Representations (Trente Ans Apr\`es) Pacific Journal of Mathematics -- to appear -- This note investigates the image of the transfer homomorphism for permutation representations of finite groups over finite fields. One obtains a number of results showing that the image of the transfer $\Im (\Tr)$ together with certain Chern classes generate the ring of invariants as an algebra. By a careful analysis of orbit sums one finds the surprising fact that the ideal $\Im (\Tr)$ is a prime ideal for cyclic $p$-groups and determines an upper bound on its height. AMS CODE: 13A50 Invariant Theory KEY WORDS: Polynomial Invariants of Finite Groups, Permutation Representation, Transfer, $p$-Regular Representation neusel.1---nd.edu 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/bertin2 AUTHOR: Mara D. Neusel TITLE: The Transfer in the Invariant Theory of Modular Permutation Representations II (Trente Ans Apr\`es, Bis) Canadian Mathematical Bulletin -- to appear -- In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals. AMS CODE: 13A50 Invariant Theory KEY WORDS: Polynomial Invariants of Finite Groups, Permutation Representation, Transfer neusel.1---nd.edu 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/kokusu AUTHOR: Mara D. Neusel TITLE: The Lasker-Noether Theorem in the Category $U(\H^*)$ (denizin kokusu) Journal of Pure and Applied Algebra -- to appear -- We prove the Lasker-Noether Theorem in the category $U(\H^*)$ of unstable $\H^*\odot \P^*$-modules. Along the way, we generalize Lam's $\J$-functor to the context of modules. AMS CODE: 55S10 Steenrod Algebra, 13A50 Invariant Theory, 13XX Commutative Rings and Algebras, 55XX Algebraic Topology KEY WORDS: Lasker-Noether Theorem, Unstable Modules, Steenrod Algebra, Dickson Algebra, Polynomial Invariants of Finite Groups neusel.1---nd.edu 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/strassen AUTHOR: Mara D. Neusel TITLE: Lots of Degree Bounds or On the Use of the T-Functor in Invariant Theory We introduce a new method employing J. Lannes's $T$-functor to describe homological properties of rings of invariants. We illustrate the power of this method by applying it to the calculation of degree bounds. We find seven bounds: two for special families of representations, two relative bounds, two general degree bounds and a general bound for $p$-groups. AMS CODE: 13A50 Invariant Theory, 55S10 Steenrod Algebra, 55XX Algebraic Topology KEY WORDS: Invariant Theory of Finite Groups, Degree Bounds, $T$-Functor, Integral Closure, $P^*$-inseparable Closure, Cohen-Macaulay, Gorenstein, Depth, Modular Invariant Theory neusel.1---nd.edu 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/uncoma AUTHOR: Mara D. Neusel TITLE: Unstable Cohen--Macaulay Algebras Mathematical Research Letters -- to appear -- We characterize Cohen--Macaulay algebras in the category $K_{fg}$ of unstable Noetherian algebras over the Steenrod algebra via the depth of the $P^*$-invariant ideals. This allows us to transfer the Cohen--Macaulay property to suitable subalgebras. We apply this to rings of invariants of finite groups and to the $P^*$-inseparable closure. AMS CODE: 55S10 Steenrod Algebra, 13XX Commutative Rings and Algebras, 55XX Algebraic Topology} KEY WORDS: Steenrod Algebra, Cohen--Macaulay Algebras, Unstable Algebras, $P^*$-Invariant Prime Ideal Spectrum, $P^*$-Inseparable Closure, Polynomial Invariants of Finite Groups neusel.1---nd.edu The re-organization of Hopf threw me off somewhat, so I might have missed a paper. Let me know if you think I missed yours. Mark Hovey New papers appearing on hopf between 3/5/01 and 5/16/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Aguade-Ruiz/mapsBKtoBK Maps between classifying spaces of Kac-Moody groups by Jaume Aguad\'e and Albert Ru\'iz (aguade---mat.uab.es, cirera---mat.uab.es) Kac-Moody groups are an important generalisation of Lie groups. Roughly speaking, they are like "Lie groups with infinite Weyl groups". Let K be the unitary form of a Kac-Moody group of rank two. In this paper we determine the self maps of BK. Contents: 1. Introduction. 2. Rank two Kac-Moody groups. 3. Relations between global and local maps. 4. Maps into BK^p and representations. 5. Admissible matrices. 6. Groups with the same classifying space. 7. Adams maps. 8. Homotopically trivial self maps. 9. Detecting maps on the maximal torus. 10. [BK,BK]. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Costenoble-May-Waner/CMWfinal Equivariant orientation theory by S.R. Costenoble, J.P. May, and S. Waner subjclass: Primary 55P91; Secondary 18B40, 20L15, 55N25, 55N91, 55P20, 55R91, 57Q91, 57R91 Hofstra University, University of Chicago, and Hofstra University Steven.R.Costenoble---Hofstra.edu, may---uchicago.edu, matszw---hofstra.edu We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal target category of G-vector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/bdi4 (This is a new version of an paper previously announced). ON THE 2-COMPACT GROUP DI(4) Author: D. Notbohm Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group $DI(4)$. The mod-2 cohomology of the associated classifying space $BDI(4)$ realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space $BDI(4)$ and that the maximal torus normalizer determines the isomorphism type of $DI(4)$ as 2-compact group. We also calculate the set of homotopy classes of self maps of $BDI(4)$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/orthogonal (This is a new version of a paper previously announced). A UNIQUENESS RESULT FOR ORTHOGONAL GROUPS AS 2-COMPACT GROUPS D. Notbohm Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizer are isomorphic. It is conjectured that this result generalizes to \pcg s. Here, we prove the generalization for orthogonal groups $O(n)$, the special orthogonal groups $SO(2k+1)$ and the spinor groups $Spin(2k+1)$ considered as 2-compact groups. There are 7 new papers this time. This is a good time to remind you that people decide whether to download your paper based on your abstract. It is therefore crucial that there be an abstract and that it be readable by humans. It is not enough to just e-mail Clarence a dvi file; you must also e-mail him an abstract, under separate cover, with minimal TeX symbols. Mark Hovey New papers appearing on hopf between 5/16/01 and 6/1/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Kitchloo/BrKi Classifying spaces of Kac-Moody groups Carles Broto and Nitu Kitchloo broto---mat.uab.es nitu---math.nwu.edu We study the structure of classifying spaces of Kac-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes' T-functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kac-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kac-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kac-Moody groups, and centralizers of finite p-subgroups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative (This is the final version, to appear in Math Proc Camb Phil Soc) Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc---julian.uwo.ca hovey---member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category of chain complexes of objects of A has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/hopfalgebroids Morita theory for Hopf algebroids and presheaves of groupoids Mark Hovey Wesleyan University Middletown, CT mhovey---wesleyan.edu 5/17/01 AMS classification nos: 14L05, 14L15, 16W30, 18F20, 18G15, 55N22 Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel change of rings theorems in algebraic topology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/ainf Author: Andrey Lazarev Title: Spaces of multiplicative maps between highly structured ring spectra. We uncover a somewhat unexpected connection between spaces of multiplicative maps between A-infinity ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory MU canonically lifts to an A-infinity map MU-->MU. This implies, in particular, that the Brown-Peterson spectrum BP splits off MU as an A-infinity ring spectrum. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/tower Towers of MU-algebras and the generalized Hopkins-Miller theorem Author: A.Lazarev Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK. email A.Lazarev---bristol.ac.uk AMS classification number 55N22 Our results are of three types. First we describe a general procedure of adjoining polynomial variables to A-infinity-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of MU, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable K(n)-localizations of such spectra. This generalizes the famous Hopkins-Miller theorem and gives strengthened versions of various splitting theorems. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localb The algebraic K-theory spectrum of a 2-adic local field by Stephen A. Mitchell mitchell---math.washington.edu (There was no abstract with this paper, so I made one up. If you don't like it, Steve, send in one!) A local field F of characteristic 0 is a finite extension of the L-adic rationals of finite degree d, where L is a prime. When L is odd, Dwyer and the author determined the homotopy type of the etale K-theory spectrum of F, but their methods fail when L=2 and -1 is not a square in F. The purpose of this paper is to study this remaining case. The recent work on the Lichtenbaum-Quillen conjecture at 2 by Rognes and Weibel allows the author to get from the etale K-theory of F to the 2-adic completion of the algebraic K-theory of F. The result essentially says that, rather than a splitting as you get in the odd primary case, there is some room for a few non-trivial extensions (which are completely determined). This is a generalization of Rognes' calculation of the 2-adic K-theory of the 2-adic rationals. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YauD/catcocat Title: Clapp-Puppe Type Lusternik-Schnirelmann (Co)category in a Model Category Donald Yau AMS Classification: Primary 55M30; Secondary 55P30, 55U35 math.AT/0104267 Department of Mathematics MIT, 2-230 77 Massachusetts Avenue Cambridge, MA 02139 USA donald---math.mit.edu We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given. There are 7 new papers this time. Mark Hovey New papers appearing on hopf between 6/1/01 and 6/21/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/Equivalences The group of homotopy equivalences of products of spheres and of Lie groups Martin Arkowitz and Jeffrey Strom AMS Classifications 55P10, 55P60, 55S37 Dartmouth College, Hanover, NH 03755 Martin.Arkowitz---Dartmouth.edu Jeffrey.Strom---Dartmouth.edu Abstract We investigate the group E_#(X) of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of E_#(X) provided the p-localization X_(p) of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that E_#(X)_(p) is a semidirect product of certain homotopy groups pi_n(X_(p)). We also show that E_#(X)_(p) has a central series whose successive quotients are pi_n(X_(p)), which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of E_#(X) and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of E_\#(G) and give numerous explicit calculations using MAPLE. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Kitchloo/BrKiCorregit This is a corrected version (the diagrams are better) of the paper announced last time, so I will just give the title: Classifying spaces of Kac-Moody groups by Carles Broto and Nitu Kitchloo 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Costenoble-May-Waner/CMWFinal Equivariant orientation theory by S.R. Costenoble, J.P. May, and S. Waner subjclass: Primary 55P91; Secondary 18B40, 20L15, 55N25, 55N91, 55P20, 55R91, 57Q91, 57R91 Hofstra University, University of Chicago, and Hofstra University Steven.R.Costenoble---Hofstra.edu, may---uchicago.edu, matszw---hofstra.edu We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal target category of G-vector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/ethtpy Etale realization on the A^1-homotopy theory of schemes Daniel C. Isaksen 14F42 (primary), 14F35 (secondary) Department of Mathematics University of Notre Dame Notre Dame, IN 46556 isaksen.1---nd.edu We compare Friedlander's definition of etale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for etale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the residue fields of S, we get a functor from the Morel-Voevodsky A^1-homotopy category of schemes to the homotopy category of pro-spaces. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/prospace (This is a revised version of a paper announced 6/99) A model structure on the category of pro-simplicial sets Daniel C. Isaksen 18E25, 55Pxx, 55U35 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 Abstract: We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/AddJan01 The additivity of traces in triangulated categories J.P. May University of Chicago may---math.uchicago.edu This paper is a much expanded version of the Appendix of the previously posted paper entitled "Picard groups, Grothendieck rings, and Burnside rings of categories. In it, we explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2 Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics, Purdue University, West Lafayette, IN 47907--1395 mcclure---math.purdue.edu jhs---math.purdue.edu In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture. There are 8 new papers this time. Mark Hovey New papers appearing on hopf between 6/21/01 and 7/13/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Hovey/relative This is the final version of the paper "Quillen model structures for relative homological algebra" by J. Daniel Christensen and Mark Hovey. There are only minor corrections and fairly major spacing changes from the previous version. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/FangF-PanJZ/cl-2-1 Title of Paper :Secondary Brown-Kervaire Quadratic forms and $\pi$-manifolds Author(s) :Fuquan Fang and Jianzhong Pan Addresses of Authors: Fuquan Fang Nankai Institute of Mathematics, Nankai University, Tianjin 300071, P.R.C email:ffang---sun.nankai.edu.cn and Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn In this paper we assert that for each $\Phi$-oriented $2n$-manifold (c.f : Definition 1.1) $M$ where $n\ge 4$ and $n\ne 3(mod 4)$, there is a well-defined quadratic function $\phi_M: H^{n-1}(M, \Z_4)\to \Q/\Z$, we call the secondary Brown-Kervaire quadratic forms, so that \begin{itemize} \item{ $\phi _{M}(x+y)=\phi _{M}(x)+\phi _{M}(y)+j(x\cup Sq^2y)[M]$}, \item{ the Witt class of $\phi _M$ is a homotopy invariant, if the Wu class $ v_{n+2-2^i}(\nu _M)=0$ for all $i$.} \end{itemize} where $j: \Z_2 \to \Q/\Z$ is the inclusion homomorphism and $\nu _M$ the stable normal bundle of $M$. Among the applications we obtain a complete classification of $(n-2)$-connected $2n$-dimensional $\pi$-manifolds up to homeomorphism and homotopy equivalence, where $n\geq 4$ and $n+2\neq 2^i$ for any $i$. In particular, we prove that the homotopy type of such manifolds determine their homeomorphism type. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/PanJZ/cocat1 Title of Paper :Having the H-space structure is not a generic property Author(s) : Jianzhong Pan AMS Classification numbers :55P60,55P45 Addresses of Authors: Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn In this note, we answer in negative a question posed by McGibbon about the generic property of H-space structure. In fact we verify the conjecture of Roitberg. Incidentally, the same example also answers in negative the open problem 10 in McGibbon. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/PanJZ/equivar Title of Paper :Equivariant Phantom maps Author(s) : Jianzhong Pan AMS Classification numbers :55P91,55P60 Addresses of Authors: Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/PanJZ/nonneg Title of Paper :Rational homotopy theory and nonnegative curvature Author(s) : Jianzhong Pan AMS Classification numbers :53C20 53C40 55P10 Addresses of Authors: Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn In this note , we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnegative curved manifolds admit (complete) metrics with nonnegative curvature. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/PanJZ-WooMH/genus2-1 Title of Paper :Mislin genus of maps Author(s) : Jianzhong Pan and Moo Ha Woo AMS Classification numbers :55D99 Addresses of Authors: Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn and Moo Ha Woo Department of Mathematics Education , Korea University , Seoul , Korea In this paper, we prove that the Mislin genus of a (co-)H-map between (co-)H-spaces under certain natural conditions is a finite abelian group which generalizes results in Zabrodsky, McGibbon and Hurvitz 7. http://hopf.math.purdue.edu/cgi-bin/generate?/PanJZ-WooMH/phan-elem Title of Paper :Phantom elements and its Applications Author(s) : Jianzhong Pan and Moo Ha Woo AMS Classification numbers :55P10,55P60,55P62,55R10 Addresses of Authors: Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing 100080 ,China email:pjz---math03.math.ac.cn and Moo Ha Woo Department of Mathematics Education , Korea University , Seoul , Korea In our previous work, a relation between Tsukiyama problem about self homotopy equivalence was found by using a generalization of phantom map. In this note , fundamental result is established for such a generalization. This is the first time one can deal with phantom maps to space not satisfying finite type condition. Application to Forgetful map is also discussed briefly. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Vavpetic-Viruel/f4at2 On the Homotopy type of the Classifying Space of the Exceptional Lie Group of Rank 4 A. VAVPETIC (ales.vavpetic---fmf.uni-lj.si) Fakulteta za Matematiko in Fiziko Univerza v Ljubljani Jadranska 19 1111 Ljubljana Slovenija and A. VIRUEL (viruel---agt.cie.uma.es) Departamento de Algebra, Geometria y Topologia Universidad de Malaga AP. 59 29080 Malaga Spain AMS Classification numbers: 55R35, 55P15 Previous work of several authors shows that the exceptional Lie group of rank 4, F_4, as a p-compact group, is determined up to isomorphism by the isomorphism type of its maximal torus normalizer for p>2. This paper considers the case p=2 proving that F_4 as 2-compact group is also determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows the authors to determine the integral homotopy type of F_4 among connected finite loop spaces with maximal tori. There are 7 new papers this time. Mark Hovey New papers appearing on hopf between 7/13/01 and 8/3/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/CohenR-JonesJDS/stringhtpy Title: A homotopy theoretic realization of string topology Authors: Ralph L. Cohen and John D.S. Jones AMS Classification numbers: 55N45 57R19 18D50 Addresses: Cohen: Dept. of Mathematics, Stanford University, Stanford, CA 94305 Jones: Dept. of Mathematics, University of Warwick, Coventry CV4 7AL England Email: Cohen: ralph---math.stanford.edu Jones: jdsj---maths.warwick.ac.uk Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. Chas and Sullivan have recently defined a kind of intersection product on the homology H_*(LM) of total degree -d. They then investigated other structure that this product induces, including a Lie algebra structure on H_*(LM), and an induced product on the S^1 equivariant homology, H_*^{S^1}(LM) . These algebraic structures, as well as others, came under the general heading of the ``String topology" of M. In this paper we describe a realization of the Chas - Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We show that this ring spectrum structure extends to an operad action of the the ``cactus operad", originally defined by Voronov, which is equivalent to the operad of framed disks in R^2. We then describe a cosimplicial model of this spectrum and, by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology of the cochains, HH^*(C^*(M); C^*(M)). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/dosSantos-Lima_Filho/quat Title: Quaternionic algebraic cycles and reality Authors: Pedro F. dos Santos (pedfs---math.ist.utl.pt) Instituto Superior Técnico Lisboa, Portugal and Paulo Lima-Filho (plfilho---math.tamu.edu) Texas A&M university College Station, Texas USA AMS classification: 55P91; Secondary 14C05, 19L47, 55N91 Abstract In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic K-theory, and the other one classifies and equivariant cohomology theory Z^*(-) which is a natural recipient of characteristic classes KH^*(X) --> Z^*(X) for quaternionic bundles over Real spaces X. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Hollander/Ho-Th-Stacks A Homotopy Theory for Stacks Sharon Hollander Department of Mathematics, MIT Cambridge, MA 02139 sharon---math.mit.edu AMS Classification: Primary 14A20 ; Secondary 18G55, 55U10 We give a homotopy theoretic characterization of stacks on a site $\cC$ as the {\it homotopy sheaves} of groupoids on $\cC$. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the $S^2$-nullification of Jardine's model structure on sheaves of simplicial sets on $\cC$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Huettemann-Roendigs/twisted Title: Twisted Diagrams Authors: Thomas Huettemann and Oliver Roendigs Author addresses: Thomas Huettemann Oliver Roendigs Department of Mathematical Sciences Fakultaet fuer Mathematik King's College, University of Aberdeen Universitaet Bielefeld Aberdeen AB24 3FX Postfach 10 01 31 UK D-33501 Bielefeld Germany Email: huette---maths.abdn.ac.uk (T. Huettemann) oroendig---mathematik.uni-bielefeld.de (O. Roendigs) Abstract: Twisted diagrams are generalised diagrams: the vertices are allowed to live in different categories, and the structure maps act through specified "twisting" functors between these categories. Examples include spectra (in the sense of homotopy theory) and quasi-coherent sheaves of modules on an algebraic variety. We construct a twisted version of Kan extensions and establish various model category structures (with pointwise weak equivalences). Using these, we propose a definition of ``homotopy sheaves'' and show that a twisted diagram is a homotopy sheaf if and only if it gives rise to a ``sheaf in the homotopy category''. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/presheaves Title of paper: Presheaves of chain complexes Author: J.F. Jardine AMS Classification numbers: 55P42 55U15 18G15 Address of Author: Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada Email: jardine---uwo.ca This paper gives the basic constructions for homology theory in the category of modules over a presheaf of commutative rings with unit. The category of simplicial modules inherits a proper closed simplicial model structure from the category of simplicial presheaves. The corresponding stable category is described by several different models, including infinitely graded chain complexes, spectrum objects in simplicial modules, and symmetric spectrum objects in simplicial modules. The tensor product of simplicial modules induces a symmetric monoidal tensor product on the category of symmetric spectrum objects, by analogy with the construction of the smash product for symmetric spectra. This paper is in preliminary form only, and is expected to pass through several revisions. Proofs of the displayed results are in place, but it is expected that more material on Tor functors and the relation with motivic homotopy theory will be added later. The paper is available in dvi, ps and pdf formats at Jardine's home page. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Lesh/uass-so-model A conjecture on the unstable Adams spectral sequences for SO and U Kathryn Lesh Subject Classification: 55T15, 55Q52, 55U99 Department of Mathematics Union College Schenectady, NY 12308 Telephone number: (518)388-6246 klesh---member.ams.org In this paper we give a systematic account of a conjecture suggested by Mark Mahowald on the unstable Adams spectral sequences for the groups SO and U. The conjecture is related to a conjecture of Bousfield on a splitting of the E_{2}-term and to an algebraic spectral sequence constructed by Bousfield and Davis. In this paper, we construct and realize topologically a chain complex which is conjectured to contain in its differential the structure of the unstable Adams spectral sequence for SO. A filtration of this chain complex gives rise to a spectral sequence that is conjectured to be the unstable Adams spectral sequence for SO. If the conjecture is correct, then it means that the entire unstable Adams spectral sequence for SO is available from a primary level calculation. We predict the unstable Adams filtration of the homotopy elements of SO based on the conjecture, and we give an example of how the chain complex predicts the differentials of the unstable Adams spectral sequence. Our results are also applicable to the analogous situation for the group U. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/newring Title: Rings of invariants and inseparable forms of algebras over the Steenrod algebra Author: Clarence W. Wilkerson, Jr. Purdue University wilker---math.purdue.edu This is the final version of the paper "ringall", one of the first papers on the Hopf archive. It's due to appear in the JAMI2000 proceedings. ------------------------------------------------------------------There are 5 new papers on Hopf this month. Mark Hovey New papers appearing on hopf between 8/3/01 and 9/2/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Farrell-Jones-Reich/oneiso On the Isomorphism Conjecture in algebraic K-theory Arthur Bartels, Tom Farrell, Lowell Jones and Holger Reich The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RG, where G is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and an arbitrary coefficient ring R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RG in terms of the K-theory of R and the homology of the group. AMS Classification: 19A31, 19B28, 19D35, 19D50 AT/0108139 Westfaelische Wilhelms-Universitaet, SFB 478, 48149 Muenster, Germany Department of Mathematics, SUNY, Binghamton, NY 13902, USA Department of Mathematics, SUNY, Stony Brook, NY 11794, USA Westfaelische Wilhelms-Universitaet, SFB 478, 48149 Muenster, Germany bartelsa---math.uni-muenster.de farrell---math.binghamton.edu lejones---math.sunysb.edu reichh---math.uni-muenster.de 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Berger-Fresse/CochainModel Tittle: Combinatorial operad actions on cochains Authors: Clemens Berger and Benoit Fresse Abstract: A classical E-infinity operad is formed by the bar constructions associated to the symmetric groups. Such an operad is introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop-spaces. The purpose of the article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We prove also that the differential graded algebras over the Barratt-Eccles operad form a closed model category. We have similar results for the normalized Hochschild cochain complex associated to an associative algebra. More precisely, the Hochschild cochain complex is acted on by a sub-operad of the Barratt-Eccles operad which is equivalent to the classical little square operad. Mail address: Laboratoire J.A. Dieudonn\'e, Universit\'e de Nice, Parc Valrose, F-06108 Nice Cedex 02 (France). E-mail address: Clemens Berger Benoit Fresse 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/strict Title: Strict Model Structures for Pro-Categories Author: Daniel C. Isaksen AMS Classification: 18G55, 55U35 Address: Department of Mathematics\\University of Notre Dame\\Notre Dame, IN 46556 e-mail: isaksen.1---nd.edu Abstract: We show that if C is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of Edwards and Hastings. The strict model structure is the starting point for many homotopy theories of pro-objects. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Rudyak/PLstructures Piecewise linear structures on topological manifolds Yuli B. Rudyak MSC 57Q25 Submitted to xxx LANL archive: math.AT/0105047 Mathematisches Institut Universitaet Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany \email: rudyak---mathi.uni-heidelberg.de This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(\ZZ/2.3) and the Hauptvermutung for manifolds. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Schwede-Shipley/class.final Title: Classification of stable model categories Authors: Stefan Schwede Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de and Brooke Shipley Department of Mathematics Purdue University W. Lafayette, IN, USA 47907 bshipley---math.purdue.edu AMS Classification numbers: 55U35, 55P42 Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. Remark: Our use of lamsarrows may make the .dvi file less portable than the .ps or .pdf files. Hope all of your loved ones are alright. There are 9 new papers on Hopf in the last two weeks. Mark Hovey New papers appearing on hopf between 9/2/01 and 9/17/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ahearn-Kuhn/towers Title: Product and other fine structure in polynomial resolutions of mapping spaces Authors: Stephen T. Ahearn and Nicholas J. Kuhn AMS classification: Primary 55P35; Secondary 55P42 Authors addresses: Department of Mathematics, De Pauw University, Greencastle, IN 46135. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 Email: sahearn---depauw.edu, njk4x---virginia.edu Abstract: Let Map(K,X) denote the space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum of Map(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on the homology of the mapping space. Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_{infty}--term corresponds to the cup product in h^*(Map(K,X)) in the usual way, and the product on the E_1--term is described in terms of group theoretic transfers. We use explicit equivariant S--duality maps to show that, when K is the n sphere, our constructions at the fiber level have descriptions in terms of the Boardman--Vogt little n--cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to the suspension spectrum of its 0th space. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Dwyer-Goerss/moduli The realization space of a \Pi-algebra: a moduli problem in algebraic topology D. Blanc, W. G. Dwyer, and P. G. Goerss A \PI-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R(A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with \pi_*(X)=A, of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R(A), in which the space at the bottom is BAut(A) and the successive fibres are determined by \Pi-algebra cohomology. (This cohomology is the analog for \Pi-algebras of the Hochschild cohomology of an associative ring or the Andre-Quillen cohomology of a commutative ring.) The main technical tool involves working with simplicial resolutions of spaces rather than with spaces themselves. It seems clear that the deformation theory can be applied with little change to study other moduli questions in topology. In the course of working out the details, we find a simple homotopy theoretic way to identify the space that results from taking a functor from finite sets to sets and applying it dimensionwise to a simplicial set. This gives an easy way to reprove and generalize many classical connectivity theorems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/E6 The K-completion of E6 Donald M. Davis 55T15, 55Q52, 57T20 Department of Mathematics Lehigh University Bethlehem, PA 18015 dmd1---lehigh.edu Abstract: We compute the 2-primary v1-periodic homotopy groups of the exceptional Lie group E6. This is done by computing the Bendersky-Thompson spectral sequence of E6. We conjecture that the natural map from E6 to its K-completion induces an isomorphism in v1-periodic homotopy, and discuss issues related to this conjecture. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Green-Hunton-Schuster/cccGHS Title: Chromatic characteristic classes in ordinary group cohomology Authors: David J. Green John R. Hunton Bj"orn Schuster MSC: 20J06 (primary), 16W30 55P47 55R40 (secondary) arXiv: math.AT/0109019 Status: Submitted for publication, Aug. 2001 Abstract: We study a family of subrings, indexed by the natural numbers, of the mod-p cohomology of a finite group G. These subrings are based on a family of v_n-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H^*(BG)), the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings give rise to a "chromatic" (co)filtration, which has both topological and algebraic definitions, of var(H^*(BG)) whose final quotient is the variety var(Ch(G)). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Gdansknotes Title: Braids, trees, and operads Author: Jack Morava AMS classification: 55R810, 14N35, 20F36 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack---math.jhu.edu Abstract: The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties. This talk [at the Gdansk conference on algebraic topology 05-06-01] is an introduction to their study. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Looptangent Title: The tangent bundle of an almost-complex free loopspace Author: Jack Morava AMS classification: 58Dxx; 53C29, 55P91 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack---math.jhu.edu Abstract: The space LV of free loops on a manifold V inherits an action of the circle group \T. When V has an almost-complex structure, the tangent bundle of the free loopspace, pulled back over a certain infinite cyclic cover \tilde LV, has an equivariant decomposition as a completion of \TV \otimes (\oplus \C(k)), where \TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evaluation at the basepoint (and \oplus \C(k) denotes an algebra of Laurent polynomials). On a flat manifold, this analog of Fourier analysis is classical. This construction uses a model for the universal cover of the space of conjugacy classes in the unitary group (also known as a symmetric product of copies of the circle) which may be of independent interest. This paper appears in the proceedings of the Stanford workshop on equivariant homotopy theory, in Homology, Homotopy and Applications, 3 (2001) 407-415. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/PGGravityfinal Title: A rudimentary theory of topological 4D gravity Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack---math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes some relatively accessible examples of such a thing, suggested by the wall-crossing formulas of Donaldson theory. This is the final version of the paper, to appear in Advances in Theoretical and Mathematical Physics. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/TateHeisenberg Title: Tate cohomology of circle actions as a Heisenberg group Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack---math.jhu.edu Abstract: This is a revision of an earlier posting, with a similar name; the paper has been reorganized, and some howlers related to the Segal conjecture have been eliminated: We study the Madsen-Tillman spectrum \CP^\infty_{-1} as a quotient of the Mahowald pro-object \CP^\infty_{-\infty}, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Virasoro Title: An algebraic analog of the Virasoro group Author: Jack Morava AMS classification: 81R10, 55S25 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack---math.jhu.edu Abstract: The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it behaves as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of two-dimensional topological gravity through the homotopy theory of moduli spaces. This is a version of a talk on 23 June 2001 at the Prague Conference on Quantum Groups and Integrable Systems, published in the Czechoslovak J. Physics 51 (2001). There are 2 new papers on Hopf this tiem. Mark Hovey New papers appearing on hopf between 9/17/01 and 10/17/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Dwyer-Isaksen/obstruction Obstruction theory in model categories J. Daniel Christensen, William G. Dwyer and Daniel C. Isaksen MSC: 55S35, 55U35, 18G55 (primary); 18G30, 55P42 (secondary) Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 jdc---uwo.ca Department of Mathematics University of Notre Dame South Bend, IN 46556 dwyer.1---nd.edu Department of Mathematics University of Notre Dame South Bend, IN 46556 isaksen.1---nd.edu Keywords: obstruction theory, closed model category, simplicial set, spectrum Working in an arbitrary pointed proper model category, we describe the cofibrations that have an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Toen-Vezzosi/agmod-web Title: Algebraic Geometry over model categories (a general approach to derived algebraic geometry) Authors: Bertrand Toen and Gabriele Vezzosi AMS Classification numbers: 14A20; 18G55; 55P43; 55U40;18F10 Submitted to the xxxLANL as math.AG/0110109, October 10, 2001 Addresses of authors: Bertrand Toen, Laboratoire J. A. Dieudonne, UMR CNRS 6621, Universite' de Nice-Sophia Antipolis, France. toen---math.unice.fr Gabriele Vezzosi, Diprtimento di Matematica, Universita' di Bologna, Italy, vezzosi---dm.unibo.it Included gzipped .ps file ABSTRACT: For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\infty}-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version. There are 9 new papers on Hopf this time, 7 with Jeffrey Strom as one of the authors, one by Bill Dwyer and Clarence Wilkerson, and one by G. Meigneiz. Mark Hovey New papers appearing on hopf between 10/17/01 and 11/13/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/H-Inverses The Inverses of an H-Space Martin Arkowitz, Hideaki Oshima and Jeffrey Strom MSC: 55P45, 55P62 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom---Dartmouth.edu Martin.Arkowitz---Dartmouth.edu Ibaraki University Mito, Ibaraki 310-8512 JAPAN ooshima---mito.ipc.ibaraki.ac.jp ABSTRACT A multiplication on an H-space X has a left inverse \lambda and a right inverse \rho. They are mutual inverses and \lambda = \rho if and only if \lambda^2 = id. In this paper we investigate the order |\lambda| of \lambda. We give an example of a multiplication with |\lambda|=6, and prove that for any finite H-complex X there are finitely many left inverses of finite order. Conditions are given for there to be infinitely many multiplications on X with the same left inverse. We then give conditions for a left inverse to have infinite order. We apply these results to specific Lie groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/NonComm Non commutativity of the group of self homotopy classes of classical simple Lie groups Martin Arkowitz, Hideaki Oshima and Jeffrey Strom MSC: 55Q05 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Ibaraki University Mito Ibaraki 310-8512 Japan Martin.Arkowitz---Dartmouth.edu ooshima---mito.ipc.ibaraki.ac.jp Jeffrey.Strom---Dartmouth.edu ABSTRACT For a large class of simple Lie groups G we prove that [G,G] is nonabelian. For certain special Lie groups we show that \nil [G,G] > 2. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/NearlyTrivial Nearly Trivial Homotopy Classes Between Finite Complexes Martin Arkowitz and Jeffrey Strom 2000 MSC: Primary 55P99; Secondary 55M30, 55P60 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Martin.Arkowitz---Dartmouth.edu Jeffrey.Strom---Dartmouth.edu ABSTRACT We construct examples of essential maps of finite complexes f : X --> Y which are trivial of order at least n. This latter condition implies that for any space K with cone length at most n, the induced map f_* = 0:[K,X] --> [K,Y]. The main result establishes a connection between the skeleta of the infinite dimensional domains of essential phantom maps and the finite dimensional domains of maps which are trivial of order at least n. In particular, there are essential maps f: \Sigma^2i ( CP^t / S^2 ) --> M( Z/p^s, 2l+3) which are trivial of order at least n. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/normalizers/tits-final Title: Cartan Involutions and the normalizer of the maximal torus Authors: William G. Dwyer and Clarence W. Wilkerson Email: dwyer.1---nd.edu cwilkers---purdue.edu Classification codes: 22E15 (55R35 55S40) One consequence of Tits' well known work \cite{rTits} on the structure of the normalizer of the maximal torus in a connected compact Lie group is that twice the $k$-invariant classifying the extension $$\{e\} \to T_G \to N_G(T_G) \to W(G) \to \{e\}$$ is zero. In this note we observe that this conclusion follows directly from the existence of an unstable Adams map of type $\Psi^{-1}$ on the classifying space $BG$. Work from the 1970's using etale methods or more recent diagramatic methods produce a $\Psi^{\alpha}$ self-map of $BG$ whenever $\alpha$ is relatively prime to the order of $W(G)$, so the $k$-invariant bound follows. However, the Lie algebra version of ${\Psi^{-1}}$ (the Cartan involution) is classical. This note discusses the Cartan involution, and shows how for a connected compact Lie group it gives rise to a self map of type $\Psi^{-1}$.\\ Analogues of $\{\Psi^{-1}\}$ are not known for the general $2$-compact group context of Dwyer-Wilkerson \cite{rDW1}. While this could be a possible divergence point for $2$-compact group theory from classical Lie theory, the authors speculate that it is not. { This was written for the Grand Lake, CO Bastille Day 2001 conference in honor of Brooke Shipley and Kevin Corlette. It has been submitted to Publ. Res. Inst. Math. Sci., RIMS, Kyoto. } 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fernandez-Suarez-Gomez-Tato-Strom-Tanre/Sp(3) The Lusternik-Schnirelmann Category of Sp(3) Lucia Fernandez-Suarez, Antonio Gomez-Tato, Jeffrey Strom and Daniel Tanre MSC: 55M30, 22E20 Departamento de Matematica (CMAT) Universidade do Minho (Gualtar) 4710 Braga, Portugal lfernandez---math.uminho.pt Departamento de Xeometria e Topoloxia Universidade de Santiago de Compostela 15706 Santiago de Compostela Espana agtato---zmat.usc.es Department of Mathematics Dartmouth College Hanover, NH 03755 U.S.A. Jeffrey.A.Strom---Dartmouth.edu Departement de Mathematiques UMR 8524 Universite de Lille 1 59655 Villeneuve d'Ascq Cedex, France Daniel.Tanre---agat.univ-lille1.fr ABSTRACT We show that the Lusternik-Schnirelmann category of the symplectic group Sp(3) is 5. This L-S category coincides with the cone length and the stable weak category. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Meigniez/sfb Title: Submersions, fibrations & bundles. Author: G. Meigniez Abstract --- When does a submersion have the homotopy lifting property ? When is it a locally trivial fibre bundle ? We establish characterizations in terms of consistency in the topology of the neighbouring fibres. -- Universite de Bretagne Sud, Centre de Recherche, Campus de Tohannic, B.P. 573, F-56017 Vannes, France. Phone: (33)6.87.49.79.45. Fax: (33)2.97.68.42.12. http://www.univ-ubs.fr/lmam/meigniez/ 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/Diagonal Decomposition of the Diagonal Map Jeffrey Strom 2000 MSC: Primary: 55M30 Secondary: 55Q25 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom---Dartmouth.edu ABSTRACT This paper presents a new method for using cup product information to draw conclusions about the Lusternik-Schnirelmann category of a space. The key idea is that of the Hopf set in X of a map f : S^{n-1} --> L; if K = L \cup_f D^n is a subcomplex of X, then cat_X (K) = cat_X (L) if and only if * is in the Hopf set in X of f. The main result explicitly constructs elements of the Hopf set in X of f in terms of members of the Hopf set in X of the attaching maps of lower dimensional cells. Applications include: a calculation of the category of Sp(2) without higher order cohomology operations; new, easily used upper bounds for Lusternik-Schnirelmann category that apply to any space; and new information about the category of the CW skeleta of loop spaces and free loop spaces on even-dimensional spheres. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/MillerSpaces Miller Spaces and Spherical Resolvability of Finite Complexes Jeffrey Strom MSC: 55Q05, 55P50 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom---Dartmouth.edu ABSTRACT We show that if K is a nilpotent finite complex, then the loop space of K can be built from spheres using fibrations and homotopy (inverse) limits. This is applied to show that if map_*(X,S^n) is weakly contractible for all n, then map_*(\s X,K) is weakly contractible for any nilpotent finite complex K. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/S1xS1 The Lusternik-Schnirelmann Category of S^1_\QQ\cross S^1 and S^1_\QQ\cross S^1_\QQ Jeffrey Strom MSC: 55M30 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom---Dartmouth.edu (From Mark: My guess is that the subscript \QQ indicates the rationalization). ABSTRACT We answer a question of Rudyak by showing that cat(S^1_\QQ\cross S^1) = cat(S^1_\QQ\cross S^1_\QQ) = 3. The second formula shows that X= S^1_\QQ is an example of a space for which \cat(X\cross X) < 2 \cat(X). These calculations are derived from a general formula for the category weight of elements of H^*(BG;\pi) that is of independent interest. 5 new papers this time. There is also a corrected version of the paper I announced last time on the Lusternik-Schnirelmann category of Sp(3), by Fernandez-Suarez, Gomez-Tato, Strom, and Tanre. Mark Hovey New papers appearing on hopf between 11/13/01 and 12/12/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/hypercover Hypercovers in topology Daniel Dugger, Daniel C. Isaksen 55U35, 14F20, 14F42 Department of Mathematics Purdue University West Lafayette, IN 47907 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 ddugger---math.purdue.edu isaksen.1---nd.edu We show that if U is a hypercover of a topological space X then the natural map from hocolim U to X is a weak equivalence. This fact is used to construct topological realization functors for the A^1-homotopy theory of schemes over real and complex fields. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Scannell-Sinha/knotss A one-dimensional embedding complex by Kevin P. Scannell and Dev P. Sinha St. Louis University and Brown University scannell---slu.edu dps---math.brown.edu We give the first explicit computations of rational homotopy groups of spaces of "long knots" in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E^1 term is defined in terms of Lie algebras related to braid groups. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E^1 term is zero, and make calculations of E^2 in a finite range. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/localcohom The geometry of the local cohomology filtration in equivariant bordism by Dev P. Sinha Brown University dps---math.brown.edu Local cohomology techniques in equivariant homotopy theory, introduced by John Greenlees, may be applied to understand homology of classifying spaces through other equivariant data. In this paper we relate the local cohomology filtration to the families filtration. By doing so, we may identify geometry codified by the local cohomology filtration in the setting of equivariant bordism. The constructions which arise are naturally analyzed by localized K-theory machinery due to Atiyah and Segal, which we review. This paper has appeared in Homology, Homotopy and Applications. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/WuJ/coHspacewu On co-H maps to the suspension of the projective plane by Jie Wu Department of Mathematics National University of Singapore Singapore 117543 Republic of Singapore matwuj---nus.edu.sg We study co-H-maps from a suspension to the suspension of the projective plane and provide examples of non-suspension 3-cell co-H-spaces. These (infinitely many) examples are related to the homotopy groups of the 3-sphere. For each element of order 2 in $\pi_n(S^3)$, there is a corresponding non-suspension co-H-space of cells in dimensions 2, 3 and n+2. Our ideas are to study Hopf invariants in combinatorial way by using the Cohen groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/WuJ/mod2Moore2-2 Homotopy Theory of the suspensions of the projective plane by Jie Wu Department of Mathematics National University of Singapore Singapore 117543 Republic of Singapore matwuj---nus.edu.sg The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds. This paper is essentially from my Ph. D. thesis at Rochester under the supervise of Fred Cohen, and my joint works with Fred Cohen and Paul Selick. The group representation theory, particularly the modular representation theory of symmetric groups, is used much in this article. The table of the homotopy groups computed in this article have been announced without proofs in Cohen's paper in the Handbook of Algebraic Topology by James. Happy New Year! 4 new papers this time, from Bendersky-Hunton, Chorny (2), and Hunton-Schuster. Mark Hovey New papers appearing on hopf between 12/12/01 and 01/02/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-Hunton/BH2 On the coalgebraic ring and Bousfield-Kan spectral sequence for a Landweber exact spectrum Martin Bendersky and John R. Hunton We construct a Bousfield-Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum $E$ with unit and which is related to the homotopy groups of a certain unstable $E$ completion $\xe$ of a space $X$. For $E$ an S-Algebra this completion agrees with that of the first author and R. Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) $E_*(\EE_*)$ for an arbitrary Landweber exact spectrum $E$, extending work of the second author and M. Hopkins\cite and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the $E_2$-term of the $E$-theory Bousfield-Kan spectral sequence when $E$ is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a $v_n$-periodic theory for all~$n$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/diag An example of a non-cofibrantly generated model category Boris Chorny AMS Classification numbers Primary 55U35; Secondary 55P91, 18G55 Centre de Recerca Matematica, Apartat 50, E-08193 Bellaterra (Barcelona), Spain cboris---crm.es We show that the model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated. In particular the category of maps between spaces may be given a non-cofibrantly generated model structure. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/ehomology Equivariant cellular homology and its applications Boris Chorny AMS Classification numbers Primary 55N91; Secondary 55P91, 57S99 Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel chorny---math.huji.ac.il In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hunton-Schuster/subalg Title: Subalgebras of group cohomology defined by infinite loop spaces Authors: John R. Hunton Bj"orn Schuster MSC: 20J06 55N20 55P47 (primary), 55R40 19A22 55P60 (secondary) arXiv: math.AT/0112169 Addresses: The Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, England Department of Mathematics, University of Wuppertal, Gaussstr.~20, D-42097 Wuppertal, Germany. Abstract: We study natural subalgebras Ch_E(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS^0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP^*(BG) in H^*(BG;F_p) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra from E to HF_p which is non-trivial in homotopy. We also extend the constructions to define subalgebras of H^*(X;F_p) for any space X; when X is finite we show that the subalgebras Ch_{E(n)}(X) give a natural unstable chromatic filtration of H^*(X;F_p). 6 papers this by time, by Ando, Bakuradze-Priddy, Bousfield, Kuhn, Martino-Priddy, and Zhou. Note that the paper by Zhou claims to prove that V(n) exists for all n and all p >= 5, contradicting Ravenel's proof that V(3) does not exist at p=5. Zhou claims that the Toda relation alpha_1 beta_1^p =0 is false, giving some reasons why Toda's proofs are wrong, and therefore Ravenel's argument does not apply. I am hoping one of you will clear this up, but in the meantime I should remind you that papers on the Hopf archive are not edited for correctness or anything else. Mark Hovey New papers appearing on hopf between 01/02/02 and 02/11/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ando/ando-aeso Title: The sigma orientation for analytic circle-equivariant elliptic cohomology Author: Matthew Ando MSC: 55N34 (Primary); 55N22, 57R91 (Secondary) Arxiv: math.AT/0201092 Address: Department of Mathematics University of Illinois at Urbana-Champaign E-mail: mando---math.uiuc.edu Abstract: Let T be the circle group. We construct a canonical Thom isomorphism in T-equivariant analytic elliptic cohomology, for T-oriented virtual vector bundles bundles whose Borel-equivariant second Stiefel-Whitney and second Chern classes vanish. The construction is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the rational case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bakuradze-Priddy/bp3b TRANSFER AND COMPLEX ORIENTED COHOMOLOGY RINGS MALKHAZ BAKURADZE AND STEWART PRIDDY Keywords: transfer, Chern class, classifying space, complex cobor- dism, Morava K-theory 55N22, 55R12. 1. Introduction Let p be a prime and let G be a subgroup of the symmetric group S_p. In this paper we use the transfer to study homotopy orbit spaces X^p_hG= EG x_G X^p in complex oriented cohomology. We are particularly interested in computing the ring structure. Thus we are led to consider the relation between cup products and transfer known as Fröbenius reciprocity by analogy with representation theory Tr*(x)y = Tr*(x rho*(y)) (formula (i) of Section 2) where rho : EG x X^p --> X^p_hG is the covering projection and Tr* : E*(X^p) ---> E*(X^p_hG) is the associated transfer homomorphism. It is worth noting that the multiplicative structure of the cohomology groups we consider is com- pletely determined by this formula. In case E = K(s) is Morava K-theory, G is cyclic of order p, and X is the classifying space of a finite group, Hopkins-Kuhn-Ravenel [11 ] have studied these cohomology groups as modules over the coefficient ring. Our paper builds on their approach by extending their notion of a good group to spaces. For X = CP^infty we determine the algebra K(s)*(X^p_hS_p) for Morava K-theory; for complex cobordism we compute the ring MU*(X^p_hS_p) making additional use of the formal group law. This enables us to make explicit computations of the transfer in both cases. In an analogous fashion we compute the algebra BP *(X^p_hS_p). The starting point and original motivation for our work comes from Quillen's famous formula for Tr*(1), the stable Euler class, for the uni- versal Z/p covering. As explained in Section 2, our results for CP^infty provide a universal example which enable us to compute the stable Eu- ler classes and the transfer in general for many other cases. For example universal coverings for some nonabelian p-groups, namely those with cyclic subgroups of index p and those which are semi-direct products of elementary abelian p-groups with Z/p. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/cosim Cosimplicial resolutions and homotopy spectral sequences in model categories A.K. Bousfield Mathematics Subject Classification. Primary 55U35; Secondary 18G55, 55P60, 55T15. Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607 bous---uic.edu We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain fiber squares up to homotopy. We also deduce that the Bendersky-Thompson completions over connective ring spectra are equivalent to Bousfield-Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the Bendersky-Thompson homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/kuhn-mc Title: The McCord model for the tensor product of a space and a commutative ring spectrum. Author: Nicholas J. Kuhn AMS classification: Primary 55P43; Secondary 18G55 Author's address: Department of Mathematics, University of Virginia, Charlottesville, VA 22904 Email: njk4x---virginia.edu Abstract: This paper begins by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal's very special Gamma--spaces, and then to a more modern situation: (K tensor R) where K is a based space and R is a unital, augmented, commutative, associative S--algebra. The model comes with an easy-to-describe filtration. If one lets K = S^n, and then stabilize with respect to n, one gets a filtered model for the Topological Andre--Quillen Homology of R. When R = Omega^{infty} Sigma^{infty} X, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0th space. Another example comes by letting K be a finite complex, and R the S--dual of a finite complex Z. Dualizing again, one arrives at G.Arone's model for the Goodwillie tower of the functor sending Z to the suspension spectrum of Map(K,Z). Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E_1--terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S--algebras not necessarily augmented. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Martino-Priddy/mobiushopf Minami-Webb type decompositions for compact Lie groups John Martino and Stewart Priddy We extend to compact Lie groups some stable classifying space decompositions of Minami, following Webb. One notable feature of Webb's work is the use of a combinatorial Möbius function to encode p-local information about the cohomology of a finite group. We wish to show similar phenomena hold for compact Lie groups. However, for a compact Lie group G one is faced with the problem of an infinite number of conjugacy classes of p-toral subgroups, that is, extensions of tori by finite p-groups. These groups are the analogs of p-groups for finite groups. We circumvent this problem by considering a certain finite G-complex which allows us to introduce combinatorial methods in the compact Lie group case. This complex is based on the notion of p-stubborn subgroups which arose earlier in modular representation theory of finite groups (where they were called p-radical groups) in connection with Alperin's conjecture in group cohomology and in the study of homotopy classes of maps between classifying spaces of compact Lie groups. We also derive a decomposition based on the corresponding complex for elementary abelian p-subgroups. Several examples are given to illustrate the various decompositions. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zzhou (See the disclaimer at the top of this announcement). Smith-Toda Spectrum $V(\infty)$ exists for all $p\geqslant 5$} Zhou Xueguang AMS classification numbers: 55Q Address of author: Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb---eyou.co Abstract In this paper, we prove that the Smith-Toda spectrum $V(n)$ exists for all non-negative integers $n$. 4 papers this by time, one by Bauer and three by Jim Turner. Mark Hovey New papers appearing on hopf between 02/11/02 and 03/05/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BauerK/bauer1 Title: Higher Hochschild homology and its decompositions Author: Kristine Bauer Department of Mathematics Johns Hopkins University E-mail: kbbauer---math.jhu.edu Let k be a field of characteristic 0, A a k-algebra and M an A-module. In this paper we seek to provide a decomposition of a generalization of Hochschild homology. The construction is as follows: Let F_A be the functor from the category of finite pointed sets to k-vector spaces which takes [n]={0,1,...,n} to the tensor product of M with the n-fold tensor product of A with itself. Now consider the homology of the chain complex associated to F_A(S^1\wedge Y) where S^1\wedge Y is a simplicial finite pointed set. The special case where the realization of Y is an (n-1)-dimensional sphere is the n-th order higher Hochschild homology. To obtain the decomposition, we show that F_A(S^1\wedge Y) is a Hopf algebra under maps whose existence is suggested by the pinch and fold maps on the circle. We are then able to apply the methods which Loday and Gerstenhaber and Schack used to obtain a decomposition of Hochschild homology, which is the case F_A(S^1). Finally, we show that this decomposition recovers the decomposition of higher Hochschild homology recently obtained by Pirashvili. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/nilpotence Title: Nilpotency in the homotopy of simplicial commutative algebras Authors: James M Turner Address: Calvin College E-mail: jturner---calvin.edu ArXiv id. no.: math.AT/0201064 MSC-class: 13D03, 13D05, 13H10, 18G30, 55S99 Abstract: In this paper, we study simplicial commutative algebras with finite Andr\'e-Quillen homology. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of results established by the author. Our goal is to replace the finiteness condition on homotopy with a weaker condition expressed in terms of nilpotency for the action of the homotopy operations. Coupled with the finiteness assumption on Andr\'e-Quillen homology, this nilpotency condition provides a way to bound the height at which the homology vanishes. As a consequence, we establish a special case of an open conjecture of Quillen. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/Noetherian Title: On simplicial commutative algebras with Noetherian homotopy Authors: James M Turner Address: Calvin College E-mail: jturner---calvin.edu ArXiv id. no.: math.AT/0201063 MSC-class: 13D03, 13D05, 18G30, 55S45, 55U99 Abstract: In this paper, a strategy is developed for studying a simplicial commutative algebra A whose zeroth homotopy group is a Noetherian ring B and whose higher homotopy groups are finite over B. The strategy replaces A with a connected simplicial supplemented k(q)-algebra, for each prime ideal q in B, which preserves much of the Andre-Quillen homology of A. The methods for this construction involves a mixture of methods of homotopy theory (e.g. Postnikov towers) with methods of commutative algebras (e.g. completions, Cohen factorizations). We finish by indicating how these methods resolve a more general form of a conjecture posed by Quillen. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/vanishing (This is the final version of a paper that has been annnounced before, the last time in 1998) Title: Simplicial commutative algebras with vanishing Andre-Quillen homology Author: James M Turner Address: Calvin College E-mail: jturner---calvin.edu MSC-class: 13D03, 13D40, 18G30, 18G55 Journal-ref: Inventiones mathematicae 142 (3) (2000), pp. 547-558 Abstract: In this paper, we study the Andr\'e-Quillen homology of simplicial commutative $\ell$-algebras, $\ell$ a field, having certain vanishing properties. When $\ell$ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, which have non-zero characteristic. 7 papers this by time, from Ghienne, Goerss-Henn-Mahowald, Ishiguro-LeeHS, McAuley, Panov-Ray-Vogt, Pengelley-Williams, and Sinha. Mark Hovey New papers appearing on hopf between 03/05/02 and 04/03/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ghienne/ghiennephsnt Title of paper: Phantom maps, SNT-theory, and natural filtrations on lim^1 sets. Author: Pierre GHIENNE. AMS Classification: 55Q05, 55S37, 55P15. Adress of author: Matematisk Institut, Universitetsparken 5, DK--2100 Křbenhavn. E-mail adress: ghienne---math.ku.dk Text of abstract: We study the so-called Gray filtration on the set of phantom maps between two spaces. Using both its algebraic characterization and the Sullivan completion approach to phantom maps, we generalize some of the recent results of Le, McGibbon and Strom. We particularly emphasize on the set of phantom maps with infinite Gray index, describing it in an original algebraic way. We furthermore introduce and study a natural filtration on SNT-sets (that is sets of homotopy types of spaces having the same $n$-type for all $n$), which appears to have the same algebraic characterization of the Gray one on phantom maps. For spaces whose rational homotopy type is that of an $H$-space or a co-$H$-space, we establish criteria permitting to determinate those subsets of this filtration which are non trivial, generalizing work of McGibbon and M\o ller. We finally describe algebraically the natural connection between phantom maps and SNT-theory, associating to a phantom map its homotopy fiber or cofiber. We use this description to show that this connection respect filtrations, and to find generic examples of spaces for which the filtration on the corresponding SNT-set consists of infinitely many strict inclusions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Henn-Mahowald/18-02-ghm Title: The homotopy of L_2V(1) for the prime 3 Authors: Paul Goerss, Hans-Werner Henn and Mark Mahowald Adresses: Northwestern University, Universite Louis Pasteur, Northwestern University ABSTRACT Let V(1) be the Toda-Smith complex for the prime 3. We give a complete calculation of the homotopy groups of the L_2-localization of V(1) by making use of the higher real K-theory EO_2 of Hopkins and Miller and related homotopy fixed point spectra. In particular we resolve an ambiguity which was left in an earlier approach of Shimomura whose computation was almost complete but left an unspecified parameter still to be determined. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Ishiguro-LeeHS/2_21_02 Homotopy fixed point sets and actions on homogeneous spaces of $p$--compact groups Kenshi Ishiguro (kenshi---cis.fukuoka-u.ac.jp) Fukuoka University, Fukuoka 814-0180, Japan and Hyang-Sook Lee (hsl---mm.ewha.ac.kr) Ewha Womans University, Seoul, Korea We generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture. As an application, we discuss extension problems considering actions on homogeneous spaces of $p$--compact groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/McAuley/mcauleypaper This is another new version of Louis McAuley's paper titled "A proof of the Hilbert-Smith conjecture". 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Panov-Ray-Vogt/0202081 Title: Colimits, Stanley-Reisner algebras, and loop spaces Authors: Taras Panov, Nigel Ray, and Rainer Vogt Addresses: Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russia; Department of Mathematics, University of Manchester, Manchester M13 9PL, England; Fachbereich Mathematik/Informatik, Universitaet Osnabrueck, D-49069 Osnabrueck, Germany. E-mail addresses: tpanov---mech.math.msu.su nige---ma.man.ac.uk rainer---mathematik.uni-osnabrueck.de Arxiv: math.AT/0202081 Abstract: We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops $\Omega DJ(K)$. We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for $\Omega DJ(K)$ for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/toclarence The global structure of odd-primary Dickson algebras as algebras over the Steenrod algebra David J. Pengelley New Mexico State University Las Cruces, NM 88003 davidp---nmsu.edu Frank Williams New Mexico State University Las Cruces, NM 88003 frank---nmsu.edu Primary 55S05; Secondary 13A50, 16W30, 16W22, 16W50, 55S10 We prove a conjecture made by Frank Peterson on the global structure of the Dickson algebras arising as odd primary general linear group invariants. The Dickson algebra $W_{n}$ of invariants in a rank $n$ polynomial algebra over $% \mathbb{F}_{p}$ is an unstable algebra over the mod $p$ Steenrod algebra. We prove that $W_{n}$ is a free unstable algebra on a certain cyclic module, modulo just one additional relation. The result is both similar to and different from the corresponding result we previously obtained with Frank Peterson at the prime $2$. We also extend our characterization to the algebras of invariants under the special linear groups. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/knots Title: The topology of spaces of knots. Author: Dev P. Sinha AMS Class: 57R40 (primary); 55T35, 57Q45 (secondary). LANL ID: math.AT/0202287 Addresses: Department of Mathematics, University of Oregon, Eugene OR and Department of Mathematics, Brown University, Providence RI Email: dps---math.brown.edu Included EPS files: smallpenta.eps, smalltreepenta.eps Abstract: We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of mapping spaces and another which is cosimplicial. These models are homotopy equivalent to the corresponding knot spaces when the dimension of the ambient manifold is greater than three, and there are spectral sequences with identifiable $E^1$ terms which converge to their cohomology and homotopy groups. The combinatorics of the spectral sequences is comparable to combinatorics which arises in finite-type invariant theory. 5 new papers this time, from Ando-Hopkins-Strickland, Christensen-Dwyer-Isaksen, Dwyer-Greenlees-Iyengar, Kitchloo-Laures-Wilson, and McClure-Smith (a new version of a previously announced paper). Mark Hovey New papers appearing on hopf between 04/03/02 and 05/01/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ando-Hopkins-Strickland/sigma-hinfty-4.26 Title: The sigma orientation is an H-infinity map Authors: Matthew Ando Michael J. Hopkins Neil P. Strickland AMS subject classification: 55N34 arXiv number: math.AT/0204053 Adresses: Department of Mathematics, University of Illinois at Urbana-Champaign mando---math.uiuc.edu Department of Mathematics, Massachusetts Institute of Technology mjh---math.uiuc.edu Department of Pure Mathematics, University of Sheffield N.P.Strickland---sheffield.ac.uk Abstract: In "Elliptic spectra, the Witten genus, and the Theorem of the cube" (Invent. Math. 146 (2001)), the authors constructed a natural map from the Thom spectrum MU<6> to any elliptic spectrum, called the "sigma orientation". MU<6> is an H-infinity ring spectrum, and in this paper we show that if E is a K(2)-local H-infinity elliptic spectrum, then the sigma orientation is a map of H-infinity spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Dwyer-Isaksen/DC Obstruction theory in model categories J. Daniel Christensen, William G. Dwyer and Daniel C. Isaksen MSC: 55S35, 55U35, 18G55 (primary); 18G30, 55P42 (secondary) Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 jdc---uwo.ca Department of Mathematics University of Notre Dame South Bend, IN 46556 dwyer.1---nd.edu Department of Mathematics University of Notre Dame South Bend, IN 46556 isaksen.1---nd.edu Keywords: obstruction theory, closed model category, simplicial set, spectrum Working in an arbitrary pointed proper model category, we describe the cofibrations that have an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Greenlees-Iyengar/duality Duality in Algebra and Topology W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar We take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in a topological setting. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to differential graded algebras or more generally to structured ring spectra. This framework allows us to view all of the following dualities o Poincare duality for manifolds o Gorenstein duality for commutative rings o Benson-Carlson duality for cohomology rings of finite groups o Poincar duality for groups o Gross-Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon. We give a new formula for the Brown-Comenetz dual of the sphere spectrum; this turns out to be one instance of a general construction that in another setting gives the dualizing module of a Gorenstein ring. We also prove the local cohomology theorem for p-compact groups and reprove it for compact Lie groups. The key observation is that the cochain algebra on BG has a simple duality property which extends Poincare duality. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556. USA, dwyer.1---nd.edu Department of Pure Mathematics, Hick Building, Sheffield S3 7RH. UK, j.greenlees---sheffield.ac.uk 202 Mathematical Sciences Building, University of Missouri, Columbia, MO 65211. USA, iyengar---math.missouri.edu 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Laures-Wilson/kitchloo-laures-wilson The Morava K-theory of spaces related to BO Nitu Kitchloo Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 nitu---math.jhu.edu Gerd Laures Mathematisches Institut der Universitaet Heidelberg Im Neuenheimer Feld 288 D-69120} Heidelberg, Germany gerd---laures.de W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu Abstract: We calculate the (p=2) Morava K-theory of all of the spaces in the connective Omega spectra for ZxBO, BO, BSO, and BSpin. This leads to a description of the (p=2) BP cohomology of many of these spaces. Of particular interest is the space BO<8> and its relationship to BSpin. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2 (This is the same abstract as before, but a new version of the paper itself) Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics, Purdue University, West Lafayette, IN 47907--1395 mcclure---math.purdue.edu jhs---math.purdue.edu In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture. --------------------------------------------- Hola from Barcelona! Sorry for the lack of updates recently. There are 7 new papers listed here (and there are a few more that have been submitted and should be announced soon), from Baker-May, Bruner-Ha-Hung, Gaudens-Schwartz, Fausk-Hu-May, Hu-Kriz-May, and 2 from May. Mark Hovey New papers appearing on hopf between 05/01/02 and 06/29/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Baker-May/CoresMay30 Title: Minimal atomic complexes Authors: A.J. Baker and J.P. May Classification: 55P15 55P42 (55P60) Address: Math. Dept., University of Glasgow, Glasgow G12 8QW, Scotland. E-mail: a.baker---maths.gla.ac.u Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may---math.uchicago.edu Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of MU considered as a commutative S-algebra, although it is a core of MU considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, $eo_2$, and BoP at the prime 2, all $BP$ at any prime p, and the indecomposable wedge summands of the suspension spectra of $CP^\infty$ and $HP^\infty$ at any prime p are minimal atomic. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bruner-Ha-Hung/alg-trans Title: On behavior of the algebraic transfer Authors: Robert R. Bruner, Le Minh Ha, and Nguyen H. V. Hung MSC-class: 55P47, 55Q45, 55S10, 55T15 Paper: math.AT/0205170 Addresses: Robert R. Bruner Department of Mathematics Wayne State University Detroit, MI 48202 USA rrb---math.wayne.edu Le Minh Ha IHES, F-91440, Bures-sur-Yvette France lha---ihes.fr Nguyen H. V. Hung Department of Mathematics Wayne State University Detroit, MI 48202 USA nhvhung---math.wayne.edu Abstract: Let V be a mod 2 vector space of rank k. W. Singer defined a transfer homomorphism from the GL(k,2) coinvariants of the primitives in the homology of BV to the cohomology of the Steenrod algebra, as an algebraic version of the geometric transfer from the stable homotopy of BV to the stable homotopy of spheres. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that it is an isomorphism for k=1, 2, or 3. However, Singer showed that it is not an epimorphism for k=5. In this paper, we prove that it also fails to be an epimorphism when k=4. Precisely, it does not detect the non zero elements in the g family, in stems 20, 44, 92, and in general, 12*2^s - 4, for each s > 0. The transfer still fails to be an epimorphism even after inverting Sq^0, thereby giving a negative answer to a prediction by Minami. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Hu-May/FormalFeb16 Title: Isomorphisms between left and right adjoints authors: H. Fausk, P. Hu, and J.P. May Classification: 14A99, 18F99, 55P91 (18D10, 55U30) Address: Dept. Math., Northwestern University, Evanston, IL 60208-2730, USA. E-mail: fausk---math.northwestern.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: poh---math.uchicago.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may---math.uchicago.edu There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate ``dualizing object''. Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topology. Another is to make clear which parts of the proofs of such results are formal. The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gaudens-Schwartz/GS Title Sur les sous-modules instables des alg\`ebres instables Authors G\'erald Gaudens et Lionel Schwartz gerald.gaudens---math.univ-nantes.fr Département de Mathématiques 2, rue de le Houssiničre - BP 92208 44322 NANTES Cédex 3 FRANCE schwartz---math.univ-paris13.fr UMR 7539 du CNRS Institut Galil\'ee Universit\'e Paris 13 Av. J. B. Cl\'ement 93430 Villetaneuse FRANCE 55S10 Cet article fait suite \`a une pr\'epublication de Laurent Piriou et du second auteur. Il contient des r\'esultats reli\'es \`a la conjecture de finitude, plus pr\'ecisement \`a la structure du treillis des sous-modules instables d'une alg\`ebre instable r\'eduite. Le premier r\'esultat, d\^u au second auteur, montre que les sous-modules instables de l'alg\`ebre de Dickson sont, soit l'alg\`ebre toute enti\`ere, soit petits vis \`a vis de l'alg\`ebre. Le second r\'esultat, d\^u au premier auteur, montre que la s\'erie des socles d'une alg\`ebre instable connexe r\'eduite non-triviale est infinie, ceci avait \'et\'e conjectur\'e par le second auteur dans [13].Un outil important, d\^u au second auteur, est la construction et l'action de certaines op\'erations de Steenrod sur des classes appartenant \`a des alg\`ebres instables. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hu-Kriz-May/99April1 Title: Cores of spaces, spectra, and $E_{\infty}$ ring spectra Authors: P. Hu, I. Kriz, and J.P. May Classification: 55P15, 55P42, 55P43, 55S12 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: pohu---math.uchicago.edu Address: Dept. Math., University of Michigan, Ann Arbor, MI 48109-1107, USA E-mail: ikriz---math.lsa.umich.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may---math.uchicago.edu In a paper that has attracted little notice, Priddy showed that the Brown-Peterson spectrum at a prime p can be constructed from the p-local sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum Y that is p-local and (n_0-1)-connected and has $\pi_{n_0}(Y)$ cyclic, there is a p-local, $(n_0-1)$-connected ``nuclear'' CW complex or CW spectrum X and a map $f: X\to Y$ that induces an isomorphism on $\pi_{n_0}$ and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a self-map that induces an isomorphism on $\pi_{n_0}$ must be an equivalence. The construction of X from Y is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of MU at p, the construction yields BP. {Appeared: Homology, homotopy, and applications 3(2001), 341--354} 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/97April1 Title: Idempotents and Landweber exactness in brave new algebra Author: J.P. May Classification: 55N20, 55N91, 55P43 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may---math.uchicago.edu We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative $S$-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to $MU$-modules. {Appeared in Homology, homotopy, and applications 3(2001), 355--359} 7. http://hopf.math.purdue.edu/cgi-bin/generate?/May/WirthRev Title: The Wirthmuller isomorphism revisited author: J.P. May Classification: 55P91, 55U30 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may---math.uchicago.edu We show how the formal Wirthmuller isomorphism theorem proven in "Isomorphisms between left and right adjoints", by Fausk, Hu, and May, simplifies the proof of the Wirthmuller isomorphism in equivariant stable homotopy theory. Other examples from equivariant stable homotopy theory show that the hypotheses of the formal Wirthmuller and formal Grothendieck isomorphism theorems in the cited paper cannot be weakened. There are 8 new papers this time, from BauerT, Blanc-Markl, Casacuberta-Gutierrez, Dugger-Hollander-Isaksen, Dugger-Isaksen, Maltsiniotis, Toen-Vezzosi, and ZhengQb. Note that papers sent by e-mail take much longer to appear on the archive than papers submitted by ftp. If ftp is an option, it will be quicker for you and make Clarence's life much easier if you use it. Mark Hovey New papers appearing on hopf between 06/29/02 and 07/18/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BauerT/pcfm Title: p-compact groups as framed manifolds author: Tilman Bauer Address: Department of Mathematics, Rm. 2-492, Massachusetts Institute of Technology, Cambridge (MA) 02139 E-mail: tilman---mit.edu We describe a natural way to associate to any p-compact group an element of the p-local stable stems, which, applied to the p-completion of a compact Lie group G, coincides with the element represented by the manifold G with its left-invariant framing. To this end, we construct a d-dimensional sphere SG with a stable G- action for every d-dimensional p-compact group G, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with SG . 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Markl/blanc-markl Title: Higher Homotopy Operations Authors: David Blanc and Martin Markl Posted to xxx.lanl.gov as math.AT/0207082 DB: Dept. of Mathematics, Univ. of Haifa, 31905 Haifa, Israel blanc---math.haifa.ac.il MM: Mathematical Inst. of the Academy, Zitna, 115 67 Prague 1, Czech Republic markl---math.cas.cz Abstract: We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Casacuberta-Gutierrez/hloc_modspc Title of Paper: Homotopy Localizations of Module Spectra Authors: Carles Casacuberta and Javier J. Gutierrez AMS Classification numbers: 55P42, 55P43, 55P60. Adresses of Authors: Carles Casacuberta Departament d'Algebra i Geometria Universitat de Barcelona, Gran Via 585 E-08007 Barcelona, Spain Javier J. Gutierrez Departament de Matematiques Universitat Autonoma de Barcelona E-08193 Bellaterra, Spain e-mail: casac---mat.ub.es jgutierr---mat.uab.es Text of Abstract: We prove that stable homotopical localizations preserve ring spectrum structures and module spectrum structures under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-Mac Lane spectrum HZ. More generally, we describe the main features of localizations of HZ-modules (i.e., stable GEMs), motivated by similar results in unstable homotopy. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Hollander-Isaksen/hypspre Title: Hypercovers and simplicial presheaves Authors: Daniel Dugger Sharon Hollander Daniel C. Isaksen AMS subject classification: 55U35, 18F20 Addresses: Department of Mathematics, Purdue University ddugger---math.purdue.edu Department of Mathematics, University of Chicago sjh---math.uchicago.edu Department of Mathematics, University of Notre Dame isaksen.1---nd.edu Abstract: We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects can be explicitly identified in terms of a hypercover descent condition. Another is a very simple approach to change-of-site functors. In an appendix, we discuss how this hypercover localization compares to the more naive process of localizing at the Cech complexes; the two are not the same in general, but agree in some cases of interest. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/wesp Title: Weak equivalences of simplicial presheaves Authors: Daniel Dugger Daniel C. Isaksen AMS subject classification: 55U35, 18F20 Addresses: Department of Mathematics, Purdue University ddugger---math.purdue.edu Department of Mathematics, University of Notre Dame isaksen.1---nd.edu Abstract: The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic results about the local homotopy theory of simplicial presheaves. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Maltsiniotis/Groth-homot-th Title: La théorie de l'homotopie de Grothendieck Authors: G. Maltsiniotis, with two appendices by D.-C. Cisinski AMS Classification Numbers: 18F20, 18G30, 18G50, 18G55, 55P10, 55P15, 55P60 Addresses: Université Paris 7 Denis Diderot Case Postale 7012 2, place Jussieu F-75251 PARIS CEDEX 05 Email addresses: maltsin---math.jussieu.fr cisinski---math.jussieu.fr Abstract: This paper is an introduction to the homotopy theory of Grothendieck as developed in "Pursuing Stacks". The aim is to study "Elementary modelizers" i.e. presheaf categories modelizing the homotopy types, thus generalizing the theory of simplicial sets. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Toen-Vezzosi/agmod-I-fin-web Title: Homotopical Algebraic Geometry I: Topos theory Authors: Bertrand Toen and Gabriele Vezzosi AMS Classification: 14A20; 18G55; 55P43; 55U40; 18F10. Submitted to the xxx.lanl archive as math.AG/0207028 Addresses: Bertrand Toen, Laboratoire J. A. Dieudonn\'e, UMR CNRS 6621, Universit\'e de Nice Sophia-Antipolis, France; Gabriele Vezzosi, Dipartimento di Matematica, Universit\`a di Bologna, Italy. E-mail addresses: toen---math.unice.fr vezzosi---dm.unibo.it ABSTRACT: This is the first of a series of papers devoted to the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this paper we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of $\infty$-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove that for an S-site T, there is a model category of stacks over T, generalizing Joyal-Jardine structure on simplicial presheaves on a Grothendieck site. We also shows, as an analog of the relation between topologies and localizing subcategories of the categories of presheaves, that there is a bijection between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Then we study the notion of model topos due to C. Rezk, and relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that Dwyer-Kan simplicial localization provides a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites. As an application, we propose a definition of \'etale K-theory of ring spectra. An appendix gives an alternative approach to the theory which uses Segal categories. We define Segal topologies, Segal sites, stacks over Segal sites and Segal topoi. The existence of internal Hom's in this context allows us to define the Segal category of geometric morphisms between Segal topoi. An application to the reconstuction of a space via its Segal category of stacks is given. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhengQb/extgroup Title of the Paper: A Subspace of Ext$_A(Z_p,Z_p)$ Author: Zheng Qibing AMS Classification Number: 55 18G Address of Author Zheng Qibing Department of Mathematics Nankai University Tianjin, 300071, P.R.China Email Address of Author: zhengqb---eyou.com Abstract In this paper, we compute the cohomology of some Hopf algebras and find a subspace of the cohomology of the Steenrod algebra that includes the representative of the Greek letter families. -------------------------------- Notice that Hopf now has a web form for submitting papers. As one of the maintainers, I can tell you that it is much easier for me if you use this web form (or ftp) to submit your papers to Hopf rather than email. The human factor (i.e., me) still causes the most delays in announcements of papers. There are 9 new papers this time, from BrownR-Janelidze, BrownR-Wensley, Cisinski, Devinatz-Hopkins, Dugger-Shipley, Kitchloo-Notbohm, Libman, Mauger, and Morava. Mark Hovey New papers appearing on hopf between 07/18/02 and 09/11/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Janelidze/dgpsmap Title: Galois theory and a new homotopy double groupoid of a map of spaces Author(s): R. Brown, G.Janelidze AMS Classification numbers: 18D05, 20L05, 55 Q05, 55Q35 R. Brown, Mathematics Division, School of Informatics, University of Wales, Dean St., Bangor, Gwynedd LL57 1UT, U.K. G.Janelidze, Mathematics Institute, Georgian Academy of Sciences, Tbilisi, Georgia. r.brown---bangor.ac.uk,george_janelidze---hotmail.com The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat1-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Wensley/crossed-modules Title of Paper: Computation and Homotopical Applications of Induced Crossed Modules Authors: Ronald Brown \\ Christopher D Wensley AMS Classification numbers: 55P10,55Q2,20L05 Addresses of Authors: Mathematics Division, School of Informatics, University of Wales, Bangor Gwynedd, LL57 1UT U.K. {r.brown,~c.d.wensley}---bangor.ac.uk We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/top Théories homotopiques dans les topos Denis-Charles Cisinski Primary 18G55 (Homotopical Algebra) 18F20 (Presheaves and Sheaves) Secondary 18E35 (Localization of Categories) 18B25 (Topoi) 18G30 (Simplicial Objects) Submitted to the J. Pure Appl. Algebra Address Institut de Mathématiques de Jussieu Université Paris 7 2, place jussieu 75251 Paris cedex O5 France cisinski---math.jussieu.fr The purpose of these notes is to give an ad hoc construction of a closed model category structure on a topos inverting an arbitrary small set of arrows. Moreover, a necessary and sufficient condition for those structures to be proper is given. As an example, the Joyal closed model category structure on the category of simplicial objects of a topos is constructed without the use of (boolean) points. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz-Hopkins/homotopy-fixed-point This is an updated version of the paper whose abstract follows. Title: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups Author: Ethan S. Devinatz and Michael J. Hopkins Addresses of Authors: Ethan S. Devinatz Department of Mathematics University of Washington Seattle, WA 98195 Michael J. Hopkins Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02135 Email: devinatz---math.washington.edu mjh---math.mit.edu Let G be a closed subgroup of the semi-direct product of the nth Morava stabilizer group with the Galois group of the field extension of degree n of the field of p elements. We construct a "homotopy fixed point spectrum" whose homotopy fixed point spectral sequence involves the continuous cohomology of G. These spectra have the expected functorial properties and agree with the Hopkins-Miller fixed-point spectra when G is finite. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Shipley/kdeqDS Title: K-theory and derived equivalences Authors: Daniel Dugger and Brooke Shipley AMS Math. Subj. Class. 19D99, 18E30, 55U35 Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Department of Mathematics, Purdue University, West Lafayette, IN 47907 email: bshipley---math.purdue.edu Abstract: We show that two rings have the same algebraic K-theory if their derived categories are triangulated-equivalent. Similar results are given for G-theory, and for the `compact K-theory' of a large class of abelian categories. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Notbohm/loopspacemanifold Authors: Nitu Kitchloo and Dietrich Notbohm Ttile: Quasi finite loop spaces are manifolds It is an old conjecture, that finite $H$-spaces are homotopy equivalent to manifolds. Here we prove that this conjecture is true for loop spaces. Actually, we show that every quasi finite loop space is equivalent to a stably parallelizable manifold. The proof is conceptual and relies on the theory of p-compact groups. On the way we also give a complete classification of all simple 2-compact groups of rank 2. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Libman/towers Title: Tower techniques for cofacial resolutions author: A. Libman Classification: 55U35,55T15,18A25 Address: Dept. of Math. Sciences, University of Aberdeen, Aberdeen AB24 3UE, UK. E-mail: assaf---maths.abdn.ac.uk Let $J$ be a continuous coaugmented functor on spaces. For every space $X$ one constructs a cofacial resolution $X \to J^\bullet X$ (namely a cosimplicial resolution without its codegeneracy maps) in the usual way. Following Bousfield and Kan, one defines $J_s(X) = tot_s J^\bullet X$. Suppose $D$ is a small category and that $X$ is a $D$-diagram of $J$-injective spaces, namely $X(d) \to JX(d)$ admits a left inverse for every object $d$ in $D$, but in a way which need not be compatible, namely a map $JX \to X$ cannot be constructed out of this data. We show that for many free diagrams $F$, the spaces $hom_D(F,X)$ are $J_s$-injective for $s<\infty$. Thus, the functors $\mathbb{Z}_s$ of Bousfield and Kan capture a large class of polyGEMs as their injective spaces. This generalises earlier results by the author. Our methods use pro-object arguments, which are originally due to Farjoun. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Mauger/hopf_alg_pgroups The Cohomology of certain Hopf Algebras Associated with p-Groups Justin Mauger AMS Classification numbers: 16E40, 16S37 2033 Sheridan Road Northwestern University Evanston, IL 60208 justin---math.northwestern.edu In this paper, we study the cohomology H^*(A)=Ext_A^*(k,k) of a locally finite, connected, cocommutative Hopf algebra A over k=F_p. Specifically, we are interested in those algebras A for which H^*(A) is generated as an algebra by H^1(A) and H^2(A). We shall call such algebras \emph{semi-Koszul}. Given a central extension of Hopf algebras $F\lra A\lra B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be semi-Koszul. Special attention is given to the case in which $A$ is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central series of a $p$-group. We show that the algebras arising in this way from extensions by $\z$ of an abelian $p$-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 $p$-groups. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/orbiHKR Author: Jack Morava Title: HKR characters and higher twisted sectors This is the writeup of an expository talk, presented at the ChengDu (Sichuan) ICM Satellite conference on stringy orbifolds. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call higher twisted sectors in the theory of orbifolds. --------------- The web form for submission to Hopf is a big success! So much so that I will have to send out these letters more frequently. There are 17 new papers this time! So I will break this letter up into two parts. This first part contains 9 new papers this time, from Anton, Bendersky-DavisD, Blanc-Peschke, 3 from Cisinski, Devinatz, Ferland-Lewis, and Hovey. Mark Hovey New papers appearing on hopf between 09/11/02 and 10/07/02, part 1 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/morava Title of Paper: On Morava K-theories of an S-arithmetic group Author: Marian F. Anton AMS Classification numbers: 55N20,19F27,11F75 Address of Author: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK Email address of Author: Marian.Anton---imar.ro Text of Abstract: We completely describe the Morava K-theories with respect to the prime p for the etale model of the classifying space of the general linear group GL(m) over the ring Z[u,1/p] when p is an odd regular prime and u a primitive p-th root of unity. For p=3 and m=2 (and conjecturally in the stable range) these K-theories are the same as those of the classifying space itself. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/SON v1-periodic homotopy groups of SO(n) Martin Bendersky and Donald M. Davis 55Q52, 55T15, 57T20 Hunter College, CUNY, NY, NY 10021 Lehigh University, Bethlehem, PA 18015 Abstract We compute the 2-primary v1-periodic homotopy groups of the special orthogonal groups SO(n). The method is to calculate the Bendersky-Thompson spectral sequence, a K*-based unstable homotopy spectral sequence, of Spin(n). The E2-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres. The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly [log_2(2n/3)] copies of Z/2. As the spectral sequence converges to the v1-periodic homotopy groups of the K-completion of a space, one important part of the proof is that the natural map from Spin(n) to its K-completion induces an isomorphism in v1-periodic homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Peschke/BlancPeschke1 Authors: David Blanc and George Peschke Title: The plus construction, Postnikov towers and universal central module extensions. Given a connected space $X$, we consider the effect of Quillen's plus construction on the homotopy groups of $X$ in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequence \ $AX\to X\to X^+$, \ we explain the contribution of \ $\pi_nX$ \ to \ $\pi_nX^+$, \ $\pi_{n+1}X^+$ \ and \ $\pi_nAX$, \ $\pi_{n+1}AX$ \ explicitly in terms of the low dimensional homology of $\pi_nX$ regarded as a module over $\pi_1X$. \ Key ingredients developed here for this purpose are universal $\Pi$-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/der2 Images directes cohomologiques dans les categories de modeles Denis-Charles Cisinski AMS Classification numbers 55U3, 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski---math.jussieu.fr Abstract Show that every complete model category M admits homotopy limits, and more generaly that every functor between small categories has a cohomological direct image in M (that is a homotopy right Kan extension). Furthermore, we study the local behavor of such constructions. For this purpose, we introduce Grothendieck's notion of derivator. Derivators correspond to the intuition of ``a homotopy complete category'' without speaking about models. Forthcoming papers will show that this setting is rich enough to define classical homotopy theory by a simple universal property. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/propuni Proprietes universelles et extensions de Kan derivees Denis-Charles Cisinski AMS Classification numbers 55U3, 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski---math.jussieu.fr Abstract We show that for all small category A, the derivator associated to the homotopy theory of presheaves in categories (or in simplicial sets) on A is the solution of a universal problem (and a similar statement about the pointed versions of such derivators is proved). When A is the final category, this shows that the derivator HOT associated to the classical homotopy theory is canonically endowed with a monoidal structure, and that every derivator admit a canonical action of HOT. As every model category defines a derivator, Hovey's homotopy coherence conjectures are then a consequence of these constructions. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/winfax Le localisateur fondamental minimal Denis-Charles Cisinski AMS Classification numbers 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski---math.jussieu.fr Abstract Basic localizors were introduced by Grothendieck in Pursuing Stacks. These are classes of arrows in the category Cat of small categories satisfying nice properties of descent (like Quillen's theorem A).For example, every cohomology theory defines a basic localizor. In particular, classical weak equivalences (i.e. those induced from the simplicial weak equivalences from th nerve functor) form a basic localizor. In this paper, we show Grothendieck's conjecture that Cat's usual weak equivalences are the smallest basic localizor. This gives in particular a combinatorial/algebraic way to define classical homotopy theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/LHSspectral Title: A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra Author: Ethan S. Devinatz AMS Subject Classification: 55N20, 55P43, 55T15 Address: Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195 e-mail: devinatz---math.washington.edu Abstract: Let H and K be closed subgroups of the n th Morava stabilizer group with H normal in K. We construct a spectral sequence of the expected form connecting the homotopy of the continuous homotopy H fixed points of the Landweber exact spectrum E_n with the homotopy of the continuous K fixed points of E_n. These continuous homotopy fixed point spectra are the spectra constructed by Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in an appropriate category of module spectra. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Ferland-Lewis/FerlandLewis Title: The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G = \mathbb{Z}/p$ Authors: Kevin K. Ferland and L. Gaunce Lewis, Jr. AMS Classification numbers: Primary 55M35, 55N91, 57S17; Secondary 14M15 55P91 Addresses: Department of Mathematics, Bloomsburg University, Bloomsburg, PA 17815 and Department of Mathematics, Syracuse University, Syracuse NY 13244-1150 email: kferland---bloomu.edu lglewis---syr.edu Abstract: It is well known that the homology of a CW-complex with cells only in even dimensions is free. The equivariant analog of this result for generalized $G$-cell complexes is, however, not obvious, since \roG-graded homology cannot be computed using cellular chains. We consider $G = \mathbb{Z}/p$ and study $G$-cell complexes constructed using the unit disks of finite dimensional $G$-representations as cells. Our main result is that, if $X$ is a $G$-complex containing only even-dimensional representation cells and satisfying certain finiteness assumptions, then its \roG-graded equivariant ordinary homology \HoeX{G}{X}{A} is free as a graded module over the homology \HoPt of a point. This extends a result due to the second author about equivariant complex projective spaces with linear $\mathbb{Z}/p$-actions. Our new result applies more generally to equivariant complex Grassmannians with linear $\mathbb{Z}/p$-actions. Two aspects of our result are particularly striking. The first is that, even though the generators of \HoeX{G}{X}{A} are in one-to-one correspondence with the cells of $X$, the dimension of each generator is not necessarily the same as the dimension of the corresponding cell. This shifting of dimensions seems to be a previously unobserved phenomenon. However, it arises so naturally and ubiquitously in our context that it seems likely that it will reappear elsewhere in equivariant homotopy theory. The second unexpected aspect of our result is that it is not a purely formal consequence of a trivial algebraic lemma. Instead, we must look at the homology of $X$ with several different choices of coefficients and apply the Universal Coefficient Theorem for \roG-graded equivariant ordinary homology. In order to employ the Universal Coefficient Theorem, we must introduce the box product of \roG-graded Mackey functors. We must also compute the $RO(G)$-graded equivariant ordinary homology of a point with an arbitrary Mackey functor as coefficients. This, and some other, basic background material on \roG-graded equivariant ordinary homology is presented in a separate part at the end of the paper. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/comodule Author: Mark Hovey Title: Homotopy theory of comodules over a Hopf algebroid Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma ). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the basic properties of Stable(Gamma), showing that it is a compactly generated tensor triangulated category. ---------------- This second part contains 8 new papers, 2 from Moller, 1 from Oliver, and 5, count 'em 5, from YauD. Mark Hovey New papers appearing on hopf between 09/11/02 and 10/07/02, part 2 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Moller/ndet Title of paper: N-determined p-compact groups Author: Jesper M. Moller AMS Classification numbers: 55R35, 55P15 Email address of Author: moller---math.ku.dk Abstract: We consider p-compact groups where p is an odd primes. The paper contains a classification of p-compact groups, excluding the E-family, in terms of maximal torus normalizers. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Moller/twocgs Author: Jesper Moller Title: The 2-compact groups in the A-family are N-determined Let G be compact Lie group locally isomorphic to SU(n) for some n. The 2-completion of the classifyong space BG is a 2-compact group in the A-family. We show that these 2-compact groups are determined up to isomorphism by their maximal torus normalizers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver/limz-odd Author: Bob Oliver Title: Equivalences of classifying spaces completed at odd primes We prove here the Martino-Priddy conjecture for an odd prime p: the p-completions of the classifying spaces of two groups G and G' are homotopy equivalent if and only if there is an isomorphism between their Sylow p-subgroups which preserves fusion. A second theorem is a description for odd p of the group of homotopy classes of self homotopy equivalences of the p-completion of BG, in terms of automorphisms of a Sylow p-subgroup of G which preserve fusion in G. These are both consequences of a technical algebraic result, which says that for an odd prime p and a finite group G, all higher derived functors of the inverse limit vanish for a certain functor on the p-subgroup orbit category of G. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/adic_genus2 Title: On adic genus, Postnikov conjugates, and lambda-rings Author: Donald Yau MSC: 55P15; 55N15, 55P60, 55S25 Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 dyau---math.uiuc.edu Sufficient conditions on a space are given which guarantee that the $K$-theory ring and the ordinary cohomology ring with coefficients over a principal ideal domain are invariants of, respectively, the adic genus and the SNT set. An independent proof of Notbohm's theorem on the classification of the adic genus of $BS^3$ by $KO$-theory $\lambda$-rings is given. An immediate consequence of these results about adic genus is that for any positive integer $n$, the power series ring $\bZ \lbrack \lbrack x_1, \ldots , x_n \rbrack \rbrack$ admits uncountably many pairwise non-isomorphic $\lambda$-ring structures. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/moduli2 Title: Moduli space of filtered lambda-ring structures over a filtered ring Author: Donald Yau MSC: 16W70, 13K05, 13F25 Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 dyau---math.uiuc.edu Motivated by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered $\lambda$-ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings $R \llbrack x \rrbrack$, where $R$ is between $\bZ$ and $\bQ$, with the $x$-adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered $\lambda$-ring structures over $R \llbrack x \rrbrack$ is canonically isomorphic to the set of ring maps from some ``universal'' ring $U$ to $R$. From a local perspective, we demonstrate the existence of uncountably many mutually non-isomorphic filtered $\lambda$-ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial and powers series rings over torsionfree $\bQ$-algebras. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/nonexistence_final_2 Title: Maps to spaces in the genus of infinite quaternionic projective space Author: Donald Yau MSC: 55S37, 55S25 Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 dyau---math.uiuc.edu Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An interpretation of these results in the context of Adams-Wilkerson embedding in integral $K$-theory is also given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/steenrod_kuhn Title: Algebra over the Steenrod algebra, lambda-ring, and Kuhn's Realization Conjecture Author: Donald Yau Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 dyau---math.uiuc.edu In this paper we study the relationships between operations in $K$-theory and ordinary mod $p$ cohomology. In particular, conditions are given under which the mod $p$ associated graded ring of a filtered $\lambda$-ring is an unstable algebra over the Steenrod algebra. This result partially extends to the algebraic setting a topological result of Atiyah about operations on $K$-theory and mod $p$ cohomology for torsionfree spaces. It is also shown that any polynomial algebra that is an algebra over the Steenrod algebra can be realized as the mod $p$ associated graded of a filtered $\lambda$-ring. Another observation is that Atiyah's result gives rise to a $K$-theoretic analogue of Kuhn's Realization Conjecture concerning the size of spaces in cohomology. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/unstable Title: Unstable $K$-cohomology algebra is filtered lambda-ring Author: Donald Yau MSC: 55N20,55N15,55S05,55S25 Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 dyau---math.uiuc.edu Boardman, Johnson, and Wilson gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complex $K$-theory by taking into account its periodicity, we prove that an unstable algebra for complex $K$-theory is precisely a filtered $\lambda$-ring, and vice versa. --------------- 6 new papers this time, from Goerss-Henn-Mahowald-Rezk, 2 from Kadeishvili-Saneblidze, Klein, Levi-Oliver, and Rodriguez-Scherer-Viruel. Also, I fixed a stupid error in my paper Hovey/comodule so if you downloaded that before Oct. 15, you might want to download a new copy. Mark Hovey New papers appearing on hopf between 10/07/02 and 11/04/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Henn-Mahowald-Rezk/ghmr-14-10-02 Title: A resolution of the K(2)-local sphere Authors: Paul Goerss, Hans-Werner Henn, Mark Mahowald and Charles Rezk Adresses: Northwestern University, Universite Louis Pasteur, Northwestern University, University of Illinois at Urbana ABSTRACT At the prime p=3, we write the spectrum L_{K(2)}S^0 as the inverse limit of a short tower of fibrations where the fibers are (suspensions of) explicit homotopy fixed point spectra E_2^{hF} with F a finite subgroup of the Morava stabilizer group. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Kadeishvili-Saneblidze/cubmodel A cubical model for a fibration by TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE In the paper the notion of truncating twisting function $\tau :X\to Q$ from a simplicial set $X$ to a cubical set $Q$ and the corresponding notion of twisted Cartesian product of these sets $X\times_{\tau }Q$ are introduced. The latter becomes a cubical set whose chain complex coincides with the standard twisted tensor product $C_*(X)\otimes_{\tau_*}C_*(Q)$. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kadeishvili-Saneblidze/permuto The twisted Cartesian model for the double path space fibration Tornike Kadeishvili and Samson Saneblidze 55R05, 55P35, 55U05, 52B05, 05A18, 05A19 math.AT/0210224 A. Razmadze Mathematical Institute Georgian Academy of Sciences M. Aleksidze st., 1 380093 Tbilisi, Georgia kade---rmi.acnet.ge A. Razmadze Mathematical Institute Georgian Academy of Sciences M. Aleksidze st., 1 380093 Tbilisi, Georgia sane---rmi.acnet.ge The paper introduces the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets. The latter becomes a permutocubical set that models in particular the path space fibration on a loop space. The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Klein/susp-spectra Moduli of Suspension Spectra by John R. Klein Wayne State University klein---math.wsu.edu For a 1-connected spectrum E, we study the moduli space of suspension spectra which come equipped with a weak equivalence to E. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Oliver/sol Construction of 2-local finite groups of a type studied by Solomon and Benson by Ran Levi and Bob Oliver A $p$-local finite group is an algebraic structure with a classifying space which has many of the properties of $p$-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of $\Spin_7(q)$ ($q$ an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the $2$-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space $BDI(4)$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Rodriguez-Scherer-Viruel/notsimple3 Jose L. Rodriguez, Jerome Scherer, and Antonio Viruel 55P60, 20E32, 20D45 math.AT/0210405 Universidad de Almeria, Universitat Autonoma de Barcelona, and Universidad de Malaga, Spain jlrodri---ual.es, jscherer---mat.uab.es, viruel---agt.cie.uma.es Often a localization functor (in the category of groups) sends a finite simple group to another finite simple group. We study when such a localization also induces a localization between the automorphism groups and between the universal central extensions. As a consequence we exhibit many examples of localizations of finite simple groups which are not simple. ----------------- Hopf received nine new papers in just nine days, so its time to announce them again already. There are papers from Anton, Broto-Levi-Oliver, Christensen-Dwyer-Isaksen, Jardine (3), and Strickland (3). Also, I just recently found out that it seems to be impossible to put files on Hopf using anonymous ftp. We are trying to fix this, but in the meantime I suggest using the web form. Mark Hovey New papers appearing on hopf between 11/04/02 and 11/13/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/elementary.invariant Title of Paper: An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup Author: Marian F. Anton AMS Classification numbers: 57T10, 20J05, 19D06, 55R40 Address of Author: University of Sheffield, Department of Pure Mathematics, Hicks Building, Sheffield, S3 7RH, U.K. Email address of Author: Marian.Anton---imar.ro Conjecturally, for p an odd prime and R a certain ring of p-integers, the stable general linear group GL(R) and the etale model for its classifying space have isomorphic mod p cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if p is regular and certain homology classes for SL(2,R) vanish. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo-surv The theory of $p$-local groups: a survey by C. Broto, R. Levi, and B. Oliver This paper is a survey of recent results by the three authors, results which describe how the p-local fusion in a finite group G determines and is determined by the homotopy type of the p-completion of its classifying space BG. This connection then suggested to us the construction of certain spaces (classifying spaces of ``p-local finite groups'' and ``p-local compact groups'') which have many of the same properties as have p-completed classifying spaces of finite and compact Lie groups, and which can be characterized in homotopy theoretic terms. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Dwyer-Isaksen/obstruction (This is an update) Obstruction theory in model categories J. Daniel Christensen, William G. Dwyer and Daniel C. Isaksen MSC: 55S35, 55U35, 18G55 (primary); 18G30, 55P42 (secondary) Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 jdc---uwo.ca Department of Mathematics University of Notre Dame South Bend, IN 46556 dwyer.1---nd.edu Department of Mathematics University of Notre Dame South Bend, IN 46556 isaksen.1---nd.edu Keywords: obstruction theory, closed model category, simplicial set, spectrum Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/cubical2 Abstract: "Cubical homotopy theory: a beginning", by. J.F. Jardine This paper gives a closed model structure for the category of cubical sets, suitably defined, and displays an equivalence of the associated homotopy category with the ordinary homotopy category of topological spaces, or simplicial sets. Cubical complexes appeared in the early descriptions of homology theory and combinatorial homotopy theory in the middle of the twentieth century, but development of the subject area effectively stopped as simplicial sets became the dominant combinatorial model for homotopy theory as a result of the work of Kan and later Quillen. Cubical complexes have recently resurfaced as objects of fundamental interest in Pratt's theory of higher dimensional automata in concurrency theory. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada URL: http://www.math.uwo.ca/~jardine/papers/ 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/gen-shea Abstract: "Generalised sheaf cohomology theory", by. J.F. Jardine This is an expanded version of notes for a set of lectures given at the Newton Institute during a NATO ASI Workshop entitled ``Homotopy Theory of Geometric Categories'' on September 23 and 24, 2002. The paper presents some of the basic features of the homotopy theory of simplicial presheaves and the stable homotopy theory of presheaves of spectra, and then displays their use in the course of giving an outline of proof of Thomason's descent theorem for Bott periodic K-theory, in the context of equivariant stable categories for profinite groups. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada URL: http://www.math.uwo.ca/~jardine/papers/ 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/int-str Abstract: "Intermediate model structures for simplicial presheaves", by J.F. Jardine This note (it is not really a finished paper) shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense. A condition is given for these new model structures to be cofibrantly generated; this condition is met by Blander's local projective theory. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada URL: http://www.math.uwo.ca/~jardine/papers/ 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Strickland/mcurve Multicurves and equivariant cobordism Neil Strickland 55N20,55N22,55N91,14L05 Department of Pure Mathematics University of Sheffield Sheffield S3 7RH UK Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Strickland/rfg Realising formal groups Neil Strickland 55N20,55N22 Department of Pure Mathematics University of Sheffield Sheffield S3 7RH UK We show that a large class of formal groups can be realised functorially by even periodic ring spectra. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Strickland/st-csi Common subbundles and intersections of divisors Neil P. Strickland 55N20 14L05 14M15 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that the intersection of V_0 and V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory. -------------- Anonymous ftp is now fixed, so you can use this method to put papers on Hopf if you prefer it to the web form. Both are better than e-mail. 4 new papers this time, from McClure-SmithJH, Nam, Palmieri, and Saneblize-Umble. Mark Hovey New papers appearing on hopf between 11/13/02 and 12/01/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClure-Smith3 Cosimplicial Objects and little $n$-cubes. I. James E. McClure and Jeffrey H. Smith AMS Classification Numbers 18D50; 55P48 Submitted to arXiv: math.QA/0211368 Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 mcclure---math.purdue.edu jhs---math.purdue.edu In this paper we show that if a cosimplicial space or spectrum $X^\bullet$ has a certain kind of combinatorial structure (we call it a $\Xi^n$-structure) then the total space of $X^\b$ has an action of a certain operad which is weakly equivalent to the little $n$-cubes operad. The $n\leq 2$ case was proved by a more complicated argument in our earlier paper A Solution of Deligne's Hochschild Cohomology Conjecture (http://front.math.ucdavis.edu/math.QA/9910126). In the special case $n=\infty$, we define a symmetric monoidal structure $\boxtimes$ on cosimplicial spaces and show that if $X^\b$ is a commutative $\boxtimes$-monoid then the total space of $\X^\b$ is an $E_\infty$ space. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Nam/namInvent A-generateurs generiques pour l'algebre polynomiale by Tran Ngoc Nam Nous résolvons génériquement le problčme ``hit'' (posé en 1986 par Franklin P. Peterson) par la découverte en degrés génériques d'un systčme générateur minimal explicite pour l'algčbre polynomiale comme module sur l'algčbre de Steenrod mod 2. Cette solution implique en particulier un résultat de J. Repka-P. Selick, une partie de celui de M. C. Crabb-J. R. Hubbuck et nous permet en męme temps de vérifier une conjecture due ŕ M. Kameko. Ce systčme générateur sera appliqué ŕ l'étude du transfert algébrique de W. M. Singer et de la représentation modulaire du groupe linéaire général. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Palmieri/quotient Some quotient Hopf algebras of the dual Steenrod algebra by J. H. Palmieri Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation P^0 on cohomology, and in this paper, we investigate this operation. We point out that if p=2, then for any element in the cohomology of A, if one applies P^0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that "enough times" should be "once". The bulk of the paper is a study of some quotients of A in which the Frobenius is an isomorphism of order n. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P^0. The dual complete Steenrod algebra makes an appearance. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Saneblidze-Umble/PMAfnl Title: Diagonals on the Permutahedra, Multiplihedra and Associahedra Authors: Samson Saneblidze, Ronald Umble MSC: 55P35, 55U05 ArXive: math.AT/0209109 Abstract: We construct an explicit diagonal on the permutahedra {P_n}. Related diagonals on the multiplihedra {J_n} and the associahedra {K_n} are induced by Tonks' projection P_n --> K_{n+1} and its factorization through J_n. We use the diagonal on {K_n} to define the tensor product of A_infty-(co)algebras. We introduce the notion of a permutahedral set Z, observe that the double cobar construction Omega^{2}C_*(X) is a naturally occurring example and lift the diagonal on {P_n} to a diagonal on Z. --------------- Happy New Year! I remind you that your abstracts must contain your name and the title of the paper at a minimum. I have had to add these in by hand in a couple of recent cases. 7 new papers this time, from BrownR-Higgins, Jiang, Luo, Madsen-Weiss (the proof of the Mumford conjecture!), MauerOats, McClure-SmithJH, and Symonds. Mark Hovey New papers appearing on hopf between 12/01/02 and 01/08/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Higgins/cubabgp3 Title of Paper: Cubical abelian groups with connections\\ are equivalent to chain complexes Author(s): Ronald Brown and Philip J. Higgins AMS Classification numbers xxx LANL archive: math.AT/0212157 Addresses of Authors: Ronald Brown Mathematics Division \\ School of Informatics, \\ University of Wales, Bangor \\Gwynedd LL57 1UT, U.K. Philip J. Higgins, Department of Mathematical Sciences, \\ Science Laboratories, \\ South Rd., \\ Durham, DH1 3LE, U.K Email address of Authors r.brown---bangor.ac.uk p.j.higgins---durham.ac.uk Abstract: The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Jiang/realization Title of Paper: On the realization of the unstable modules Author: JIANG Dong Hua AMS Classification numbers: 55N99, 55S10 math.AT/0212054 Address of Author: LAGA, Institut Galilee, UMR 7539 University Paris Nord, Avenue Jean-Baptiste Clement 93430 VILLETANEUSE, FRANCE Email address of Author: donghua.jiang---m4x.org In this article, we give some restrictions about the structure of an unstable module, which should be verified providing this module is the reduced mod 2 cohomology of a space or a spectrum. We begin by studing the structure of the sub-modules of \Sigma^s H^\ast(B(Z/2)^{\oplus d}; Z/2)^{\oplus \alpha_d} (s \geq 0, \alpha_d > 0), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generelise this result for the unstable modules whose nilpotent filtration has a finite length, and who verified an additional condition. The result says that under some hypothesis, the reduced mod 2 cohomology of a space or a spectrum does not have arbitrary big gaps in its structure. This result is obtained by applying the famous Adams' theorem about the Hopf invariant and the classification of the injective unstable modules. For the unstable modules satisfing the condition of the theorem 3 (for example, any suspension of a sub-module of H^\ast(B(Z/2)^{\oplus d}; Z/2)^{\oplus \alpha_d}, the theorem 3 gives the upper bound of the length of the gaps in the modules, which means the module does not contain arbitrary big gaps. So when the module is reduced satisfing the condition of the theorem 4, its weight should be infinite. This gives us so many examples of the non-realizable unstable modules: F(n), any tensor product of F(n_i), etc. (These examples can also be proved by the theorem of Lionel Schwartz about the Kuhn conjecture, which was generalised by F-X. Dehon - G. Gaudens.) This article is written in french and the work is done under the direction of L. Schwartz. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Luo/pre Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids Zhi-ming Luo We prove that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Madsen-Weiss/mumf The stable moduli space of Riemann surfaces: Mumford's conjecture Ib Madsen and Michael Weiss AMS classification numbers 57R50; 14H15, 32G15, 57R45, 57M99 Submitted to arXiv: math.AT/0212321 Institute for the Mathematical Sciences Aarhus University 8000 Aarhus C Denmark Department of Mathematics University of Aberdeen Aberdeen AB24 3UE United Kingdom imadsen---imf.au.dk m.weiss---maths.abdn.ac.uk The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group (i.e, the group of isotopy classes of automorphisms of a connected oriented surface of "large" genus). In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group. It is part of a more recent development in the field which began with Ulrike Tillmann's result (Invent. Math., 1997) that the plus construction makes the classifying space of the stable mapping class group into an infinite loop space. This led to a stable homotopy theory version of Mumford's conjecture, stronger than the original (Madsen and Tillmann, Invent. Math., 2001). We prove the extended version of Mumford's conjecture by a mixture of techniques from singularity theory and from homotopy theory. The stability theorem of J.Harer (Annals of Math., 1985) and the "First Main theorem" of V.Vassiliev ("Complements of Discriminants of smooth maps: Topology and Applications", Trans. of Math. Monographs Vol.98, revised edition, Amer. Math. Soc. 1994) are prominent components of our proof. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/MauerOats/algebraic-calc Algebraic Goodwillie calculus and a cotriple model for the remainder Andrew Mauer-Oats We define an ``algebraic'' version of the Goodwillie tower, P_n^alg F(X), that depends only on the behavior of F on coproducts of X. When F is a functor to connected spaces or grouplike H-spaces, the functor P_n^alg F is the base of a fibration whose fiber is the simplicial space associated to a cotriple built from the (n+1) cross effect of the functor F. When the connectivity of X is large enough (for example, when F is the identity functor and X is connected), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor F in many interesting cases. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2_1 Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 mcclure---math.purdue.edu jhs---math.purdue.edu This is a revision of a paper first posted June 4, 2001. It will appear in the Journal of the AMS. In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Symonds/morava The Tate-Farrell cohomology of the Morava Stabilizer Group $S_{p-1}$ with coefficients in $E_{p-1}$ Peter Symonds We calculate the Tate-Farrell cohomology of the Morava stabilizer group $S_{p-1}$ with coefficients in the moduli space $E_{p-1}$ for odd primes $p$. ------------ ------------------------------ The dvipdf and dviselect programs don't seem to be working quite right on Hopf. Only a very few papers are affected, but if you have any trouble with pdf files, use the dvi file instead. 13 new papers this time, from BrownR, BrownR-Higgins, Chataur-Rodriguez-Scherer, Hovey, Hovey-Strickland (2 papers), Hung (4), Hung-Nam (2), and Marzantowicz-Prieto. Mark Hovey New papers appearing on hopf between 1/08/03 and 01/21/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-artxx Title of Paper: Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems Author(s): Ronald Brown AMS Classification numbers: 01-01,16E05,18D05,18D35,55P15,55Q05 Already submitted to the xxx LANL archive, include the id. no., math.AT/0212271 Addresses of Authors: Ronald Brown Mathematics Division, School of Informatics, University of Wales, Bangor Gwynedd LL57 1UT, U.K. Email address of Authors r.brown---bangor.ac.uk Abstract: We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Higgins/orbitgpdxx Title of Paper: The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Author(s): Ronald Brown and Philip J. Higgins AMS Classification numbers: 0F34, 20L13, 20L15, 57S30 Already submitted to the xxx LANL archive, include the id. no., math.AT/0212271 Addresses of Authors: Ronald Brown Mathematics Division, School of Informatics, University of Wales, Bangor Gwynedd LL57 1UT, U.K. Philip J. Higgins Department of Mathematical Sciences, Science Laboratories, South Rd., Durham, DH1 3LE, U.K. Email address of Authors r.brown---bangor.ac.uk p.j.higgins---durham.ac.uk Text of Abstract (try for 20 lines or less) The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (1988 edition). This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader. It is expected that this material will be part of a new edition of the book. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chataur-Rodriguez-Scherer/operadplus Plus-construction of algebras over an operad, cyclic and Hochschild homologies up to homotopy David Chataur, Jose L. Rodriguez, and Jerome Scherer math.AT/0301130 CRM Barcelona, dchataur---crm.es Universidad de Almeria, jlrodri---ual.es Universidad Autonoma de Barcelona, jscherer---mat.uab.es The aim of this paper is to show how to apply the machinery of homotopical localization to the framework of differential graded algebras over an operad. By performing nullification with respect to a universal acyclic algebra one obtains a plus-construction, which doesn't affect Quillen homology and quotients out the maximal perfect ideal of $\pi_0$. For any associative algebra the general linear Lie (resp. Leibniz) algebra is a Lie (resp. Leibniz) algebra up to homotopy. The plus-construction yields then two new homology theories, closely related to cyclic and Hochschild homology (they coincide with the classical cyclic and Hochschild homology over the rational). We also compute the first homology groups of these theories, in analogy with the computation of the first $K$-theory groups of a ring. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/barcelona Chromatic phenomena in the algebra of BP_{*}BP-comodules Mark Hovey Wesleyan University mhovey---wesleyan.edu This paper begins with an exposition of the author's research on the category of BP_*BP-comodules, much of which is joint with Neil Strickland. We give an overview of the results obtained in the papers Hovey/comodule, Hovey-Strickland/torsion-comod, and Hovey-Strickland/derived-ln. The main result of that work is that the category of E(n)_*E(n)-comodules is equivalent to a localization of the category of BP_*BP-comodules (the localization is L_n, analogous to the topological L_n). The main new result in this paper is that, analogously, the stable homotopy category of E(n)_*E(n)-comodules is equivalent to a localization (the finite localization L_n^f this time, not L_n) of the stable homotopy category of BP_*BP-comodules. These stable homotopy categories were constructed in Hovey/comodule, and are supposed to model stable homotopy theory; it is like stable homotopy theory where there are no differentials in the Adams-Novikov spectral sequence. Our result embeds the Miller-Ravenel and Hovey-Sadofsky change of rings theorems as special cases of isomorphisms like [X,Y]=[L_n^f X, Y] for L_n^f-local objects Y. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/torsion-comod Comodules and Landweber exact homology theories Mark Hovey and Neil Strickland Wesleyan University University of Sheffield mhovey---wesleyan.edu N.P. Strickland---sheffield.ac.uk We show that, if E is a Landweber exact ring spectrum, then the category of E_*E-comodules is equivalent to the localization of the category of BP_*BP-comodules with respect to the hereditary torsion theory of v_n-torsion comodules, where n is the height of E. In particular, the category of E(n)_*E(n)-comodules is equivalent to the category of (v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems for E_*E-comodules; we show every E_*E-comodule has a primitive, we classify the invariant radical ideals, and we prove a version of the Landweber filtration theorem. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/derived-ln Local cohomology of BP_*BP-comodules Mark Hovey and Neil Strickland Wesleyan University University of Sheffield mhovey---wesleyan.edu N.P. Strickland---sheffield.ac.uk In the paper torsion-comod (announced above) on this archive, we showed that the category of E(n)_*E(n)-comodules is a localization of the category of BP_*BP-comodules. In this paper, we study the resulting localization functor L_n on the category of BP_*BP-comodules. It is an algebraic analogue of the usual topological localization L_n. It is left exact, so has right derived functors L_n^i. We show that these derived functors are closely related to the local cohomology groups of BP_*-modules studied by Greenlees and May; in fact, they coincide with Cech cohomology with respect to I_{n+1}. We also construct a spectral sequence of comodules analogous to the Greenlees-May spectral sequence (of modules) converging to BP_*(L_n X) whose E_2-term involves L_n^i(BP_*X). The proofs require getting a partial understanding of injective objects in the category of BP_*BP-comodules. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2001 Title of Paper: On triviality of Dickson invariants in the homology of the Steenrod algebra Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Current Address: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore MD 21218 - 2689 E-mail address: nhvhung---math.jhu.edu Permanent Address: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$ the Dickson algebra of $k$ variables. We study the Lannes-Zarati homomorphisms $$ \varphi_k: Ext_{\cal A}^{k,k+i}(F_2,F_2)\to (F_2\otimes_{\cal A} D_k)_i^*, $$ which correspond to an associated graded of the Hurewicz map $ H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)$. An algebraic version of the long-standing conjecture on spherical classes predicts that $\varphi_k=0$ in positive stems, for $k>2$. That the conjecture is no longer valid for $k=1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes. This conjecture has been proved for $k=3$ by Hung [Trans AMS 349 (1997), 3893-3910]. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k>2$ [Hung and Peterson, Math. Proc. Camb. Phil. Soc. 124 (1998), 253-264] and on the image of Singer's algebraic transfer for $k>2$ [Hung, 1997; Hung and Nam, Trans AMS 353 (2001), 5029-5040]. In this paper, we establish the conjecture for $k=4$. To this end, our main tools include (1) an explicit chain-level representation of $\varphi_k$ and (2) a squaring operation $Sq^0$ on $(F_2\otimes_{\cal A} D_k)^*$, which commutes with the classical $Sq^0$ on $Ext_{\cal A}^k(F_2,F_2)$ through the Lannes-Zarati homomorphism. (To appear in Math. Proc. Camb. Phil. Soc. 134 (2003).) 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2002h Title of Paper: The cohomology of the Steenrod algebra and representations of the general linear groups Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Current Address: Department of Mathematics, Wayne State University 656 W. Kirby Street, Detroit, MI 48202 (USA) E-mail address: nhvhung------math.wayne.edu Permanent Address: Department of Mathematics, Vietnam National University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung------vnu.edu.vn Abstract: Let $Tr_k$ be the algebraic transfer that maps from the coinvariants of certain $GL_k$-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\V _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$ and that $Tr= \oplus_k Tr_k$ is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree $d$, and apply $Sq^0$ repeatedly at most $(k-2)$ times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the $GL_k$-representation. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(k-2)$ non zero elements. Two applications are exploited. The first main theorem is that $Tr_k$ is not an isomorphism for $k\geq 5$. Furthermore, $Tr_k$ is not an isomorphism in infinitely many degrees for each $k > 5$. We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, $Tr_k$ is not a monomorphism in infinitely many degrees for each $k>\ell$. The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(k-2)$ elements, are either all detected or all not detected by $Tr_k$, for every $k$. Applications of this study to the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three families $g$, $D_3$, $p'$ and $Tr_5$ does not detect the family $\{h_{n+1}g_n |\; n\geq 1\}$. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS01 Title of Paper: Spherical classes and the Lambda algebra Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let $\Gamma^{\wedge}= \oplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the Lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\cal A}(F_2, F_2)$, where ${\cal A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, $D_k$, into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes--Zarati dual homomorphism $$ \varphi_k^*: F_2\otimes_{\cal A} D_k \to Tor^{\cal A}_k(F_2,F_2) \cong H_k(\Gamma^{\wedge}). $$ The Lannes--Zarati homomorphisms themself, $\varphi_k$, correspond to an associated graded of the Hurewicz map $$ H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)\,. $$ Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that {\it Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism.} One of these algebraic conjectures predicts that every Dickson element, i. e. element in $D_k$, of positive degree represents the homology class $0$ in $Tor^{\cal A}_k(F_2, F_2)$ for $k>2$. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS97 Title of Paper: Spherical classes and the algebraic transfer Author: Nguy\^{e}n H. V. Hung 1991 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: We study a weak form of the classical conjecture which predicts that there are no spherical classes in $Q_0S^0$ except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer. We prove that the weak conjecture is equivalent to the following one: Every positive degree Dickson invariant of at least 3 variables belongs to the image of the Steenrod algebra acting on the corresponding polynomial algebra. This conjecture is proved for the case of 3 variables in two different ways. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamJA01 Title of Paper: The hit problem for the modular invariants of linear groups Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam 2000 Mathematics Subject Classification: Primary 55S10, Secondary 55Q45. Address of authors: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn E-mail address: namtn---vnu.edu.vn Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the general linear group, $GL_k:=GL(k, F_2)$, act on $P_{k}:=F_2[x_{1},...,x_{k}]$ with $\deg(x_{i})=1$ in the usual manner. We prove that, for a family of some rather small subgroups $G$ of $GL_k$, every element of positive degree in the invariant algebra $P_{k}^G$ is hit by ${\cal A}$ in $P_{k}$. In other words, $(P_{k}^G)^+ \subset {\cal A}^+\cdot P_{k}$, where $(P_{k}^G)^+$ and ${\cal A}^+$ denote respectively the submodules of $P_{k}^G$ and ${\cal A}$ consisting of all elements of positive degree. This family contains most of the parabolic subgroups of $GL_k$. It should be noted that the smaller the group G is the harder the problem turns out to be. Remarkably, when $G$ is the smallest group of the family, the invariant algebra $P_{k}^G$ is a polynomial algebra in $k$ variables, whose degrees are $\leq 8$ and fixed while $k$ increases. It has been shown by Hung [Trans AMS 349 (1997), 3893-3910] that, for $G=GL_k$, the inclusion $(P_{k}^{GL_k})^+\subset {\cal A}^+\cdot P_{k}$ is equivalent to a week algebraic version of the long-standing conjecture stating that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamTAMS01 Title of Paper: The hit problem for the Dickson algebra Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam 2000 Mathematics Subject Classification: Primary 55S10, Secondary 55P47, 55Q45, 55T15. Address of authors: Department of Mathematics, Vietnam National University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn E-mail address: namtn---vnu.edu.vn Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the general linear group, $GL(k, F_2)$, act on $P_{k}:= F_2[x_{1},...,x_{k}]$ with $|x_{i}|=1$ in the usual manner. We prove the conjecture of the first-named author in {\it Spherical classes and the algebraic transfer}, (Trans. AMS 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra $D_{k}:=(P_{k})^{GL(k,F_2)}$ is ${\cal A}$-decomposable in $P_{k}$ for arbitrary $k>2$. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/Marzantowicz-Prieto/Marprieto The unstable equivariant fixed point index and the equivariant degree by Waclaw Marzantowicz and Carlos Prieto A correspondence between the equivariant degree introduced by Ize, Massab\'o, and Vignoli and an unstable version of the equivariant fixed point index defined by the second author and Ulrich is shown. With the help of conormal maps and properties of the unstable index, we prove a sum decomposition formula for the index and consequently also for the degree. As an application, we decompose equivariant homotopy groups as direct sums of smaller groups of fixed orbit types, and we give a geometric interpretation of each summand in terms of conormal maps. --------------- ------------------------------- 7 new papers this time, from Bokstedt-Ottosen, Chataur-Scherer, Hung, Jardine, Rosu (2), and Ruiz-Viruel. Mark Hovey New papers appearing on hopf between 1/21/03 and 03/01/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/stringV4 Title: A spectral sequence for string cohomology Authors: Marcel Bokstedt and Iver Ottosen AMS Classification numbers: 55N91, 55P35, 18G50 Address of Authors: Institut for Matematiske Fag Aarhus Universitet Ny Munkegade DK-8000 Aarhus C Matematisk Afdeling Koebenhavns Universitet Universitetsparken 5 DK-2100 Koebenhavn OE Email address of Authors: marcel---imf.au.dk iver---math.ku.dk Abstract: Let $X$ be a 1-connected spaces with free loop space $\Lambda X$. We introduce two spectral sequences converging towards $H^*(\Lambda X;\ZZ /p)$ and $H^*((\Lambda X)_{hS^1};\ZZ /p)$. The $E_2$-terms are certain non Abelian derived functors applied to $H^*(X;\ZZ /p)$. When $H^*(X;\ZZ /p)$ is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If $X$ is a sphere it is a surprising fact that the spectral sequences collapse for $p=2$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chataur-Scherer/fibrewise Fibrewise nullification and the cube theorem David Chataur and Jerome Scherer CRM Barcelona, dchataur---crm.es Universidad Autonoma de Barcelona, jscherer---mat.uab.es Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mather's cube theorem. Working with model categories in which the cube theorem holds, we characterize completely those who admit a fibrewise nullification. As an application we get fibrewise plus-construction and fibrewise Postnikov sections for algebras over an operad. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungMZ99 Title of Paper: The weak conjecture on spherical classes Author: Nguy\^{e}n H. V. Hung 1991 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let $A$ be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, $$ Tr_k^*: Tor^A_k(F_2,F_2) \to F_2\otimes_A F_2[x_1,...,x_k], $$ which maps Singer's invariant-theoretic model of the lambda algebra to $F_2[x_1^{\pm},...,x_k^{\pm}]$ and is the inclusion of the Dickson algebra into the polynomial algebra $F_2[x_1,...,x_k]$. Based on this chain-level representation, we study some aspects of the weak conjecture on spherical classes and prove it in some special cases. (Address of Paper: Math. Zeit. 231 (1999), 727-743) 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/simpset3 Abstract: "Simplicial approximation", by J.F. Jardine This paper displays an approach to the construction of the homotopy theory of simplicial sets and the corresponding equivalence with the homotopy theory of topological spaces which is based on simplicial approximation techniques. The required simplicial approximation results for simplicial sets and their proofs are given in full. Subdivision behaves like a covering in the context of the techniques displayed here. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada E-mail: jardine---uwo.ca URL: http://www.math.uwo.ca/~jardine/papers/ 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Rosu/ellc Title: Equivariant Elliptic Cohomology and Rigidity Author: Ioanid Rosu, AMS Classification numbers: 55N34; 55N91 xxx LANL archive ID number: AT/9912089 Addresses and emails of Authors: Ioanid Rosu, M.I.T., Cambridge, MA. ioanid---math.mit.edu Equivariant elliptic cohomology with complex coefficients was defined axiomatically by Ginzburg, Kapranov and Vasserot and constructed by Grojnowski. We give an invariant definition of S^1-equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Rosu/kt Title: Equivariant K-theory and Equivariant Cohomology Author: Ioanid Rosu, with an appendix by Allen Knutson and Ioanid Rosu AMS Classification numbers: 55N91 xxx LANL archive ID number: AT/9912088 Addresses and emails of Authors: Ioanid Rosu, M.I.T., Cambridge, MA. ioanid---math.mit.edu Allen Knutson, University of California at Berkeley, CA allenk---math.berkeley.edu For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ruiz-Viruel/rv TITLE: The classification of $p$-local finite groups over the extraspecial group of order $p^3$ and exponent $p$. AUTHORS: Albert Ruiz, LAGA Universit{\'e} Paris XIII 99av J.B.\ Cl{\'e}ment 93430 Villetaneuse France ruiz---math.univ-paris13.fr Antonio Viruel Dpto de {\'A}lgebra, Geometr{\'\i}a y Topolog{\'\i}a Universidad de M{\'a}laga Apdo correos 59 29080 M{\'a}laga Spain viruel---agt.cie.uma.es ABSTRACT: The concept of $p$-local finite group arise in the work of Broto-Levi-Oliver as a generalization of the classical concept of finite group. Therefore, the classification of $p$-local finite groups has interest, not only by itself but, as an opportunity to enlighten one of the highest mathematical achievements in the last decades: The Classification of Finite Simple Groups. In this work we classify all $p$-local finite group over the $p$-groups of type $p^{1+2}_+$. In this classification three new exotic $7$-local finite groups arise. ------------------------------------------------------ 10 new papers this time, from Anderson-Grodal-Moller-Viruel, Anjos-Granja, Bauer-McCarthy, Behrens-Pemmaraju, Budney-Conant-Scannell-Sinha, Donadze-Inassaridze-Porter, Dorabiala-Johnson, Kitchloo-Laures-Wilson, Salvatore, and Sinha. Mark Hovey New papers appearing on hopf between 3/01/03 and 4/09/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anderson-Grodal-Moller-Viruel/classificationpodd Title: The classification of p-compact groups for p odd Authors: Kasper K. S. Andersen, Jesper Grodal, Jesper M. M{\o}ller, Antonio Viruel Subj-class: AT Algebraic Topology (GR Group theory; RT Representation Theory) MSC-class: 55R35 (Primary) 55P35, 57T10, 20G20 (Secondary) Comments: 87 pages \\ A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our method however leads to a largely self-contained proof of the entire classification theorem. \\ 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Anjos-Granja/homotopy.decomp.symplect Title: Homotopy decomposition of a group of symplectomorphisms of S^2\times S^2 Authors: Silvia Anjos and Gustavo Granja AMS Classification numbers: 57S05, 57R17, 55R35 Address of Authors: Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais 1049-001 Lisboa Portugal Email address of Authors: sanjos---math.ist.utl.pt ggranja---math.ist.utl.pt Abstract: We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bauer-McCarthy/mcbauer3 Kristine Bauer Department of Mathematics Johns Hopkins University 3400 N. Charles St. Baltimore, MD 21218 USA kbbauer---math.jhu.edu Randy McCarthy Department of Mathematics University of Illinois 1409 W. Green St. Urbana, IL 61801 USA randy---math.uiuc.edu On vanishing Tate cohomology and decompositions in Goodwillie calculus Mathematical Subject Classification: 55P65 (55P45, 13D03) Our main result is that if F is a functor from a pointed category C to spectra, the Goodwillie tower of F evaluated at X splits rationally when X is a co-H-object of C. We show that the layers of F(X) in this case are easy to identify. The splitting of the Goodwillie tower gives a decomposition of F(X) into a product of its layers. We use this to recover the rational decompositions of Hochschild and higher Hochschild homology by Pirashvili, Loday,and Gerstenhaber-Schack. Finally, we extend the main theorem to include dual calculus to recover the Poincar\'e-Birkhoff-Witt theorem, and improve the theorem in the special case in which the comultiplication map is cocommutative. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Pemmaraju/v2 On the existence of the self map v_2^9 on the Smith-Toda complex V(1) at the prime 3 Mark Behrens Department of Mathematics University of Chicago Chicago, IL 60637, U.S.A. mbehrens---math.uchicago.edu Satya Pemmaraju Fixed Income Derivatives UBS Warburg Stamford, CT 06901, U.S.A. Satya.Pemmaraju---ubsw.com AMS Classification: 55Q51; 55Q45, 55T15 math.AT/0303223 submitted to proceedings of the Northwestern University conference on algebraic topology, March 2002 Included EPS files: assE2.eps bss.eps eo_2V1.eps eo_2V1ASS.eps extP.eps splitting.eps Note: there is one chart created using the landscape package in LaTeX. On some dvi viewers, this chart does not display properly, but is viewable when converted to Postscript. Abstract Let V(1) be the Smith-Toda complex at the prime 3. We prove that there exists a map v_2^9: \Sigma^{144}V(1) \to V(1) that is a K(2) equivalence. This map is used to construct various v_2-periodic infinite families in the 3-primary stable homotopy groups of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Budney-Conant-Scannell-Sinha/selflink Title: New perspectives on self-linking Author: Ryan Budney, James Conant, Kevin P. Scannell, Dev P. Sinha AMS Class: 57M27; 55R80; 57R40; 57M25; 55P99 LANL ID: math.AT/0303034 Addresses: Departments of Mathematics, Rochester University, Cornell University, St. Louis University, University of Oregon Email: rybu---math.rochester.edu, jconant---polygon.math.cornell.edu, scannell---slu.edu, dps---math.uoregon.edu Abstract: We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. Note: The .dvi version is missing many (fun) figures - we strongly recommend downloading the .pdf file. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Donadze-Inassaridze-Porter/Hopfder N-fold Cech derived functors and generalized Hopf type formulas by Guram Donadze, Nick Inassaridze, and Timothy Porter, In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k. The paper ends with an application to algebraic K-theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Dorabiala-Johnson/torsion The product theorem for parametrized topological Reidemeister torsion Wojtek Dorabiala Mark W. Johnson Primary: 19D10; Secondary: 18F25, 19Exx, 55R70 Department of Mathematics Penn State Altoona Altoona, PA 16601-3760 wud2---psu.edu mwj3---psu.edu The goal of this article is to prove the product formula for parametrized topological Reidemeister torsion. The theorem states that the product of the parametrized Euler characteristic of one fibration with the parametrized Reidemeister torsion class of another fibration yields the parametrized Reidemeister torsion class of the product fibration. In the process of establishing the theorem, several new products must be defined involving (derivative theories of) parametrized $\Aof$-theory and a detailed description of the coassembly map for parametrized $\Aof$-theory is included. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Laures-Wilson/klw2 Splittings of bicommutative Hopf algebras Nitu Kitchloo, Gerd Laures and W. Stephen Wilson Department of Mathematics Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA Mathematisches Institut der Universit\"at Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany wsw---math.jhu.edu, nitu---math.jhu.edu, gerd---laures.de Abstract: We use the theory of Dieudonne modules to show that certain types of short exact sequences of Hopf algebras split. Several examples occur naturally with Morava K-theory. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Salvatore/config Title: Configuration spaces on the sphere and higher loop spaces Author: Paolo Salvatore AMS classification numbers: 55P48, 55R80, 55S12 xxx number: math.AT/0303290 Address: Dipartimento di Matematica, Universita` di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: salvator---mat.uniroma2.it Abstract: We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/semifree Title: Bordism of semi-free $S^1$-actions. Author: Dev P. Sinha AMS Class: 57R85 (primary); 55R40 (secondary). LANL ID: math.AT/0303100 Addresses: Department of Mathematics, University of Oregon, Eugene OR Email: dps---math.uoregon.edu Abstract: We calculate the geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. To calculate the geometric theory, we prove a case of the geometric realization conjecture, which in general would determine the geometric theory in terms of the homotopical. The determination of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry. --------------------------------------------- 4 new papers this time, from BrownR, DavisD, Dwyer, and Gottlieb. Mark Hovey New papers appearing on hopf between 4/09/03 and 5/13/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/noncommut-at Title: Towards non commutative algebraic topology Author: Ronald Brown AMS Classification numbers: 55D15, 55U40, 18D35 Address of Author: Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK. Email address of Author: r.brown---bangor.ac.uk Text of Abstract: These are the transparencies (slightly edited) for a seminar at University College, London, on May 7, 2003. They give a quick overview of some background and some directions taken for algebraic methods for higher dimensional, non commutative, local to global problems, including some algebraic models of homotopy types. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/E7E8 Representation types and 2-primary homotopy groups of certain compact Lie groups Donald M. Davis 55Q52, 55T15, 57T20 Department of Mathematics Lehigh University Bethlehem, PA 18015 dmd1---lehigh.edu Abstract: Bousfield has shown how the 2-primary v1-periodic homotopy groups of certain compact Lie groups can be obtained from their representation ring with its decomposition into types and its exterior power operations. He has formulated a Technical Condition which must be satisfied in order that he can prove his description is valid. We prove that a simply-connected compact simple Lie group satisfies his Technical Condition if and only if it is not E6 or Spin(4k+2) with k not a 2-power. We then use his description to give an explicit determination of the 2-primary v1-periodic homotopy groups of E7 and E8. This completes a program, suggested to the author by Mimura in 1989, of computing the v1-periodic homotopy groups of all compact simple Lie groups at all primes. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer/local Localization W. G. Dwyer This is a largely expository paper, which describes the concept of localization, as it usually comes up in topology, and gives some examples of it. The examples include local homology and cohomology, homological localizations of spaces and spectra, and localization with respect to a map f. For appropriate choices of the map f, this last gives constructions related to the Goodwillie calculus and to motivic homotopy theory. There's also a proof that if a localization functor exists, the higher order categorical invariants associated to inverting the local equivalences are trivial. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gottlieb/eigbndl EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE Daniel Henry Gottlieb Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions with five examples. One of those examples gives rise to a 4 by 4 matrix representation of the Complex Quaternions. This representation shows the relationship of the Biquaternions with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity Theory. The eigenstructure of this representation is very interesting, and our choice of notation produces important mathematical expressions found in those fields and in Quantum Mechanics. In particular, we show that the Doppler shift factor is analogous to Berry's Phase. ----------------------------------------------------- These are mostly papers that just made it out of Clarence's e-mail. The new policy at Hopf is that e-mail submssions are strongly deprecated. Please use the web form if at all possible. There is a significant and unpredictable delay associated with e-mail submission and it is easier for papers to get misplaced. 11 new papers this time, from Chernov-Rudyak, Dugger (3), Dwyer-Wilkerson, Ibanez-Rudyak-Tralle, Ibanez-Rudyak-Tralle-Ugarte, Oliver, Oprea-Rudyak, Rudyak, and Wilkerson. Mark Hovey New papers appearing on hopf between 5/13/03 and 5/17/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chernov-Rudyak/wavefronts Title: Affine Linking Numbers and Causality Relations for Wave Fronts Authors: Vladimir Chernov (Tshernov), Yuli Rudyak Addresses: V. Chernov, Department of Mathematics, 6199 Bradley Hall, Dartmouth College, Hanover NH 03755, USA Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105 U.S.A e-mail: rudyak---math.ufl.edu e-mail: Vladimir.Chernov---dartmouth.edu 4 figures (eps files) Abstract: Let M be an oriented manifold. We study the causal relations between the wave fronts W and W' that originated at some points of M. We introduce a numerical topological invariant CRI(W, W') (the so-called causality relation invariant) that, in particular, gives the algebraic number of times the wave front W passed through the point that was the W' before the front W' originated. This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of M outside of a part V of M such that W and W' are null-homotopic in V. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/krspecDD Title: An Atiyah-Hirzebruch spectral sequence for KR-theory Author: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: We construct a spectral sequence for KR-theory which is analagous to the spectral sequence relating motivic cohomlogy to algebraic K-theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult1DD Title: Multiplicative structures on spectral sequences I Authors: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: This is mostly an expository paper, recording basic facts about towers of homotopy fiber sequences. We show that a pairing of towers induces an associated pairing of spectral sequences, for towers of spaces and towers of spectra. In the hope that this might eventually be a useful reference for people, feel free to send me suggestions for things that should be improved (with the understanding that it might be a while before I get around to implementing them). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult2DD Title: Multiplicative structures on spectral sequences II Authors: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: This paper summarizes the constructions of pairings for some of the standard spectral sequences in algebraic topology. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/NT Title: Normalizers of Tori Authors: W. G. Dwyer AND C. W. Wilkerson, Notre Dame and Purdue Suppose that G is a connected compact Lie group and that T G is a maximal torus, or in other words a maximal connected abelian subgroup. The normalizer NT of T lies in a short exact sequence (1.1) 1 -> T -> NT -> W -> 1 in which W is a finite group called the Weyl group of G. In this pa- per we reformulate some ideas of Tits [27 ] in order to describe exactly which groups appear as such an NT . This leads to an analogous deter- mination of which groups appear as the normalizer NT~ of a maximal 2-discrete torus in a connected 2-compact group (1.16). In the compact Lie group case, NT determines G up to isomorphism [3], and so in listing the possible NT 's we are giving an alternative approach to the classification of connected compact Lie groups them- selves. In contrast, it is not known that the normalizer of a maximal 2-discrete torus in a connected 2-compact group X determines X up to equivalence. However, this seems likely to be true [23 ] [19 ], and we hope that the results of this paper will eventually contribute to a classification of connected 2-compact groups. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle/aspherical Title: On the fundamental groups of symplectically aspherical manifolds Authors: R. Ibanez, Yu. Rudyak, A. Tralle Adresses of Authors: R. Ibanez, Departamento de Matematicas, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain Yu. Rudyak, Department of Mathematics, Universoty of Florida, 358 Little Hall, Gainesville, FL 32601, USA A. Tralle, Department of Mathematics, University of Warmia and Mazura, 10561 Olsztyn, Poland email: mtpibtor---lg.ehu.es rudyak---math.ufl.edu tralle---matman.uwm.edu.pl In this paper we are interested in the fundamental groups of closed symplectically aspherical manifolds; i.e., of symplectic manifolds whose symplectic form vanishes on 2-dimensional spherical homology classes. Motivated by some results of Gompf, we consider two classes of fundamental groups of symplectically aspherical manifolds: with trivial and-non-trivial second homotopy group. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes. Also, we notice that there are some dimensional phenomena in the realization problem. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle-Ugarte/HomotopySymplecticKahler Title: On certain geometric and homotopy properties of closed symplectic manifolds Authors: R. Ibanez, Yu. Rudyak, A. Tralle, L. Ugarte Adresses of Authors: R. Ibanez, Departamento de Matematicas, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain Yu. Rudyak, Department of Mathematics, Universoty of Florida, 358 Little Hall, Gainesville, FL 32601, USA A. Tralle, Department of Mathematics, University of Warmia and Mazura, 10561 Olsztyn, Poland L. Ugarte, Departamento de Matem\'aticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain email: mtpibtor---lg.ehu.es rudyak---math.ufl.edu tralle---matman.uwm.edu.pl ugarte\---posta.unizar.es The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver/limz Equivalences of classifying spaces completed at the prime two Bob Oliver We prove here the Martino-Priddy conjecture for the prime $2$: the $2$-completions of the classifying spaces of two groups $G$ and $G'$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $2$-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group $G$, the second higher derived functor of the inverse limit vanishes for a certain functor $\calz_G$ on the $2$-subgroup orbit category of $G$. The proof of this result uses the classification theorem for finite simple groups. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Oprea-Rudyak/cat3man Title: Detecting Elements and Lusternik--Schnirelmann Category of 3-Manifolds Authors: John Oprea, Yuli Rudyak Addresses: John Oprea, Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115 U.S.A Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105 U.S.A e-mail: rudyak---math.ufl.edu Abstract: In this paper, we give a new simplified calculation of the Lusternik-Schnirelmann category of closed 3-manifolds. We also describe when 3-manifolds have detecting elements and prove that 3-manifolds satisfy the equality of the Ganea conjecture. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Rudyak/PLstructures (This is an updated version of a paper already on the archive) Title: Piecewise linear structures on topological manifolds Author: Yuli Rudyak Address: Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105, USA e-mail: rudyak---math.ufl.edu Abstract: This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2,3) and the Hauptvermutung for manifolds. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/e8-lab Lab Notes on the exceptional Lie group $E_8$ at the prime $2$ \author[C. W. Wilkerson]{Clarence W. Wilkerson, Jr.} \dedicatory{Dedicated to Morton L. Curtis (1921-1989).} \address{Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395} \thanks{Thanks to the National Science Foundation, Purdue University, Johns Hopkins University, and Fukuoka University for financial support during this research and the 2000 sabbatical of the author. Thanks to the Clay Foundation for travel support during this research.} \email{wilker---math.purdue.edu} This is an account of the author's use of computer algebra tools to explore the structure of the maximal elementary abelian $2$-subgroups of the exceptional Lie group $E_8$. The principal result obtained thus far by these methods is that any rank $8$ connected $2$-compact group $(BX,X)$ with Weyl group isomorphic to that of the exceptional Lie group $E_8$ has its normalizer of the maximal torus isomorphic to that of $E_8$ at the prime $2$. Similar results hold for the comparison of possible exotic forms of $G_2$, $DI(4)$, $F_4$, and $E_7/\Center(E_7)$ to the standard forms.\\ Corollaries of this result include that the Krull dimension of the mod $2$ cohomology of such $BX$ is $9$ and that the cohomology ring is not Cohen-Macaulay. \\ ----------------------------------------------- I had to do some hand editing of abstracts this time, so remember to include author's name and title of paper with the abstract, at the least. It is also suggested that you include author's e-mail and AMS subject classifications. 7 new papers this time, from Andersen-Bauer-Grodal-Pedersen, Baas-Dundas-Rognes, Granja, Grojnowski (this is his old paper about equivariant elliptic cohomology), Hornbostel, Kuhn, and Morava. Mark Hovey New papers appearing on hopf between 5/17/03 and 6/18/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Andersen-Bauer-Grodal-Pedersen/loopnotlie Title: A finite loop space not rationally equivalent to a compact Lie group Authors: Kasper K. S. Andersen, Tilman Bauer, Jesper Grodal, Erik K. Pedersen Subj-class: Algebraic Topology; Geometric Topology MSC-class: 55P35; 55P15, 55R35 Comments: 8 pages, arXiv : math.AT/0306234 We construct a connected finite loop space of rank $66$ and dimension $1254$ whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than $66$ is in fact rationally equivalent to a compact Lie group, extending the classical known bound of $5$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Baas-Dundas-Rognes/segal60 Title of Paper: Two-vector bundles and forms of elliptic cohomology Authors: Nils A. Baas, Bjorn I. Dundas and John Rognes Addresses of Authors: Department of Mathematical Sciences The Norwegian University of Science and Technology NO-7491 Trondheim Norway Department of Mathematical Sciences The Norwegian University of Science and Technology NO-7491 Trondheim Norway Department of Mathematics University of Oslo NO-0316 Oslo Norway Email address of Authors: baas---math.ntnu.no, dundas---math.ntnu.no and rognes---math.uio.no In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Granja/notehpn Title: Self maps of HP^n via the unstable Adams spectral sequence Authors: Gustavo Granja AMS Classification numbers: 55S35,55S36,55S37 Address of Author: Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais 1049-001 Lisboa Portugal Email address of Author: ggranja---math.ist.utl.pt Abstract: We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology is proved in two new cases. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Grojnowski/deloc Delocalized equivariant elliptic cohomology by Ian Grojnowski This is an old paper, which has been circulating quietly for almost a decade. It contains a definition of an equivariant elliptic cohomology theory for compact connected Lie groups and reasonable topological spaces. The theory is defined over Q, i.e. neglects torsion completely, and yet was still interesting. This is because of the well known heuristic identifying elliptic cohomology with something like the K-theory of the loop space. The functor of "loops into" is not local---there is no Mayer-Vietoris style patching. Yet elliptic cohomology has such a property. However the equivariant elliptic cohomology defined here does not satisfy such a naive locality propery. Instead, the elliptic cohomology of a space is a non-trivial bundle on the canonical abelian variety associated to the group. The crudest invariant of such a bundle is its first Chern class. This is a combinatorial shadow of the failure of locality. These same obstruction invariants occur in the study of semi-infinite D-modules on the infinitesimal neighbourhood of formal loops in the loop space of an algebraic variety; just as one would expect. -- Since this paper was written there have been several developments. Rosu and Ando used this theory to give a new proof of Witten rigidity, and Greenlees constructed a model for that part of rational equivariant S^1 homotopy that is seen by an elliptic cohomology theory. (There has also been the extraordinary work of Hopkins et al on tmf). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel/Motchrom Chromatic motivic homotopy theory by Jens Hornbostel We construct a motivic version of the chromatic filtration and the chromatic spectral sequence. This should be used to study the stable ${\bf A}^1$-homotopy groups of the motivic sphere spectrum. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Kuhn Title: Localization of Andre--Quillen--Goodwillie towers, and the periodic homology of infinite loopspaces Author: Nicholas J. Kuhn AMS classification numbers: 55P43, 55P47, 55N20, 18G55 Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 email: njk4x---virginia.edu abstract: Let K(n) be the nth Morava K--theory at a prime p. This paper is a thorough study of questions like the following: to what extent does the K(n)--localization, or the K(n)--homology, of a spectrum X determine the K(n)--homology of its 0th space X_0? Our methods combine techniques from modern homotopical algebra with chromatic homotopy. In particular, we use the telescopic functors of Bousfield and the author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith), as well as Topological Andre--Quillen Homology and Goodwillie calculus in nonconnective settings. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/SegalMS Heisenberg groups and algebraic topology by Jack Morava This paper overlaps considerably with earlier, sketchier papers about the Tate cohomology of circle actions and its connection to Heisenberg groups. It will appear in the Segal Festschrift: We study the Madsen-Tillmann spectrum $\C P^\infty_{-1}$ as a quotient of the Mahowald pro-object $\C P^{\infty}_{-\infty}$, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten. ------------------------------------------------ 6 new papers this time, from Baas-Dundas-Rognes (an update), Richter, Sinha, Strickland, and (Jim) Turner (2), Mark Hovey New papers appearing on hopf between 6/18/03 and 7/11/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Baas-Dundas-Rognes/segal60 Title of Paper: Two-vector bundles and forms of elliptic cohomology Authors: Nils A. Baas, Bjorn I. Dundas and John Rognes Email address of Authors: baas---math.ntnu.no, dundas---math.ntnu.no and rognes---math.uio.no In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology. (This is an new version of a paper previously on Hopf). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Richter/Richter-Lambda-EHP Title: Lambda algebra unstable composition products and the Lambda EHP sequence Author: William Richter AMS Classification numbers: 55T15, 55Q40, 55Q25 Address: Math Department, Northwestern University, Evanston IL 6020 Email: richter---math.nwu.edu Abstract: Simple combinatorial proofs are given of Lambda algebra results, mostly due to Priddy & the 6 authors, but also the ``Adams filtration better'' unstable Lambda products of Wang, Mahowald and Singer: Lambda^{s,t}(n) --- Lambda(n+t ) ---> Lambda(n) which imply the folklore Lambda EHP sequence Lambda(n) >-E--> Lambda(n+1) -H-->> Lambda(2n+1) The 6 authors proved Lambda(n) is a chain complex, but not that H is a chain map. A careful reader could deduce a proof from the papers of Wang, Mahowald and Singer, but Singer, who best stated the formulas, gave no proofs. New results: combinatorial proofs of the Lambda admissible monomial basis; the differential d is well-defined. The paper should be accessible to geometers interested in forthcoming applications with Mahowald on 3-cell Poincare complexes. Perhaps the Lambda algebra is undergoing a Renaissance, as 2 young people, Mark Behrens and Mizuho Hikida are doing interesting new work in it. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/compactify Title: Manifold theoretic compactifications of configuration spaces. Author: Dev P. Sinha AMS Class: 55R80; 32J05 LANL ID: math.GT/0306385 Addresses: Departments of Mathematics, University of Oregoni, Eugene, OR 97403 Email: dps---math.uoregon.edu Abstract: We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Strickland/axsurv Axiomatic stable homotopy--a survey by N. P. Strickland We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We focus mainly on representability theorems, localisation, Bousfield classes, and nilpotence. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/finite On simplicial commutative algebras with finite Andre-Quillen homology by James M. Turner L. Avramov, following D. Quillen, posed a conjecture to the effect that if $R \to A$ is a homomorphism of Noetherian rings then the Andr\'e-Quillen homology on the category of A-modules satisfies: $D_{s}(A|R;-) = 0$ for $s\gg 0$ implies $D_{s}(A|R;-) = 0$ for s>2. In an earlier paper, the author posed an extended version of this conjecture which considered A to be a simplicial commutative R-algebra with Noetherian homotopy such that the characteristic of $\pi_{0}A$ is non-zero. In addition, a homotopy characterization of such algebras was described. The main goal of this paper is to develop a strategy for establishing this extended conjecture and provide a complete proof when R is Cohen-Macaulay of characteristic 2. Note: this paper replaces "Nilpotency in the homotopy of simplicial commutative algebras". 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/Gorenstein Characterizing Simplicial Commutative Algebras with Vanishing Andr'e-Quillen Homology by James M. Turner The use of homological and homotopical devices, such as Tor and Andr\'e-Quillen homology, have found substantial use in characterizing commutative algebras. The primary category setting has been differentially graded algebras and modules, but recently simplicial categories have also proved to be useful settings. In this paper, we take this point of view up a notch by extending some recent uses of homological algebra in characterizing Noetherian commutative algebras to characterizing simplicial commutative algebras having finite Noetherian homotopy through the use of simplicial homotopy theory. These characterizations involve extending the notions of locally complete intersections and locally Gorenstein algebras to the simplicial homotopy setting. --------------------------------------------------------- 4 new papers this time, from Bartels-Reich, Goodwillie (Calc III!), Gorbounov-Malikov, and Kuhn. Mark Hovey New papers appearing on hopf between 7/11/03 and 8/21/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Reich/isoIIhopf Title: On the Farrell-Jones Conjecture for higher algebraic K-theory Authors: Arthur Bartels, Holger Reich e-mail adresses: bartelsa---math.uni-muenster.de, reichh---math.uni-muenster.de arxiv: math.AT/0308030 Abstract: We prove the Farrell-Jones Isomorphism Conjecture about the algebraic K-theory of a group ring RG in the case where the group G is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring R is an arbitrary associative ring with unit and the result applies to all dimensions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Goodwillie/calculus3 Title: Calculus III, Taylor series Author: Thomas G. Goodwillie Author's e-mail address: tomg---math.brown.edu Abstract: We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gorbounov-Malikov/LG-CY-try Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence Vassily Gorbounov and Fyodor Malikov We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Tate Title: Tate cohomology and periodic localization of polynomial functors Author: Nicholas J. Kuhn AMS classification numbers: Primary 55P65; Secondary 55N22, 55P60, 55P91 Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 email: njk4x---virginia.edu abstract: In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. ------------------------------------------------ 12 new papers this time, from Bendersky-DavisD-Mahowald, Dugger-Isaksen, Jessup-Lupton, KrauseH, Lupton, Lupton-SmithSB (2 papers), Nofech, Pengelley-Williams, Pitsch-Scherer, Toen-Vezzosi, and ZhouXueguang. Mark Hovey New papers appearing on hopf between 8/21/03 and 9/26/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD-Mahowald/sgd2 Stable geometric dimension of vector bundles over even-dimensional real projective spaces Martin Bendersky, Donald M. Davis, and Mark Mahowald mbenders---shiva.hunter.cuny.edu dmd1---lehigh.edu mark---math.northwestern.edu Abstract In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order 2^e over RP^{2n} if e > 74 and n is sufficiently large. In this paper, we use the Bendersky-Davis computation of v1-periodic homotopy groups of SO(m) to determine this geometric dimension for all values of e (still provided that n is sufficiently large). The same formula that worked for e>74 works for e>5, but for e \le 5 the geometric dimension is often different due to anomalies in the v1-periodic homotopy groups of SO(m) when m<11. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/hopfDI The Hopf condition for bilinear forms over arbitrary fields Daniel Dugger (ddugger---math.uoregon.edu) Daniel C. Isaksen (isaksen---math.wayne.edu) We settle an old question about the existence of certain "sums-of-squares" formulas over a field F. A classical result, due originally to Hopf and proven via topological methods, says that if such a formula exists over a field of characteristic 0 then certain binomial coefficients must be even. We use motivic methods to prove that the result also holds for fields of characteristic p. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Jessup-Lupton/JessLup Title: Free Torus Actions and Two-Stage Spaces Author(s): Barry Jessup, Gregory Lupton Author's e-mail address: Bjessup---sciences.uottawa.ca, G.Lupton---csuohio.edu AMS classification number: 55P62, 57S99 Other useful information: math.AT/0309434. To appear, Math. Proc. Camb, Philos. Soc. Abstract: We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/KrauseH/quotient Title: Cohomological quotients and smashing localizations Author: Henning Krause Email: henning---maths.leeds.ac.uk Abstract: The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton/Catell Title: The Rational Toomer Invariant and Certain Elliptic Spaces Author(s): Gregory Lupton Author's e-mail address: G.Lupton---csuohio.edu AMS classification number: Primary 55P62, 55M30; Secondary 55T10 Other useful information: math.AT/0309392. Contemporary Mathematics, Vol. 316 (2002), 135--146 Abstract: We give an explicit formula for the rational category of an elliptic space whose minimal model has a homogeneous-length differential. We also show that for such a space, there are no gaps in the sequence of integers realized as the rational Toomer invariant of some cohomology class. With an additional hypothesis, we show a result from which we deduce the relation dim(H^*(X;Q)) >= 2 cat_0(X). 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/Cyclic Title: Cyclic Maps in Rational Homotopy Theory Author(s): Gregory Lupton, Samuel Bruce Smith Author's e-mail address: G.Lupton---csuohio.edu, smith---sju.edu AMS classification number: 55P62, 55Q05 Other useful information: math.AT/0309423 Abstract: The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/Gseq Title: Rationalized Evaluation Subgroups of a Map and the Rationalized G-Sequence Author(s): Gregory Lupton, Samuel Bruce Smith Author's e-mail address: G.Lupton---csuohio.edu, smith---sju.edu AMS classification number: 55P62, 55D23 Other useful information: math.AT/0309432 Abstract: Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also allows for the identification of several long exact sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, we obtain an identification of the rationalization of the so-called G-sequence of the map f. This is a sequence---in general not exact---of groups and homomorphisms that includes the Gottlieb groups of X and the evaluation subgroups of f. We use these results to study the G-sequence in the context of rational homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence and negative derivations of rational cohomology. We also relate the splitting of the rational G-sequence of a fibre inclusion to a well-known conjecture in rational homotopy theory. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Nofech/e2 An $E^2$-type closed model category for bisimplicial groups Alexander Nofech anofech---shaw.ca A closed model category structure is defined on the category of bisimplicial groups in which the weak equivalences are isomorphisms on bigraded homotopy groups $\pi_{k,l}$ and at the same time isomorphisms on the $E^2$ term of the Quillen spectral sequence. There is an analogue of the spiral exact sequence of Dwyer-Kan-Stover. One of the reasons for looking specifically at groups rather than at a general construction of a $E^2$-type model category is that it is easier to find the abelianization of a cofibrant group. This structure is considered as a convenient setting for a study of the relation between bigraded homotopy and hyperhomology. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/subsmalg Global Structure of the mod 2 Symmetric Algebra over the Steenrod algebra. David J. Pengelley (davidp---nmsu.edu) Frank Williams (frank---nmsu.edu) The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., a minimal set of generators and a minimal set of relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^n - 1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the mod 2 Dickson algebras, and we speculate about possible related topological realizability. Our methods also produce a related simple A-module presentation of the cohomology of infinite-dimensional real projective space, with a filtration having well-known filtered quotients. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Pitsch-Scherer/completion Title: Homology fibrations and group completion revisited Authors : Jerome SCHERER and Wolfgang PITSCH e-mail : jscherer---mat.uab.es and Wolfgang.Pitsch---math.unige.ch AMS classification : Primary 55U10; Secondary 19D06 arXiv : math.AT/0307339 Abstract : We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Toen-Vezzosi/bravenew Title: ``Brave New'' Algebraic Geometry and global derived moduli spaces of ring spectra Authors: Bertrand Toen, Gabriele Vezzosi Author's e-mail address: toen---picard.ups-tlse.fr ; vezzosi---dm.unibo.it Other useful information: arXive submission numbermath.AT\0309145 Abstract: We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, "brave new algebraic geometry". We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures as geometric S-stacks. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhouxin title of the paper: A reply author: Zhou Xueguang AMS classification numbers: Q55 Address of author:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb---eyou.com Abstract: In this paper, we answer the question why V(n) exists for all non-negative integers $n$. ----------------------------------------------------------------------- 10 new papers this time, from Dugger-Isaksen, Flores, Gaudens, Kitchloo-Wilson, Klein, LinJP, Luo, Nam, Sauvageot, and Schwede. Mark Hovey New papers appearing on hopf between 9/26/03 and 10/24/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/motcell Title: Motivic cell structures Authors: Daniel Dugger and Daniel C. Isaksen Authors' e-mail address: ddugger---math.uoregon.edu and isaksen---math.wayne.edu Abstract: An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores/draft1 NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS by RAM'ON J. FLORES Departamento de Matem'aticas, Universidad Aut'onoma de Barcelona, E-08193 Bellaterra, Spain E-mail address: ramonj---mat.uab.es Mathematical subject classification: 55P20, 55P80. Abstract. In this note we discuss the effect of the BZ/p-nullification and the BZ/p-cellularization functors over classifying spaces of finite groups, and we compare them with the corresponding ones with regard to Moore spaces, that have been intensively studied in the last years. We describe the BZ/p- nullification of BG by means of a Postnikov fibration, and we classify all finite groups G for which BG is BZ/p-cellular. In particular, we relate the effect these (co)localizations have over the fundamental group with the analogous functors in the category of groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gaudens/bocksteinnul Title: A remark on N. Kuhn's unbounded strong realization conjecture Author(s): Gerald Gaudens Author's e-mail address: gaudens---math.univ-nantes.fr AMS classification number: 55S10; 57S35 Abstract: N. Kuhn has given several conjectures on the special features satisfied by the singular cohomology of topological spaces with coefficients in a finite prime field, as modules over the Steenrod algebra. The so-called Realization conjecture was solved in special cases By N. Kuhn and in complete generality by L. Schwartz. The more general Strong realization conjecture has been settled at the prime 2, as a consequence of the work of L. Schwartz, and the subsequent work of F.-X. Dehon and the author. In this note, we are interested in the even more general Unbounded strong realization conjecture. We shall prove that it holds at the prime $2$ for the class of spaces whose cohomology has a trivial Bockstein action in high degrees. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/kitchloo-wilson Title: On fibrations related to real spectra Authors: Nitu Kitchloo and W. Stephen Wilson E-mail addresses: nitu---math.jhu.edu, wsw---math.jhu.edu Address: Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 Abstract: We consider real spectra, collections of Z/(2)-spaces indexed over Z direct sum Z_\alpha with compatibility conditions. We produce fibrations connecting the homotopy fixed points and the spaces in these spectra. We also evaluate the map which is the analogue of the forgetful functor from complex to reals composed with complexification. Our first fibration is used to connect the real 2^{n+2}(2^n-1)-periodic Johnson-Wilson spectrum ER(n) to the usual 2(2^n-1)-periodic Johnson-Wilson spectrum, E(n). Our main result is the fibration \Sigma^{\lambda(n)} ER(n) -> ER(n) -> E(n), where \lambda(n) = 2^{2n+1}-2^{n+2}+1. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Klein/embclass Title: On embeddings in the sphere Author: John R. Klein Author's e-mail address: klein---math.wayne.edu Abstract: We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification such embeddings in a wide range. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/LinJP/Lin=HspaceAnalog1 (This abstract was sent in dvi form; the program we use to convert is not perfect). H-spaces analogous to E8 mod 3 Dedicated to the memory of Masahiro Sugawara James P. Lin Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112, U.S.A. email:jimlin---euclid.ucsd.edu Abstract: Let p be an odd prime. Let X0 be a finite, p-local, simply connected homotopy associative H-space. Suppose H* (X0; Zp) contains the subalgebra Zp [x0,_z0]_xp p(r0, P1 r0, Pp P1 r0, y0) 0, z0 satisfying z0 = Pp x0 = Q0Pp P1 r0, Pp P1 r0 = P1 y0 for r0 2 H3 (X0; Zp). The only known examples occur for p = 3 and involve the Lie group E8. In this note we prove that if X0 exists, then p must be 3. Thus there are no homotopy associative H-space analogues of E8mod 3 for primes bigger than 3. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Luo/pre (This is an updated version) Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids Zhi-ming Luo We prove that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Nam/transfer Title : Transfert alg'ebrique et repr'esentation modulaire du groupe lin'eaire Author : Tran Ngoc Nam Author's e-mail address : trngnam---hotmail.com Author's mailing address : LAGA, Universit'e Paris 13, 93430 Villetaneuse, France Abstract : On se propose de d'eterminer la dimension d'une repr'esentation du groupe lin'eaire d'efinie par un sous-espace vectoriel de l'alg`ebre `a puissances divis'ees, d'expliciter l'image du transfert alg'ebrique en degr'e g'en'erique et celle du transfert alg'ebrique quadruple, d'identifier les ind'ecomposables de degr'e pair de l'alg`ebre polynomiale `a 4 variables, vue comme module sur l'alg`ebre de Steenrod mod 2. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/thesis STABILISATION DES COMPLEXES CROISES Orin R. Sauvageot orin.sauvageot---epfl.ch Ecole Polytechnique Federale de Lausanne Institute of Mathematics This is my PhD thesis in FRENCH, 158 pages. The graphic files C-tensor-I.eps, pi-delta-4.eps and pi-xc.eps are included in the zip archives thesis-print.dvi.zip and thesis-screen.dvi.zip. (Note from Mark; you should get these eps files individually if you get the dvi file. The file thesis.dvi is thesis-print.dvi; thesis-screen.dvi is in case you have trouble viewing the diagrams in thesis.dvi on your screen. The files thesis.ps and thesis.pdf already have the eps files embedded) Abstract In this doctoral thesis we present a stabilization of the category of crossed complexes. Our work is motivated by the difficulty one has in performing algebraic calculations in Boardman's stable homotopy category, since products and actions are defined only up to homotopy in the underlying category of spectra, as defined by Bousfield and Friedlander. To correct this lack of precision, a number of new models of the stable homotopy category have been developed in which algebraic constructions are exactly defined. One such model is the category of symmetric spectra on simplicial sets, the manipulation of which is still not easy, however. The idea behind this thesis is to stabilize the category of crossed complexes, as it is an interesting approximation to the category of simplicial sets, reflecting certain, though not all, nonabelian homotopical information concerning simplicial sets. We have stabilized it according to the procedure codified in Hovey's "Spectra and symmetric spectra in general model categories". Stabilization requires that the category of crossed complexes satisfies certain properties. We have succeeded in proving these properties, in each case establishing a previously unknown result. For example, we have shown that it is cofibrantly generated and that it is a symmetric monoidal model category. Furthermore we have verified that it is a proper, cellular category. In proving the properness we have answered an open question posed by Brown and Golasinski. In the course of establishing these properties we have established a nonabelian version of the 5-Lemma. A crossed complex is a generalization of a chain complex of abelian groups. We have shown, however, that the stabilization of crossed complexes is homotopy equivalent to that of the category of chain complexes. On the other hand, the situation of unpointed crossed complexes is different, and it is very likely that their stabilization is not that of chain complexes. In order to argue so, we have constructed an innovative simplicial model of the Hopf map. It remains then to give a topological meaning to an unpointed stabilization. An attempt of answer is sketched. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Schwede/Morita Title: Morita theory in abelian, derived and stable model categories Author: Stefan Schwede e-mail address: sschwede---math.uni-muenster.de This is a survey paper, based on lectures given at the Workshop on "Structured ring spectra and their applications" which took place January 21-25, 2002, at the University of Glasgow. The term `Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two `rings' have `equivalent' module categories ? We discuss this question in different contexts and illustrate it by examples: (Classical) When are the module categories of two rings equivalent as categories ? (Derived) When are the derived categories of two rings equivalent as triangulated categories ? (Homotopical) When are the module categories of two ring spectra Quillen equivalent as model categories ? There is always a related question, which is in a sense more general: What characterizes the category of modules over a `ring' ? The answer is, mutatis mutandis, always the same: modules over a `ring' are characterized by the existence of a `small generator', which plays the role of the free module of rank one. The precise meaning of `small generator' depends on the context, be it an abelian category, a derived category or a stable model category. --------------------------------------- 12 new papers this time, from Aouina-Klein, Chalupnik (3), Fausk-Oliver, Grandis (2), Knudson-Walker, Notbohm-Ray, Sauvageot, Troesch, and ZhouXueguang. Mark Hovey New papers appearing on hopf between 10/24/03 and 11/25/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aouina-Klein/config_stable Title: On the homotopy invariance of configuration spaces Author(s): Mokhtar Aouina and John R. Klein Author's e-mail address: aouina---math.wayne.edu, klein---math.wayne.edu AMS classification number: Primary 55R80; Secondary 57Q35, 55R70. Abstract: For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of embedding theory and fiberwise algebraic topology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/cohsdr Title of Paper: Schur_De-Rham complex and its cohomology Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We associate to a Young diagram a complex of strict polynomial functors which we call the Schur-De-Rham complex. Its cohomology turns out to reflect deep combinatorial properties of a diagram. We show that if a ground field is of characteristic p, the Schur-De-Rham complex is acyclic when the p-core of a diagram is nontrivial. We also compute its cohomology for a diagram with a trivial p-core and p-quotient consisting of a single diagram. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extpol Title of Paper: Extensions of strict polynomial functors Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We compute Ext-groups between various strict polynomial functors important in representation theory (eg. between twisted Weyl and Schur functors). Our method utilizes: computation of the Ext-groups between twisted divided and symmetric powers due to Franjou-Friedlander-Scorichenko-Suslin, resolutions of functors by divided and symmetric powers, interplay between functors and representations of the symmetric group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extws Title of Paper: Extensions of Weyl and Schur functors Author: Marcin Chalupnik Abstract: We use here the Schur-De-Rham complex to extend calculations of the Ext-groups between twisted Weyl and Schur functors initiated in the paper ``Extensions of strict polynomial functors''. The main result is a full calculation of those groups in the case of a pair of diagrams which can be obtained from diagrams of the same weights by the operation F described in ``Schur-De-Rham complex and its cohomology''. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Oliver/piperfect Title: Continuity of p-perfection for compact Lie groups Authors: Halvard Fausk and Bob Oliver Author's e-mail address: fausk---math.uio.no and bob---math.univ-paris13.fr AMS classification number: 55P91 Abstract: Let G be a compact Lie group, and let pi be any prime or set of primes. We construct a ``pi-perfection map'': a continuous function from the space of conjugacy classes of all closed subgroups of G to the space of conjugacy classes of pi-perfect subgroups with finite index in their normalizer. We use this to show that the idempotent elements of the Burnside ring of G localized at pi are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of pi-perfect subgroups of G with finite index in their normalizer. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Bsy2 (Note: this paper is only available in pdf format) Normed combinatorial homology and noncommutative tori Marco Grandis Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups. Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis---dima.unige.it http://www.dima.unige.it/~grandis/ Notes: Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p. Abstract. Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of "noncommutative topology", agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets associated with "irrational" rotations have thus the same classification up to isomorphism as the well-known irrational rotation C*-algebras. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Dht1 (Note: this paper is only available in pdf format) Directed homotopy theory, I. The fundamental category Marco Grandis Key words: homotopy theory, homotopical algebra, directed homotopy, fundamental category. Dipartimento di Matematica Universita di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis---dima.unige.it http://www.dima.unige.it/~grandis/ Notes: to appear in: Cahiers Topologie Geometrie Differentielle Categoriques Preprint: Dip. Mat. Univ. Genova, Preprint 443 (2001), 26 p. Revised version: 5 Nov 2001. Abstract. Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for the foundations of such a theory, a d-space, is a topological space equipped with a family of directed paths, closed under some operations. This allows for directed homotopies, generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the fundamental category of a d-space is introduced and a 'Seifert-van Kampen' theorem proved; its homotopy invariance rests on directed homotopy of categories. In the process, new shapes appear, for d-spaces but also for small categories, their elementary algebraic model. Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Knudson-Walker/hom11-19 Title: Homology of linear groups via cycles in BG x X Author1: Kevin P. Knudson Author2: Mark E. Walker email1: knudson---math.msstate.edu email2: mwalker---math.unl.edu Abstract: Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups H_i(X,G) by considering cycles in the simplicial scheme BG x X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups H_i(\Delta^j,G), which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups H_i and the Z/2-equivariant cohomology of the classifying space of the discrete group G(R). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm-Ray/djthrational On Davis-Januszkiewicz Homotopy Types I; Formality and Rationalisation by Dietrich Notbohm} and Nigel Ray For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of $K$. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space $X$ is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of $X$ is formal as a differential graded noncommutative algebra. We then specialise to the rationals, by proving the corresponding property for Sullivan's {\it commutative\/} cochain algebra; this confirms that the rationalisation of $X$ is unique. In a sequel, we will consider the uniqueness of $X$ at each prime separately, and apply Sullivan's arithmetic square to produce global results in special families of cases. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/simpl-hopf-model A simplicial model for the Hopf map Orin R. Sauvageot Ecole Polytechnique Federale de Lausanne orin.sauvageot---epfl.ch We give an explicit simplicial model for the Hopf map S^3 -> S^2. For this purpose, we construct a model of S^3 as a principal twisted cartesian product K x_{eta} S^2, where K is a simplicial model for S^1 acting by left multiplication on itself, S^2 is given the simplest simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We construct a Kan complex for the simplicial model K of S^1. The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Troesch/troesch_Resolution_of_symmetric_powers Title: A propos d'une question de Friedlander et Suslin I -- Une r'esolution injective des puissances sym'etriques twist'ees (in French) Author: Alain Troesch Address of Author: Institut de Mathematiques de Jussieu, Case 82 4 place Jussieu, F-75252 PARIS CEDEX 05 e-mail address: troesch---math.jussieu.fr Abstract. Some years ago, Friedlander and Suslin constructed an explicit injective resolution of twisted symmetric powers in the category of strict polynomial functors over a ground field of characteristic 2. The factors in this resolutions are given by direct sums of tensor products of (non twisted) symmetric powers. The case of a symmetric power twisted only once is a well-known result: it is some kind of Koszul complex. In characteritic p>2, nothing similar was known up to now, even for a single twist. In this paper, we construct such injective resolutions. The resolutions we construct are in fact "p-resolutions", that is, the differential does not vanish when composed twice, but only when composed p times. This result should unable us to constuct an injective resolution of any twisted functor if we know an injective resolution of the corresponding non twisted functor. This will be the subject of another paper. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhou2 title of the paper: The answer to an email of Mr. Douglas C. Ravenel author: Zhou Xueguang Address of author:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb---eyou.com Abstract: In this paper, we answer the question why V(n) exists for all non-negative integers n. ----------------- --------------------------------- Happy New Year! This is the beginning of the 10th year I have been doing this. 10 new papers this time, from Bubenik, ChornyB (2), Gillespie, Hovey, Jardine, Lueck, Mitchell, Pengelley-Williams, and Vavpatic-Viruel. Mark Hovey New papers appearing on hopf between 11/25/03 and 1/05/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik/fsi Title: Free and semi-inert cell attachments Author: Peter Bubenik Author's e-mail address: peter.bubenik---epfl.ch AMS classification number: 55P35 (Primary) 16E45 (Secondary) arXive submission number: math.AT/0312387 Abstract: Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously studied inert condition. Under these conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an $R$-algebra respectively. Under a further condition we show that $H_*(\Omega Y;R)$ is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/Localization1 Title: Localization with respect to a class of maps I - Equivariant localization of diagrams of spaces Author: Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces. This model category is not cofibrantly generated. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the classes of maps which induce ordinary localizations on the generalized fixed-points sets. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/Localization2 Title: Localization with respect to a class of maps II - Equivariant cellularization and its application Author: Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie/sheafproblem Title: The flat model structure on Ch(O) Author: James Gillespie Email: jrg21---psu.edu Abstract: Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category O-MOD of O-modules. I.e., we have a natural way to define the functors Ext and Tor in O-MOD. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/morava-E-operations Operations and co-operations in Morava $E$-theory Mark Hovey Wesleyan University mhovey---wesleyan.edu In this paper, we revisit the calculations of the operations and co-operations in Morava E-theory. Recall that the co-operations are the continuous functions from a profinite group G that is a version of the Morava stabilizer group to E_*. The operations are the completed twisted group ring E_*[[G]]. These results have appeared in the literature before. The advantage of this paper is that it is self-contained, works out all the details that are usually skipped over, and uses a new approach, not directly dependent on Morava's Annals paper on comodules, that the author finds fairly simple and elegant. Most of all, though, the author wrote this paper because he was unable to understand the proofs in the literature. He hopes it will be useful for people in the same unhappy situation. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/cat5 Categorical homotopy theory J.F. Jardine This paper is an exposition and extension of the ideas and methods of Cisinksi, set at the level of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The model structures for the category of simplicial presheaves and all of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves with an equivalent homotopy category. The theory specializes, for example, to the homotopy theories of cubical sets, cubical presheaves, and gives a cubical model for motivic homotopy theory. The applications of Cisinski's ideas are explained in some detail for cubical sets. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_classifyingspaces1203 Title: Survey on Classifying Spaces for Families of Subgroups Author: Wolfgang Lueck AMS Classification numbers: 55R35, 57S99, 20F65, 18G99 Address: Mathematisches Institut der Westfaelischen Wilhelms Universitaet Einsteinstr. 62 48149 Muenster Germany Abstract: We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW-version E_F(G) and the numerable G-space version J_F(G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if G is totally disconnected and F is the family of compact subgroups or of compact open subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra, for the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Mitchell/sw Author: Stephen A. Mitchell Title: Stiefel-Whitney classes, united K-theory and real embeddings of number rings e-mail: mitchell---math.washington.edu We study the relations among the Stiefel-Whitney classes associated to the real embeddings of a number ring. Our results depend on a computation of the real and self-conjugate K-theory of the algebraic K-theory spectrum of the number ring. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/subsmalg Global Structure of the mod 2 Symmetric Algebra over the Steenrod algebra. David J. Pengelley (davidp---nmsu.edu) Frank Williams (frank---nmsu.edu) The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., a minimal set of generators and a minimal set of relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^n - 1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the mod 2 Dickson algebras, and we speculate about possible related topological realizability. Our methods also produce a related simple A-module presentation of the cohomology of infinite-dimensional real projective space, with a filtration having well-known filtered quotients. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Vavpetic-Viruel/PU On the mod p cohomology of BPU(p) Ales Vavpetic Fakulteta za matematiko in fiziko Univerza v Ljubljani Jadranska 19 SI-1111 Ljubljana Slovenia ales.vavpetic---fmf.uni-lj.si Antonio Viruel Dpto de Algebra, Geometria y Topologia Universidad de Malaga Apdo correos 59 E29080 Malaga Spain viruel---agt.cie.uma.es AMS Classification numbers: 55R35, 55P15 ABSTRACT: We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of PU(p). Finally, we proof that the classifying space of the projective unitary group PU(p) is determined by its mod p cohomology as an unstable algebra over the Steenrod algebra for p>3, completing previous works of Dwyer, Miller, Wilkerson at prime 2 and Broto, Viruel at prime 3. ---------------- 10 new papers this time, from BrownR-Kamps-Porter, Chalupnik, ChornyB, Clarke-Crossley-Whitehouse, GrayB, Hikida, Hornbostel, IsaksenD, Lupton-SmithSB, and Porter. Mark Hovey New papers appearing on hopf between 1/5/04 and 2/1/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Kamps-Porter/vkt7 TITLE: A van Kampen theorem for the homotopy double groupoid of a Hausdorff space AUTHORS: R. Brown, K.H. Kamps, T.Porter EMAILS: r.brown---bangor.ac.uk, heiner.kamps---fernuni-hagen.de, t.porter---bangor.ac.uk ADDRESSES: Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. Fachbereich Mathematik, FernUniversit\"at in Hagen, D-58084 Hagen, Germany Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. ABSTRACT: We show that the homotopy double groupoid of a Hausdorff space defined by the authors in a previous paper satisfies a version of the van Kampen theorem, and so is a suitable tool for non abelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. There is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/koeks Title of Paper: Koszul duality and extensions of exponential functors Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals for various functors and apply these results to the problem of calculating Ext--groups between exponential functors. The main application is a full description of the Ext--groups between twisted exterior and divided powers and between twisted symmetric and divided powers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/prospaces Title: A generalization of Quillen's small object argument Author(s): Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with the appearance of several important examples of model categories which were proven to be non-cofibrantly generated. Our current approach allows for the construction of functorial factorizations and localizations in the equivariant model category on diagrams of spaces and in two different model structures on the category of pro-spaces. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Clarke-Crossley-Whitehouse/ccwops Title: Algebras of operations in K-theory Authors: Francis Clarke, Martin Crossley, Sarah Whitehouse Authors' e-mail addresses: F.Clarke---Swansea.ac.uk, M.D.Crossley---Swansea.ac.uk, S.Whitehouse---sheffield.ac.uk Abstract: We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/decomposition On Decompositions in Homotopy Theory Brayton Gray 55P35, 5P30, 55P45 Dept. of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 brayton---uic.edu We first describe Krull-Schmidt theorems decomposing $H$ spaces and simply connected co-$H$ spaces into atomic factors in the category of pointed nilpotent $p$-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic $H$ spaces and homotopy types of atomic co-$H$ spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hikida/Acycle Title: Some acyclic relations in the lambda algebra Author: Mizuho Hikida Author's e-mail address: hikida---bus.hiroshima-pu.ac.jp Author's mailing address: Hiroshima Prefectural University, Shobara-shi, 727-0017, Japan AMS Classification numbers: 55Q40 Abstract: We consider the relations omega gamma = 0 in Lambda, and show that if omega alpha = 0 then alpha = gamma beta for some beta. These relations give the acyclic chain complex Lambda -gamma-> Lambda -omega-> Lambda . We consider various cases, e.g. omega = lambda_n and gamma = lambda_{2n+1}. Especially, we consider the case omega = w_n = d lambda_n for n=2^{e+r} + 2^{e}-1, where gamma = (h_{e+r})^r. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel/Motchrom2 Author : Jens Hornbostel Author's e-mail address : jens.hornbostel---mathematik.uni-regensburg.de Author's mailing address: Universitaet Regensburg, NWF I - Mathematik, D- 93047 Regensburg, Germany Abstract: We construct a motivic version of the chromatic filtration and the chromatic spectral sequence. This should be used to study the stable ${\bf A}^1$-homotopy groups of the motivic sphere spectrum. We also study different localization techniques both for classical and motivic spectra. This is an improved version of a preprint posted in June 2003. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/flasque Title: Flasque model structures for simplicial presheaves Author: Daniel C. Isaksen Author's e-mail address: isaksen---math.wayne.edu Abstract: By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/GseqII Title: Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps Authors: Gregory Lupton and Samuel Bruce Smith Authors' e-mail addresses: G.Lupton---csuohio.edu and smith---sju.edu Authors' mailing addresses: Department of Mathematics, Cleveland State University, 2121 Euclid Ave., Cleveland OH 44115 and Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131 AMS classification number: 55P62, 55Q52 Other useful information: 33 pages; http://arXiv.org/abs/math.AT/0401178 Abstract: Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation subgroups of a map f: X -> Y in this setting. As applications, we consider a generalization of a question of Gottlieb, within the context of rational homotopy theory. We also identify the rationalization of the G-sequence of f and make explicit computations of the homology of this sequence. In a separate result of independent interest, we give an explicit Quillen minimal model of a product AxX, in the case in which A is a rational co-H-space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Porter/s-catsv2 Title: S-categories, S-groupoids, Segal categories and quasicategories Author: Timothy Porter Author's e-mail address: t.porter---bangor.ac.uk Author's mailing address: Mathematics Department, School of Informatics, University of Wales Bangor, Bangor, Gwynedd, LL57 1UT, United Kingdom. Included ps or eps files: 5 epsi files AMS classification number: 55U35 Other useful information: arXive submission number: math.AT/0401274 Abstract: These notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguna, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions;(ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas;(iii) to introduce Joyal's quasicategories, and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and Porter; and finally to ask lots of questions of myself and of the reader. ----------------15 new papers this month, from Aguilar-Prieto (2), Arkowitz-Brown, Arkowitz-Stanley-Strom, Arkowitz-Strom, Ausoni, DJGreen, Elmendorf-Mandell, Hovey, Jardine-Luo, Marzantowicz-Prieto, McClure-SmithJH (2), and SchwartzL (2). Mark Hovey New papers appearing on hopf between 2/1/04 and 3/1/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aguilar-Prieto/transrami-1 Transfers for ramified coverings in homology and cohomology Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Transfer, ramified covering maps, classifying spaces Marcelo A. Aguilar and Carlos Prieto Abstract Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Aguilar-Prieto/transrami-2 A classification of cohomology transfers for ramified coverings Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Transfer, covering maps, ramified covering maps, classifying spaces Marcelo A. Aguilar and Carlos Prieto cprieto---math.unam.mx, marcelo---math.unam.mx Abstract We construct a cohomology transfer for $n$-fold ramified covering maps. Then, we define a very general concept of transfer for ramified covering maps and prove a classification theorem for these transfers. This generalizes Roush's classification of transfers for $n$-fold ordinary covering maps. We characterize those representable cofunctors which admit a family of transfers for ramified covering maps that have two naturality properties, as well as normalization and stability. This is analogous to Roush's characterization theorem for the case of ordinary covering maps. Finally, we classify these families of transfers and construct some examples. In particular, we extend the determinant function in $\GL(k,\C)$ to a transfer. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Brown/Lef The Lefschetz-Hopf Theorem and Axioms for the Lefschetz Number Martin Arkowitz and Robert F. Brown martin.arkowitz---dartmouth.edu rfb---math.ucla.edu 55M20 The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X --->Y and g:Y --->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X --->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X --->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X --->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Stanley-Strom/Cl&Cat The Cone Length and Category of Maps: Pushouts, Products and Fibrations Martin Arkowitz, Donald Stanley and Jeffrey Strom martin.arkowitz---dartmouth.edu stanley---math.uregina.ca Jeffrey.Strom---wmich.edu 55M30; 55P99, 55R05 For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: l_A(f), the A-cone length of f, and L_A(f), the A-category of f. When A is the collection of all spaces, these are the cone length and category of f, respectively, both of which have been studied previously. The following results have been obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for I_A and L_A of the induced map of homotopy pushouts in terms of I_A and L_A of the other maps. This has many applications including an inequality for I_A and L_A of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for I_A and L_A of the product of two maps in terms of I_A and L_A of the given maps and the A-cone length of their domains. (3) We study our invariants in a pullback square and obtain as a consequence an upper bound for the A-cone length and A-category of the total space of a fibration in terms of the A-cone length and A-category of the base and fiber. We conclude with several remarks, examples and open questions. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/Secat The Sectional Category of a Map Martin Arkowitz and Jeffrey Strom martin.arkowitz---dartmouth.edu Jeffrey.Strom---wmich.edu 55M30; 55P99 We study a generalization of the Svarc genus of a fiber map. For an arbitrary collection E of spaces and a map f:X--->Y, we define a numerical invariant, the E-sectional category of f, in terms of open covers of Y. We obtain several basic properties of E-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple properties which characterize the E-sectional category. In the final section we obtain inequalities for the E-sectional category of a composition and inequalities relating the E-sectional category to the Fadell-Husseini category of a map and the Clapp-Puppe category of a map. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ausoni/thhku-ausoni Author: Christian Ausoni Title: Topological Hochschild Homology of connective complex K-theory Email: ausoni---math.uni-bonn.de Abstract: Let ku be the connective complex K-theory spectrum, completed at an odd prime p. We present a computation of the mod (p,v_1) homotopy algebra of the topological Hochschild homology spectrum of ku. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/DJGreen/essCM Title: The essential ideal is a Cohen-Macaulay module Author: David J. Green Institution: University of Wuppertal, Germany MSC 2000: Primary 20J06; Secondary 13C14 arXiv: math.GR/0402434 Abstract: Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/RMA Title: Rings, modules, and algebras in infinite loop space theory Authors: Anthony D. Elmendorf and Michael A. Mandell Email: aelmendo---math.purdue.edu Email2: mandell---math.uchicago.edu Abstract: We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in Smith's category of symmetric spectra, with an intermediate stop at a new category that may be of interest in its own right, whose objects we call symmetric functors. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/morava-E-SS Some spectral sequences in Morava E-theory by Mark Hovey mhovey---wesleyan.edu The Morava E-theory of X is the homotopy of the K(n)-localization of E smash X, where E is the completed and extended version of E(n) on which the Morava stabilizer group acts. Because K(n)-localization is not smashing, Morava E-theory is not a homology theory; it is exact, but does not preserve coproducts. Nevertheless, it is the most important theory to use in understanding the K(n)-local stable homotopy category; for example, X is small in the K(n)-local stable homotopy category if and only if the Morava E-theory of X is degreewise finite. In the paper at hand, we show how the usual spectral sequences used with homology theories work for Morava E-theory. The most interesting such spectral sequence is a spectral sequence that converges to the Morava E-theory of an infinite coproduct. The E_2-term involves the derived functors of direct sum in the category of "L-complete" E_*-modules. There are (n-1) such derived functors (n if we try to compute filtered homotopy colimits). Thus, Morava E-theory is "n derived functors away from being a homology theory". In particular, when n=1, we see that p-completed K-theory actually commutes with coproducts, in the category of Ext-p-complete abelian groups. It follows that K(1)-local homotopy also commutes with coproducts as a functor to Ext-p-complete abelian groups. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine-Luo/cocycles6 Title: Higher order principal bundles Authors: J.F. Jardine and Z. Luo AMS Classification numbers: 14F05, 55R65, 14A20 E-mail: jardine---uwo.ca E-mail: zluo---uwo.ca Abstract: We define torsors for sheaves of simplicial groups and sheaves of groupoids enriched in simplicial sets, and give classification results for these torsors in terms of the homotopy theory of simplicial sheaves. The proofs of the classification results use a new, general approach to cocycles taking values in simplicial sheaves. We prove a homotopy classification result for gerbes locally isomorphic to a fixed sheaf of groups. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Marzantowicz-Prieto/decompAMS Computation of the equivariant $1$-stem by a decomposition of equivariant stable homotopy classes W. Marzantowicz and C. Prieto marzan---main.amu.edu.pl, cprieto---math.unam.mx Primary 54H25; Secondary 55M20, 55M25, 55N91 Equivariant stable homotopy groups, equivariant stems, equivariant fixed point index and fixed point transfer Abstract For any compact Lie group $G$, we give a decomposition of the group $\{X,Y\}_G^k$ of (unpointed) stable $G$-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the $G$-homotopy classes in terms of the generalized fixed point transfer and making use of conormal maps. Finally, we give a full computation of the first equivariant (stable) stem for $G$, $\pi\ho{G\,\rm{st}}_1=\{*,*\}_G\ho{-1}$. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClure-Smith_survey Operads and cosimplicial objects: an introduction. James E. McClure and Jeffrey H. Smith AMS Classification Numbers 18D50; 55P48 Submitted to arXiv: math.QA/0402117 mcclure---math.purdue.edu jhs---math.purdue.edu This paper is an introduction to a series of papers in which we have given combinatorial models for certain important operads (including A-infinity and E-infinity operads, the little n-cubes operads, and the framed little disks operad) and combinatorial conditions for them to act on a given space or chain complex. The paper does not assume any prior knowledge of operads---Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/mcclure-smith3 Cosimplicial Objects and little $n$-cubes. I. James E. McClure and Jeffrey H. Smith AMS Classification Numbers 18D50; 55P48 Submitted to arXiv: math.QA/0211368 mcclure---math.purdue.edu jhs---math.purdue.edu This is a revised version of a paper previously posted on Hopf. The main theorem says that if a cosimplicial space has a certain kind of combinatorial structure (called a $\Xi^n$-structure) then its total space has an action of an operad $\cal D_n$ which is weakly equivalent to the little $n$-cubes operad. There are three new sections in the revised version: Section 10 shows that $\Xi^2$-structures are essentially the same thing as operads with multiplication, Section 11 shows that the operad $\cal D_n$ acts on $n$-fold loop spaces, and Section 15 shows that the main results are still valid for the homotopy-invariant version of Tot. 14. http://hopf.math.purdue.edu/cgi-bin/generate?/SchwartzL/Erd L'alg`ebre de Steenrod, modules injectifs, et foncteurs polynomiaux Lionel Schwartz These notes come from talks made in Nantes in December 2001 at a session " Etat de la Recherche " of the french mathematic society. They are an introduction to the algebraic aspects of the theory of unstable modules over the Steenrod algebra and to the relations of the related category to functor categories. The Steenrod algebra is introduced using the additive group scheme. Reduced injective modules are described follwing the point of view of Campbell and Selick. Most of the material is classical, however there are new (at least in an accessible form) remarks concerning the odd prime case, as well as some new proofs of classical results, in particular the structure of Miller's algebra. The Adem relations are discussed following Bullett and MacDonald. 15. http://hopf.math.purdue.edu/cgi-bin/generate?/SchwartzL/Grot Sur l'anneau de Grothendieck de la cat'egorie des modules instables Lionel Schwartz 19 octobre 2003 R'esum'e Dans cet article on calcule l'anneau de Grothendieck de la cat'egorie des modules instables de type fini et de la cat'egorie obtenue par quotient par la sous-cat'egorie des modules instables nilpotents. Les r'esultats principaux montrent que la s'erie de Poincar'e, ou un substitut ad'equat d'eterminent ces groupes. On peut de plus caract'eriser les s'eries repr'esentant un module instable. Ce type de sujet a d'ej`a 'et'e abord'e par N. Kuhn dans [K2] d'un point de vue de th'eorie des repr'esentations, on retrouve ses r'esultats au long du d'eveloppement. ----------------------------------------- 4 new papers this month, from Badzioch-Chung-Voronov, Broto-Castellana-Grodal-Levi-Oliver, Christensen-Isaksen, and KrauseH. Mark Hovey New papers appearing on hopf between 3/1/04 and 4/5/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Badzioch-Chung-Voronov/bcv Title: Yet another delooping machine Authors: Bernard Badzioch, Kuerak Chung, and Alexander A. Voronov Author's e-mail address: voronov---math.umn.edu Authors' mailing address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455 Included ps or eps files: mor.eps AMS classification number: 55P48 (Primary); 18C10 (Secondary) ArXiv submission number: math.AT/0403098 Abstract: We suggest a new delooping machine, which is based on recognizing an n-fold loop space by a collection of operations acting on it, like the traditional delooping machines of Stasheff, May, Boardman-Vogt, Segal, and Bousfield. Unlike in the traditional delooping machines, which carefully select a nice space of such operations, we consider all natural operations on n-fold loop spaces, resulting in the algebraic theory Map (V_. S^n, V_. S^n). The advantage of this new approach is that the delooping machine is universal in a certain sense, the proof of the recognition principle is more conceptual, works the same way for all values of n, and does not need the test space to be connected. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo1 Subgroup families controlling $p$-local finite groups by C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ``conjugacy'' relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system $F$ over a finite $p$-group $S$ is saturated can be determined by just looking at smaller classes of subgroups of $S$. We also prove that the homotopy type of the classifying space of a given $p$-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of $F$-centric $F$-radical subgroups (at a minimum) to the set of $F$-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to $p$-constrained finite groups, and prove that they in fact all arise from groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Isaksen/duality Duality and Pro-spectra J. Daniel Christensen and Daniel C. Isaksen jdc---uwo.ca isaksen---math.wayne.edu Keywords: Spectrum, pro-spectrum, Spanier-Whitehead duality, closed model category, colocalization Arxiv: math.AT/0403451 MSC-class: 55P42 (Primary); 55P25, 18G55, 55U35, 55Q55 (Secondary) Abstract: Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/KrauseH/stable Title: The stable derived category of a noetherian scheme Author: Henning Krause E-mail: hkrause---math.upb.de Abstract: For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal Cohen-Macaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. -------------------------4 new papers this month, from Bousfield, Castellana-Crespo-Scherer, IsaksenD, and Kitchloo-Morava. Mark Hovey New papers appearing on hopf between 4/5/04 and 5/4/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/vper On the 2-primary v1-periodic homotopy groups of spaces A.K. Bousfield bous---uic.edu AMS Classification Numbers: 55Q51(Primary),55N15,55P60,55S25,57T20 We develop foundations of a general approach for calculating p-primary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomolgies with particular attention to the case p = 2. As a main application, we derive a method for calculating v1-periodic homotopy groups of simply-connected compact Lie groups using their complex, real, and quaternionic representation theories. This method has been applied very effectively by D.M. Davis in recent work. We rely heavily on the p-primary v1-stabilization functor Phi from spaces to spectra. Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space X from the p-adic KO-cohomology of Phi X, which we obtain from the p-adic KO-cohomology and K-cohomology of X by a v1-stabilization process under suitable conditions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/DeconstructH Title: Deconstructing Hopf spaces Authors: Natalia Castellana, Juan A. Crespo, Jerome Scherer email: natalia---mat.uab.es, JuanAlfonso.Crespo---uab.es, jscherer---mat.uab.es AMS classification number: 55P45; 55S10; 55P60; 55P47; 55S45 Abstract: We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We ``deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting and give a criterion to recognize connected covers of H-spaces with finite mod p cohomology. The key observation is that the module of indecomposables lies in some stage of the Krull filtration of the category of unstable modules over the Steenrod algebra. We compare this algebraic condition with a topological one, namely that some iterated loop space of X is BZ/p-local. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/completion Author: Daniel C. Isaksen Author's e-mail address: isaksen---math.wayne.edu Author's mailing address: Department of Mathematics \\ Wayne State University \\ Detroit, MI 48202 Included ps or eps files: None AMS classification number: 55P60, 55N10 (Primary); 18G55, 55U35 (Secondary) Abstract: For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Morava/Thomprospectra2 THOM PROSPECTRA FOR LOOP GROUP REPRESENTATIONS NITU KITCHLOO, JACK MORAVA We construct an S^1-equivariant prospectrum that models the Atiyah dual of a free loop space of a manifold. By applying a suitably completed S^1-equivariant K-theory to the Atiyah dual, we show how to recover the Witten genus of a manifold. The main technical tool is a Tits building for the loop group. We use the Tits building to construct a dualizing spectrum for the loop group and relate it to work of Freed, Hopkins and Teleman. ------------------2 new papers this month, from Biss-Farb and Flores. Mark Hovey New papers appearing on hopf between 5/4/04 and 6/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biss-Farb/kg Title: K_g is not finitely generated Authors: Daniel Biss and Benson Farb Author's email addresses: daniel---math.uchicago.edu, farb---math.uchicago.edu Included files: curve1.eps and curve3.eps Abstract: We prove that for any genus g>1, the subgroup K_g of the mapping class group of a closed genus g surface generated by Dehn twists about separating curves is not finitely generated. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores/draft2 Nullification functors and the homotopy type of the classifying space for proper bundles Ram'on J. Flores Departamento de Matem'aticas, Universidad Aut'onoma de Barcelona, E-08193 Bellaterra, Spain E-mail address: ramonj---mat.uab.es Abstract. Let G be a discrete group. In this note we build a bridge between the homotopy theory of BG and the theory of proper G-actions, by showing that under mild restrictions, the classifying space for proper G-bundles has the homotopy type of the W-nullification of BG for some space W. This allows us to use properties of the localization functors to obtain spaces that are homotopy equivalent to this "proper" classifying space for a wide range of groups, and on the other hand, we take profit of the existence of well-known geometrical and finite-dimensional models of it for some infinite groups to deduce homotopical information about the p-primary part of their classifying spaces. -----------------10 new papers this month, from Bergner, Broto-Moller, Bruner-Rognes, Galvez-Whitehouse, Intermont-Strom, Jardine, Rezk, and YauD (3 papers). Mark Hovey New papers appearing on hopf between 6/2/04 and 7/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/SimplicialCategoryMC Title: A model category structure on the category of simplicial categories Author: Julia E. Bergner Author's e-mail address: jbergner---nd.edu AMS Classification: 18G55, 18D20 arXiv submission number: math.AT/0406507 Author's address: Department of Mathematics University of Notre Dame Notre Dame, IN 46556 Abstract: In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Moller/Chev Title: Finite Chevalley versions of p-compact groups Authors: Carles Broto, Jesper M. Moller Author's e-mail address: broto---mat.uab.es moller---math.ku.dk Address: Carles Broto Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain Address: Jesper M. Moller Matematisk Institut Universitetsparken 5 DK-2100 Copenhagen Denmark AMS class: 55R35, 55P15, 55P10 Abstract: We describe the spaces of homotopy fixed points of unstable Adams operations acting on p-compact groups and also of unstable Adams operations twisted with a finite order automorphism of the p-compact group. We obtain new exotic p-local finite groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bruner-Rognes/bruner Title: Leibniz Formulas for Cyclic Homotopy Fixed Point Spectra Authors: Robert R. Bruner and John Rognes MSC-class: 19D55, 55P43, 55P91, 55S12, 55T05. ArXiv ID: math.AT/0406081 Addresses: Robert R. Bruner Department of Mathematics Wayne State University Detroit, Michigan 48067 USA rrb---math.wayne.edu John Rognes Department of Mathematics University of Oslo Box 1053, Blindern NO-0316 Oslo Norway rognes---math.uio.no Abstract: We analyze the homotopy fixed point spectrum of a circle-equivariant commutative S-algebra R in homological terms. There is a homological homotopy fixed point spectral sequence that converges conditionally to the continuous homology of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups of the circle, and for the Tate- and homotopy orbit spectra. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Galvez-Whitehouse/centres Title: Infinite Sums of Adams Operations and Cobordism Authors: Imma Galvez, Sarah Whitehouse E-mail: i.galvezicarrillo---londonmet.ac.uk, s.whitehouse---sheffield.ac.uk Addresses: Computing, Communications Technology and Mathematics, London Metropolitan University, Holloway Road, London N7 8DB, UK. Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK. Included ps or eps files: centrediag1.ps, centrediag2.ps AMS classification number: Primary: 55S25; Secondary: 55N22, 19L41. Abstract: In recent work by Clarke, Crossley and the second author, various algebras of stable degree zero operations in p-local K-theory were described explicitly. The elements are certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for p-local cobordism and for BP, thus identifying the "Adams subalgebra" of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Intermont-Strom/GoodSp Complexity and Good Spaces M. Intermont (Kalamazoo College) and J. Strom (Western Michigan University) intermon---kzoo.edu jeffrey.strom---wmich.edu This paper is an exploration of two ideas in the study of closed classes: the A-complexity of a space X and the notion of good spaces (spaces A for which C(A) = \overline{C(A)}). A variety of formulae for the computation of complexity are given, along with some calculations. Good spaces are characterized in terms of the functors CW_A and P_A. The main result is a countable upper bound for the complexity with respect to the suspension of A when A is a good space. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/stack-coh6 Title: Fibred sites and stack cohomology Author: J.F. Jardine AMS Classification numbers: 55P42, 18F20, 14A20 J.F. Jardine Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada E-mail: jardine---uwo.ca The usual notion of a site fibred over a stack is expanded to a definition of a site C/A fibred over a presheaf of categories A. Presheaves of simplicial sets on the site fibred over a presheaf of categories A are contravariant enriched diagrams defined on A, taking values in simplicial sets. The standard model structure for presheaves of simplicial sets induces a coarse equivariant structure for enriched contravariant A-diagrams. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site C/G is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on C/G. In particular, stack cohomology can be calculated on the fibred site for a representing presheaf of groupoids. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Rezk/rezk-units-and-logs Title: The units of a ring spectrum and a logarithmic cohomology operation Author: Charles Rezk Authors e-mail address: rezk---math.uiuc.edu Abstract: We construct a ``logarithmic'' cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E^0(K) of a space K. We obtain a formula for this map in terms of the action of Hecke operators on Morava E-theory. Our formula is closely related to that for an Euler factor of the Hecke L-function of an automorphic form. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/cohom Cohomology of $\lambda$-rings Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu A cohomology theory for $\lambda$-rings is developed. This is then applied to study deformations of $\lambda$-rings. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/lambda-rev2 On $\lambda$-ring structures over Z[[x]] Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu It is shown that the $\lambda$-ring structure over the power series ring Z[[x]] given by the $K$-theory of $CP^\infty$ is uniquely determined by the following condition: \psi^p(x) = px mod{x^2} for each prime $p$, where $\psi^p$ is the Adams operation. Applications to algebraic topology and formal group laws are given. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/truncated Extensions of filtered $\lambda$-ring structures over the dual number ring Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu We study problems related to the existence and uniqueness of filtered $\lambda$-ring structures over the truncated polynomial ring Z[x]/(x^3) that extend a given filtered $\lambda$-ring structure over Z[x]/(x^2). ---------------5 new papers this month, from Basterra-Mandell, BrownR, Diaz-Ruiz-Viruel, Dugger-Isaksen, and Kuhn. Mark Hovey New papers appearing on hopf between 7/2/04 and 8/7/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Basterra-Mandell/Basterra-Mandell-stability Title: Homology and Cohomology of E-infinity Ring Spectra Authors: Maria Basterra Department of Mathematics, University of New Hampshire, Durham, NH basterra---math.unh.edu Michael A. Mandell Department of Mathematics, University of Chicago, Chicago, IL mandell---math.uchicago.edu AMS Subject class: Primary 55P43; Secondary 55P48, 55U35 Abstract: We show that every homology or cohomology theory on a category of E-infinity ring spectra is Topological Andr'e-Quillen homology or cohomology with appropriate coefficients. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/IMA-talk Author: Ronald Brown Author's e-mail address: r.brown---bangor.ac.uk Author's web page: http://www.bangor.ac.uk/~mas010 Author's mailing address: Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom AMS classification number: 18D05,18D15,18G50,55P15,55U40,57M07 arXive submission number: math.AT/0407275 Abstract: We sketch the background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker and Rafael Sivera (Valencia). The aim is to give in one place a full account of work by R. Brown and P.J. Higgins and others since the 1970s which defined and applied crossed complexes and cubical higher homotopy groupoids to local-to-global problems and homotopy classification of maps. This yields a distinctive account of that part of algebraic topology which lies between homology theory and homotopy theory, in which the fundamental group and its actions plays an essential role, and which allows for nonabelian calculations in dimension 2. This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on `$n$-categories: foundations and applications', June 7-18, 2004, organised by John Baez and Peter May. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Diaz-Ruiz-Viruel/drv Title: All p-local finite groups of rank two for odd prime p Authors: Antonio Diaz, Albert Ruiz, Antonio Viruel Author's e-mail address: adiaz---agt.cie.uma.es, Albert.Ruiz---uab.es, viruel---agt.cie.uma.es ArXive submission number: math.AT/0407324 Abstract: In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group and the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/ksumDI Algebraic K-theory and sums-of-squares formulas Daniel Dugger and Daniel C. Isaksen Email: ddugger---math.uoregon.edu and isaksen---math.wayne.edu Addresses: Daniel Dugger Department of Mathematics University of Oregon Eugene, OR 97403 Daniel C. Isaksen Department of Mathematics Wayne State University Detroit, MI 48202 Abstract: We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to produce obstruction conditions when the field is the real numbers. Our result applies to arbitrary fields not of characteristic 2, making use of algebraic K-theory in place of topological K-theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/homologyiso Title: Mapping spaces and homology isomorphisms Author: Nicholas J. Kuhn AMS classification numbers: 55P35, 55N20, 55P42 arXiv no.: math.AT/0407146 address: University of Virginia, Charlottesville, VA USA email: njk4x---virginia.edu Abstract: Let Map(K,X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E_* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X) will send an E_*--isomorphism in either variable to a map that is monic in E_* homology. Interesting examples arise by letting E_* be K--theory, K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces. ---------------4 new papers this month, from Devinatz, Dugger, IsaksenD, and Sinha. Mark Hovey New papers appearing on hopf between 8/7/04 and 9/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/recog Title: Recognizing Hopf algebroids defined by a group action Author: Ethan Devinatz e-mail: devinatz---math.washington.edu Abstract: Let A be a complete noetherian regular local ring, and suppose that S is a profinite group acting continuously on A via ring homomorphisms. Let T be the algebra of continuous functions from S to A. Then (A,T) has a canonical structure of a complete Hopf algebroid, determined by the action of S on A. We give necessary and sufficient conditions for a general Hopf algebroid to be of this form. Applications to Morava theory are also discussed. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/milnor Title: Notes on the Milnor conjectures Author: Daniel Dugger email: ddugger---math.uoregon.edu Abstract: These are some expository notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/gencohlgy Title: Generalized cohomology of pro-spectra Author: Daniel C. Isaksen E-mail: isaksen---math.wayne.edu AMS classification: 55T25, 55P42, 55U35, 55N20, 18G55 (Primary), 19L99 (Secondary) Abstract: We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak equivalences are also detected by cohomology as in the classical Whitehead theorem for spectra. We establish an Atiyah-Hirzebruch spectral sequence in this context, which makes possible the computation of topological K-theory (and other generalized cohomology theories) of pro-spectra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/opknot Title: Operads and knot spaces Author: Dev Sinha E-mail: dps---math.uoregon.edu Abstract: Let F_m be the space of knotted intervals in I^m equipped with a trivialization through immersions. We show that the totalization of the Kontsevich operad provides a model for the embedding calculus tower for F_m. Combined with results of Goodwillie-Klein-Weiss and Volic, this resolves Kontsevich's conjecture of existence of such a model which captures the homotopy type of F_m when m>3 and which classifies finite-type framed knot invariants when m=3. We carefully develop the Kontsevich operad, which is closely related to the Fulton-MacPherson operad and weakly equivalent to the little cubes operad. In doing so we show that the standard simplicial model for the two-sphere carries an operad structure in the opposite category of pointed sets. We apply the well-developed machinery of McClure and Smith on operads with multiplication to deduce that our model has a little two-cubes action. (Note: if you want the dvi file to contain the figures, you need to download the directory Figures as well. The pdf file already has the figures built in.) ----------------As you have heard from Clarence, the Hopf Archive is now a virtual server on the Purdue Math Department's server. This means there is no more ftp access to Hopf, only web access. Also, because of this changeover, my October message was a bit delayed. 3 new papers this month, from Blanc, Devinatz, and Kuhn. Mark Hovey New papers appearing on hopf between 9/2/04 and 10/15/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/mod03 Moduli spaces of homotopy theory by David Blanc Dept. of Mathematics, Univ. of Haifa, 31905 Haifa, Israel E-mail address: blanc---math.haifa.ac.il The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970's; and general homotopy types, in the work of Dwyer-Kan and their collaborators. We here explain the two approaches, and show how they may be related to each other. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/homotopydev Title: Homotopy groups of homotopy fixed point spectra associated to E_n Author: Ethan Devinatz e-mail: devinatz---math.washington.edu Abstract: We compute the mod(p) homotopy groups of the continuous homotopy H_2 fixed points of E_2 for p>2, where E_n is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space of lifts of height n formal group laws, and H_n is the semi-direct product of the group of diagonal matrices in the nth Morava stabilizer group with an appropriate Galois group. We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K(n)_*-local spectra. We also indicate a plan for generalizing this computation to n>2. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Kinosaki Title: Goodwillie towers and chromatic homotopy: an overview Author: Nicholas J. Kuhn Email:njk4x at virginia.edu Address: University of Virginia, Charlottesville, VA 22904 arXive no: math.AT/0410342 Abstract: This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone--Mahowald. --------------- 6 new papers this month, from Boardman-Wilson, Goerss-Henn-Mahowald-Rezk, Lewis-Mandell, McClure, Turiel, and Wodarz. Mark Hovey New papers appearing on hopf between 10/15/04 and 11/04/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Boardman-Wilson/BWonPn Title: k(n)-torsion-free H-spaces and P(n)-cohomology Authors: J. Michael Boardman, W. Stephen Wilson E-mail: boardman---math.jhu.edu, wsw---math.jhu.edu Address: Dept. of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore MD 21218-2686 AMS Classifications: Primary 55N22, 55P45 Abstract: In his thesis, the second author split the H-space that represents Brown-Peterson cohomology BP^k(-) into indecomposable factors, which have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)-cohomology P(n)^k(-) in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting (i-1)-connected H-spaces Y_i have free connective Morava K-homology k(n)_*(Y_i), and may be built from the spaces in the Omega-spectrum for k(n) using only v_n-torsion invariants. We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)_*-module P(n)^*(X) is generated by elements of P(n)^i(X) for i>=0. This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava K-theory. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Henn-Mahowald-Rezk/ghmr Title: A resolution of the K(2)-local sphere at the prime 3 Authors: Paul Goerss, Hans-Werner Henn, Mark Mahowald and Charles Rezk Northwestern University, Universit\'e Louis Pasteur et CNRS, Northwestern University, University of Illinois Urbana, IL 61801 (This is an updated version) ABSTRACT We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)}S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^{hF} where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Lewis-Mandell/Lewis-Mandell-UCT Equivariant Universal Coefficient and Kunneth Spectral Sequences L. Gaunce Lewis, Jr. Department of Mathematics Syracuse University Syracuse, NY 13244-1150 lglewis---syr.edu Michael A. Mandell DPMMS, University of Cambridge Wilberforce Road Cambridge CB3 0WB UK M.A.Mandell---dpmms.cam.ac.uk AMS Classification: Primary 55N91; Secondary 55P43,55U20,55U25} Abstract We construct hyper-homology spectral sequences of Z-graded and ROG-graded Mackey functors for Ext and Tor over G-equivariant S-algebras (A-infty ring spectra) for finite groups G. These specialize to universal coefficient and Kunneth spectral sequences. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure/intersection On the chain-level intersection pairing for PL manifolds. J.E. McClure mcclure---math.purdue.edu AMS classification numbers: 57Q65; 18D50 Posted on arXiv: math.QA/0410450 Abstract: Let M be a compact oriented PL manifold and let C_*M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex G of C_*M\otimes C_*M. We prove that G is a ``full'' subcomplex, that is, the inclusion of G in C_*M \otimes C_*M is a quasi-isomorphism. An analogous result is true for the domain of the iterated intersection pairing. Using this, we show that the intersection pairing gives C_*M a structure of partially defined commutative DGA, which in particular implies that C_*M is canonically quasi-isomorphic to an E_\infty chain algebra. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Turiel/poly Polynomial Maps and Even Dimensional Spheres Javier Turiel turiel---agt.cie.uma.es Abstract: We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Wodarz/ExactHomotopyFunctors Title: Exactness of Homotopy Functors of Spaces Author: Nathan Wodarz AMS Classification: 55P65, 55T25 Address: Grand Valley State University, Allendale, MI E-mail: wodarzn---gvsu.edu Abstract: We will provide an analysis of the generalized Atiyah--Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called $n$--exact functors, which are analogous to Goodwillie's $n$--excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt--Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt--Goerss Spectral Sequence on homotopy. We close by giving an identification of the $E^2$ term of the GAHSS in the case of 2--exact functors on Moore spaces. ---------------- -------------------------------------- Sorry for the delay this month. The semester is finally over! 6 new papers this month, from Angeltveit, Bokstedt-Ottosen (2), Castellana-Crespo-Scherer, Kitchloo-Wilson, and May. Mark Hovey New papers appearing on hopf between 11/04/04 and 12/14/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Angeltveit/Ainfinity Title: $A_\infty$ obstruction theory and the strict associativity of $E/I$ Author: Vigleik Angeltveit E-mail address: vigleik---math.mit.edu Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Abstract: We prove that for a ring spectrum $K$ with a perfect universal coefficient formula, the obstructions to extending the multiplication to an $A_\infty$ multiplication lie in $Ext^{*,*}_{K_*K^{op}}(K_*,K_*)$. As a corollary, we show that if $E$ is even and $I=(x_1,x_2,\ldots)$ is a regular sequence in $E_*$, then any product on $E/I$ can be extended to an $A_\infty$ multiplication. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/hiem Ttile: An alternative approach to homotopy operations Authors: Marcel Bokstedt and Iver Ottosen Email: marcel---imf.au.dk, ottosen---imf.au.dk Address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 530, DK-8000 Aarhus C, Denmark Abstract: We give a particular choice of the higher Eilenberg-MacLane maps of a simplicial ring by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/kkp Title: A splitting result for the free loop space of spheres and projective spaces Authors: Marcel Bokstedt and Iver Ottosen Email: marcel---imf.au.dk, ottosen---imf.au.dk Address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 530, DK-8000 Aarhus C, Denmark MSC: 55P35, 18G50, 55S10 Abstract: Let X be a 1-connected compact space such that the algebra H*(X;Z/2) is generated by one single element. We compute the cohomology of the free loop space H*(LX;Z/2) including the Steenrod algebra action. When X is a projective space CP^n, HP^n, the Cayley projective plane CaP^2 or a sphere S^m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum of LX_+. The splitting is in terms of the suspension spectrum of X_+ and the Thom spaces of the q-fold Whitney sums of the tangent bundle over X for non negative integers q. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/CWPostH Title: Postnikov pieces and BZ/p-homotopy theory Authors: Natalia Castellana, Juan A. Crespo, Jerome Scherer email: natalia---mat.uab.es, JuanAlfonso.Crespo---uab.es, jscherer---mat.uab.es AMS classification number: 55R35; 55P60, 55P20, 20F18 ArXiv submission number: math.AT/0409399 Abstract: We present a constructive method to compute the cellularization with respect to K(Z/p, m) for any integer m > 0 of a large class of H-spaces, namely all those which have a finite number of non-trivial K(Z/p, m)-homotopy groups (the pointed mapping space map( K(Z/p, m), X) is a Postnikov piece). We prove in particular that the K(Z/p, m)-cellularization of an H-space having a finite number of K(Z/p, m)-homotopy groups is a p-torsion Postnikov piece. Along the way we characterize the BZ/p^r-cellular classifying spaces of nilpotent groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/kitchloo-wilson-ER2 Title: On the Hopf ring for ${ER(n)}$ Authors: Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) La Jolla, CA 92093-0112 nitu---math.ucsd.edu W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu Kriz and Hu construct a real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$ periodic. $ER(1)$ is just $KO_{(2)}$. We do two things in this paper. First, we compute the homology of the $2^{n+2}k}$ spaces in the Omega spectrum for $ER(n)$. There are $2^n-1$ of them and their double is the Hopf ring for $E(n)$. As a byproduct of this we get the homology of the zeroth spaces for the Omega spectrum for real complex cobordism and real Brown-Peterson cohomology. The second result is to compute the homology Hopf ring for all 48 spaces in the Omega spectrum for $ER(2)$. This turns out to be generated by very few elements. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/Split A note on the splitting principle J.P. May may---math.uchicago.edu 55R40, 55N99 We offer a new* perspective on the splitting principle. We give an easy proof that applies to all classical types of vector bundles and in fact to $G$-bundles for any compact connected Lie group $G$. The perspective gives precise calculational information and directly ties the splitting principle to the specification of characteristic classes in terms of classifying spaces. * Note to the list: if this is not new, please let me know --- it shouldn't be, but it was to those experts I tried it out on. -------------- ------------------------------ 4 new papers this month, from Bendersky-Churchill, Hovey, Naumann, and Zivaljevic. Mark Hovey New papers appearing on hopf between 12/14/04 and 1/10/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-Churchill/NormalForms Title: A spectral sequence approach to normal forms. Authors: Martin Bendersky & Richard C. Churchill Address: CUNY/Hunter College, Graduate Center New York, NY 10021 AMS Classification: 55T05, 34C20 Abstract: The theory of normal forms has been around since Poincare's time. An incomplete list of applications are to vector fields, Hamiltonians at equilibria, differential equations and singularity theory. In general one tries to modify a given element in a Lie algebra into a particularly useful form. The algorithm that performs the conversion (the normal form algorithm) can be a formidable computation. In this paper we generalize the notion of normal form to that of an initially linear group representation. In this general setting we are able to interpret the normal form algorithm as a calculation of a particularly simple spectral sequence. As a consequence we show that various vector spaces that appear in the process of carrying out the normal form algorithm are invariants of the orbit of the group representation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/prod-spec-seq The generalized homology of products Mark Hovey Wesleyan University We construct a spectral sequence that computes the E-homology of a product of spectra. The E_{2}-term of this spectral sequence consists of the right derived functors of product in the category of E_{*}E-comodules, and the spectral sequence always converges (with a horizontal vanishing line at E_{infty}) when E is the Johnson-Wilson theory E(n) and each factor of the product is L_{n}-local. We are able to prove some results about the E_{2}-term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra X^{\alpha} is just the comodule product of the E(n)_{*}X^{\alpha}. This spectral sequence is relevant to the chromatic splitting conjecture. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Naumann/comodlandweber Comodule categories and the geometry of the stack of formal groups N. Naumann We generalise recent results of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras using the formalism of algebraic stacks. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Zivaljevic/synergia Title: Equipartitions of measures in R^4 Author: Rade Zivaljevic AMS Class.: 52A39; 52C35; 55S40; 57R22; 57R91; 68P30 arXiv:math.CO/0412483 v1 December 2004 Address: Mathematical Institute SANU, Knez Mihailova 35/1, p.o. box 367 11001 Belgrade Serbia and Montenegro A measure in R^4 admits an equipartition by 4 hyperplanes, provided it is symmetric with respect to a 2-dimensional, affine subspace L of R^4. The computation is based on the Koschorke's exact singularity sequence for groups of normal bordisms and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4. --------------- -------------------------- 6 new papers this month, by Chacholski-Pitsch-Scherer, Ching, DavisDaniel, Dugger, Flores-Scherer, and May-Sigurdsson. Mark Hovey New papers appearing on hopf between 1/10/05 and 2/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chacholski-Pitsch-Scherer/hopullbacks Title: Homotopy pull-back squares up to localization Authors: Wojciech Chacholski, Wolfgang Pitsch, Jerome Scherer AMS classification numbers: Primary 55P60, 55R70; Secondary 55U35, 18G55 ArXiv submission number: math.AT/0501250 Abstract: We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs ``up to Bousfield localization". This yields a generalization of Puppe's theorem which allows to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Ching/operad_bar Bar constructions for topological operads and the Goodwillie derivatives of the identity Michael Ching Massachusetts Institute of Technology Includes 19 PS figures with filenames *.pstex Abstract: We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of those derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over ooperads) and show that a based space naturally gives rise to a right module over the operad formed by the derivatives of the identity. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/enhfps2 Title: Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action Author: Daniel Davis Address: Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067 Abstract: Let G be a closed subgroup of G_n, the extended Morava stabilizer group. Let E_n be the Lubin-Tate spectrum, let X be an arbitrary spectrum with trivial G-action, and define E^(X) to be L_K(n)(E_n ^ X). We prove that E^(X) is a continuous G-spectrum with a G-homotopy fixed point spectrum, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is the homotopy groups of the G-homotopy fixed point spectrum of E^(X). We show that the homotopy fixed points of E^(X) come from the K(n)-localization of the homotopy fixed points of the spectrum (F_n ^ X). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/spenrich Spectral enrichments of model categories Daniel Dugger Abstract: We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (really a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichment of model categories provided by the Dwyer-Kan hammock localization. As a particular application, we associate to every object in a stable, combinatorial model category a certain "homotopy endomorphism ring spectrum". The homotopy type of this ring spectrum is preserved by Quillen equivalences, and so serves as an invariant of model categories. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores-Scherer/cwandfusion Title: Cellularization of classifying spaces and fusion properties of finite groups Authors: Ramon J. Flores, Jerome Scherer AMS classification numbers: Primary 55P60, 20D200; Secondary 55R37, 55Q05 ArXiv submission number: math.AT/0501442 Abstract: One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their B\Z/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May-Sigurdsson/MSMaster Parametrized homotopy theory J. P. May and J. Sigurdsson University of Chicago, University of Notre Dame Primary 19D99, 55N20, 55P42; Secondary 19L99, 55N22, 55T25 Abstract: We provide rigorous modern foundations for parametrized (equivariant, stable) homotopy theory in this four part monograph. In Part I, we give preliminaries on the necessary point-set topology, on base change and other relevant functors, and on generalizations of various standard results to the context of proper actions of non-compact Lie groups. In Part II, we give a leisurely development of the homotopy theory of ex-spaces that emphasizes several issues of independent interest. It includes much new material on the general theory of topologically enriched model categories. The essential point is to resolve problems in the homotopy theory of ex-spaces that have no nonparametrized counterparts. In contrast to previously encountered situations, model theoretic techniques are intrinsically insufficient for this purpose. Instead, a rather intricate blend of model theory and classical homotopy theory is required. In Part III, we develop the homotopy theory of parametrized spectra. We work equivariantly and with highly structured smash products and function spectra. The treatment is based on equivariant orthogonal spectra, which are simpler for the purpose than alternative kinds of spectra. Again, there are many difficulties that have no nonparametrized counterparts and cannot be dealt with model theoretically. In Part IV, we give a fiberwise duality theorem that allows fiberwise recognition of dualizable and invertible parametrized spectra. This allows application of the formal theory of duality in symmetric monoidal categories to the construction and analysis of transfer maps. A construction of fiberwise bundles of spectra, which are like bundles of tangents along fibers but with spectra replacing spaces as fibers, plays a central role. Using it, we obtain a simple conceptual proof of a generalized Wirthmuller isomorphism theorem that calculates the right adjoint to base change along an equivariant bundle with manifold fibers in terms of a shift of the left adjoint. Due to the generality of our bundle theoretic context, the Adams isomorphism theorem relating orbit and fixed point spectra is a direct consequence. ------------- --------------------------------- 13 new papers this month, by Angeltveit-Rognes, Arkowitz-Oshima-Strom, Arkowitz-Stanley-Strom, Barker-Snaith, Broto-Castellana-Grodal-Levi-Oliver, Broto-Levi-Oliver, Iwase-Stanley-Strom, Jardine, Levi-Oliver, Lupton-SmithSB, Nendorf-Scoville-Strom, and Rognes (2). Mark Hovey New papers appearing on hopf between 2/5/05 and 3/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Angeltveit-Rognes/vigleik Title: Hopf algebra structure on topological Hochschild homology Author(s): Vigleik Angeltveit and John Rognes Author's e-mail address: Abstract: The topological Hochschild homology THH(R) of a commutative S-algebra (E-infinity ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bokstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, using topological cyclic homology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/Equiv Homotopy classes of self-maps and induced homomorphisms of homotopy groups Martin Arkowitz Hideaki Oshima Jeffrey Strom For a based space X, we consider the group of all self homotopy classes of $X$ such that which induce the identity on homotopy groups in dimensions 1 through n, and the group of all homotopy classes which loop to the identity. Analogously, we study the semigroups defined by replacing `identity' by `0' above. There is a chain of containments of these groups and semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that these groups are equal, or when the semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for certain low dimensional Lie groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Stanley-Strom/Length1 The A-category and A-cone length of a map Martin Arkowitz Donald Stanley Jeffrey Strom For any collection A of spaces we define two numerical invariants of maps: A-category of f and the A-cone length of f. These invariants are defined axiomatically, and our first results give equivalent constructive definitions in terms of mapping cone decompositions. We show that if A is the collection of all spaces, then the A-category of f is the category of f as defined by Fadell and Husseini and the A-category of f is the cone length of f as defined by Marcum. By specializing to the unique maps from and to a one-point space, we obtain four invariants of spaces. Each of these four invariants has its own axiomatic and constructive definitions. We compare them similar invariants defined by Scheerer and Tanr\'e. We conclude by giving lower bounds for these invariants in terms of cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Barker-Snaith/psi3triangle3 \psi^3 as an upper triangular matrix Jonathan Barker and Victor Snaith 55S25 (Primary) 55P42 (Secondary) math.AT/0502472 Jonathan Barker Building 54 (School of Mathematics) University of Southampton Highfield Southampton SO17 1BJ UK Victor Snaith Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1\wedge\psi^3, where \psi^3 is the Adams operation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo2 EXTENSIONS OF p-LOCAL FINITE GROUPS C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ``conjugacy'' relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we study and classify extensions of $p$-local finite groups, and also compute the fundamental group of the classifying space of a $p$-local finite group. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo4 A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS Carles Broto, Ran Levi, and Bob Oliver A saturated fusion system consists of a finite $p$-group $S$, together with a category which encodes ``conjugacy'' relations among subgroups of $S$, and which satisfies certain axioms which are motivated by properties of the fusion in a Sylow $p$-subgroup of a finite group. We describe here new ways of constructing abstract saturated fusion systems, first as fusion systems of spaces with certain properties, and then via certain graphs. Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: classifying space, $p$-completion, finite groups, fusion. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Iwase-Stanley-Strom/GaneaCond Implications of the Ganea Condition Norio Iwase Donald Stanley Jeffrey Strom Suppose the spaces X and X x A have the same Lusternik-Schnirelmann category: cat(X x A) = cat(X). Then there is a strict inequality cat(X x (A \halfsmash B)) < cat (X) + cat(A \halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/diagrams Author: J.F. Jardine Author's mailing address: Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada Suppose that A is a small presheaf of categories enriched in simplicial sets on a small Grothendieck site. It is shown that the homotopy theory of enriched A-diagrams taking values in simplicial sets can be identified with the homotopy theory of simplicial presheaves fibred over the diagonalized nerve dBA of A. One can also identify the set [*,dBA] of morphisms in the simplicial presheaf homotopy category with path components of the category of A-torsors, suitably defined. These statements are special cases of localized results which hold when the corresponding localized model structures are proper. Examples of the latter include the motivic homotopy category of Morel and Voevodsky, and so these results lead to a theory of motivic A-torsors which is classifiable up to equivalence by a family of morphisms in the motivic homotopy category. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Oliver/sol-corr Correction to: CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE STUDIED BY SOLOMON AND BENSON by Ran Levi and Bob Oliver A $p$-local finite group is an algebraic structure with a classifying space which has many of the properties of $p$-completed classifying spaces of finite groups. In our paper \cite{Sol}, we constructed a family of 2-local finite groups which are ``exotic'' in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of $\Spin_7(q)$ ($q$ an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. As predicted by Benson, the classifying spaces of these 2-local finite groups are very closely related to the Dwyer-Wilkerson space $BDI(4)$. An error in our paper \cite{Sol} was pointed out to us by Andy Chermak, and we correct that error here. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/pi_one(map) Title: Rank of the fundamental group of a component of a function space Authors: Gregory Lupton and Samuel Bruce Smith ArXive: math.AT/0502311 MSC-class: 55Q52; 55P15 We compute the rank of the fundamental group of an arbitrary connected component of the space map(X, Y) for X and Y nilpotent CW complexes with X finite. For the general component corresponding to a homotopy class f : X --> Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula expresses the rank in terms of classical invariants of X and Y. Among other applications and calculations, we obtain the following: Let G be a compact simple Lie group with maximal torus T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and only if f: S^2 --> G/T^n is essential. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Nendorf-Scoville-Strom/Seq1 Categorical Sequences Rob Nendorf Nick Scoville Jeffrey Strom We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skelta. The k-th term in the categorical sequence of a CW complex X, is the least integer n for which the n-skeleton of X has L-S category at least k. We show that the categorical sequence of X is a well-defined homotopy invariant. We prove that the sequence is `superadditive' which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)) (which is known to be 5). 12. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/dualizable Title: Stably dualizable groups Author: John Rognes Abstract: We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by (Dwyer and) J.R. Klein and the p-complete study for p-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the E-local stable homotopy category, for any spectrum E. The principal new examples occur in the K(n)-local category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably dualizable for all 0 <= q <= n. We show how to associate to each E-locally stably dualizable group G a stably defined representation sphere S^{adG}, called the dualizing spectrum, which is dualizable and invertible in the E-local category. Each stably dualizable group is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}. There are dimension-shifting norm- and transfer maps for spectra with G-action, again with a shift given by S^{adG}. The stably dualizable group G also admits a kind of framed bordism class [G] in the homotopy of L_E S, in degree dim_E(G) = [S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere spectrum. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/galois Title: Galois extensions of structured ring spectra Author: John Rognes Abstract: We introduce the notion of a Galois extension of commutative S-algebras (E-infinity ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and etale) extensions of commutative S-algebras, and the Goerss-Hopkins-Miller theory for E-infinity mapping spaces. We show that the global sphere spectrum~S is separably closed (using Minkowski's discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf-Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin-Tate Galois extensions. ------------ ------------------------------------------- I seem to have forgot to send this out in April. My apologies. There are 6 new papers this month, by Arone-Lesh, Bergner, DavisD-Potocka, Lawson, Lueck, and Strohm. Mark Hovey New papers appearing on hopf between 3/5/05 and 5/7/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lesh/arone-lesh-filtered-spectra Title: Filtered spectra arising from permutative categories Authors: Gregory Arone University of Virginia Kathryn Lesh Union College Abstract: Given a special Gamma-category C satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to C and the Eilenberg-Mac Lane spectrum HZ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of HZ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, A_{m}, that interpolate between bu and HZ. We think of A_{m} as a ``bu-analogue'' of the m'th symmetric power and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the m'th subquotient is contractible unless m is a power of a prime, and in v_{k}-periodic homotopy the filtration has only k+2 nontrivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {A_{m}}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of $bu$. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/ThreeModels Title: Three models for the homotopy theory of homotopy theories Author: Julia E. Bergner AMS classification number: Primary: 55U35; Secondary 18G30, 18E35 arXiv submission number: math.AT/0504334 Abstract: Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the ``homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a ``homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with their respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk's complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Potocka/sun2long 2-primary v1-periodic homotopy groups of SU(n) revisited Donald M. Davis, Lehigh University, Bethlehem, PA 18015, Katarzyna Potocka, Ramapo College of New Jersey, Mahwah, NJ 07430 Abstract In 1991, Bendersky and Davis used the BP-based unstable Novikov spectral sequence to study the 2-primary v1-periodic homotopy groups of SU(n). Here we use a K-theoretic approach to add more detail to those results. In particular, whereas only the order of the groups v1^{-1} pi_{2k-1}(SU(n)) was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for v1^{-1} pi_*(SU(n)) for n < 14. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Lawson/lawson_productformula Title: The product formula in unitary deformation $K$-theory Author: Tyler Lawson MSC classification: 19D23; 19L41; 20C99 PaperID: math.KT/0503468 Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract: We prove that "unitary deformation K-theory" takes products of finitely generated groups to coproducts of algebra spectra over ku, the connective K-theory spectrum. Additionally, we give spectral sequences for computing the homotopy groups of the unitary deformation K-theory of a group G and the cofiber of the Bott map in terms of PU(n)-equivariant K-theory and homology of spaces of G-representations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_burnside0504 Title of Paper: The Burnside Ring and Equivariant Cohomotopy for Infinite Groups Author: Wolfgang Lueck AMS Classification numbers: 55P91, 19A22. math.AT/0504051 Fachbereich Mathematik Universitaet Muenster Einsteinstr. 62 48149 Muenster Germany Abstract: After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-$G$-set-version, the inverse-limit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the equivariant zero-th cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant cohomotopy groups of finite proper equivariant CW-complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Strohm/diploma_main Title: The Proportionality Principle of Simplicial Volume Authors: Clara Strohm (=Clara Löh) Address: Einsteinstr. 62, 48143 Münster, Germany MSC: 57R19, 55N35 Abstract: The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial volume is linked to Riemannian geometry in various ways, e.g., by the proportionality principle. The proportionality principle of simplicial volume states that the simplicial volume and the Riemannian volume are proportional for oriented closed connected Riemannian manifolds sharing the same universal Riemannian covering. Thurston indicated a proof of the proportionality principle using his (smooth) measure homology. It is the purpose of this diploma thesis to provide a full proof of the proportionality principle based on Thurston's approach. In particular, it is shown that (smooth) measure homology and singular homology are isometrically isomorphic for all smooth manifolds. This implies that the simplicial volume indeed can be computed in terms of measure homology. Included eps files: fg.eps, dragon_schoon.eps ---------------- --------------------- There are 7 new papers this time, from Bendersky-DavisD, Elmendorf-Mandell, Goerss-Hopkins, Murillo-Buijs (2), Rezk, and Vistoli. Mark Hovey New papers appearing on hopf between 5/7/05 and 7/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/sgdp2 Stable geometric dimension of vector bundles over odd-dimensional real projective spaces Martin Bendersky, Hunter College, CUNY 10021, Donald M. Davis, Lehigh University, Bethlehem, Pa. 18015 55S40, 55R50, 55T15 Abstract: In a recent paper, the geometric dimension of all stable vector bundles over real projective space P^n was determined if n is even and sufficiently large with respect to the order 2^e of the bundle. Here we perform a similar determination when n is odd and e>6. The work is more delicate since P^n does not admit a v1-map when n is odd. There are a few extreme cases which we are unable to settle precisely. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/RMA2 Rings, modules, and algebras in infinite loop space theory A. D. Elmendorf and M. A. Mandell Subject classes: Primary 19D23; Secondary 55P43, 18D10 xxx-LANL identifier: math.KT/0403403 Addresses: A. D. Elmendorf Dept. of Mathematics Purdue University Calumet Hammond, IN 46323 M. A. Mandell (current) DPMMS CMS University of Cambridge Cambridge CB3 0WB England M. A. Mandell (effective Fall 2005) Department of Mathematics Indiana University Bloomington, IN 47405 This is a major revision of a previous submission of the same name. We have completely rewritten sections 5 -- 7, giving a new construction of the first part of our functor. The main abstract is as follows: We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory (elsewhere also called colored operad), a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in the Hovey-Shipley-Smith category of symmetric spectra, with an intermediate stop at a category of functors out of a particular wreath product. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Hopkins/obstruct Title: Moduli spaces for Structured Ring Spectra Authors: P.G. Goerss and M.J. Hopkins Abstract: In this document we make good on all the assertions we made in the previous paper ``Moduli spaces of commutative ring spectra'' wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of commutative ring spectra. In particular, we develop a theory of moduli spaces of algebra structures on spectra, and give a decomposition of the moduli space as a tower of fibrations wherein the successive fibers can be calculated using Andre'-Quillen cohomology. By examining the obstructions to lifting a basepoint up the tower, we then produce successively defined obstructions to the realizing an algebra structure. A point worth emphasizing is that the moduli problems here begin with algebra: for example, we may have a homology theory E and a commutative ring A in the category comodules associated to E and we wish to discuss the homotopy type of the space of all commutative (in the strict sense) ring spectra X so that the E-homology of X is A as a commutative ring. We do not, a priori, assume that this moduli space is non-empty, or even that there is a spectrum whose E-homology is A. For a variety of applications we are not simply interested in this absolute problem, but in a relative version as well. Fortunately, Andre'-Quillen cohomology is inherently relative and the theory adapts well to this case. The main idea, which goes back to Dwyer, Kan, and Stover, is to try to construct a simplicial ring spectrum, whose geometric realization will realize A. Then we use the new simplicial direction and apply Postnikov tower techniques to get the decomposition of the moduli space. Making this work requires a certain amount of technical detail. In particular, we need to be very careful with resolution model categories and their localizations at a homology theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Murillo-Buijs/mapping Title of Paper: Basic constructions in rational homotopy theory of function spaces Author(s) Aniceto Murillo and Urtzi Buijs AMS Classification numbers 55P62 Addresses of Authors Departamento de Algebra, Geometria y Topologia Universidad de Malaga, AP. 59, 29080 Malaga SPAIN Text of Abstract Via the Bousfield-Gugenheim realization functor, and starting from the Brown-Szczarba approach to the Haefliger model of a function space, we give a functorial framework to describe basic objects and maps concerning the rational homotopy type of function spaces and its path components. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Murillo-Buijs/lie_algebra Title of Paper: The rational homotopy Lie algebra of function spaces Author(s) Aniceto Murillo and Urtzi Buijs AMS Classification numbers 55P62 Addresses of Authors Departamento de Algebra, Geometria y Topologia Universidad de Malaga, AP. 59, 29080 Malaga SPAIN Text of Abstract We give a full and explicit description of the rational homotopy Lie algebra of function spaces (free or pointed) 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Rezk/rezk-units-and-logs Title: The units of a ring spectrum and a logarithmic cohomology operation Author: Charles Rezk Abstract: We construct a ``logarithmic'' cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E^0(K) of a space K. We obtain a formula for this map in terms of the action of Hecke operators on Morava E-theory. Our formula is closely related to that for an Euler factor of the Hecke L-function of an automorphic form. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Vistoli/PGL_p On the cohomology and the Chow ring of the classifying space of PGL_p Angelo Vistoli Dipartimento di Matematica Universitŕ di Bologna Piazza di Porta San Donato 5 40014 Bologna Italy arXive submission number: math.AG/0505052 Abstract: We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine their additive structures completely, and we reduce the problem of determining their multiplicative structures to a problem in invariant theory. ---------------- ------------------------------------- There are 11 new papers this time, from Behrens (3), Behrens-Lawson, Chebolu, DavisDaniel, Hovey, Lueck, Morava, Neusel, and Neusel-Wisniewski. Mark Hovey New papers appearing on hopf between 7/1/05 and 8/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/rootkin/rootkin Title: Some root invariants at the prime 2 Author(s): Mark Behrens Abstract: The first part of this paper consists of lecture notes which summarize the machinery of filtered root invariants. A conceptual notion of "homotopy Greek letter element" is also introduced, and evidence is presented that it may be related to the root invariant. In the second part we compute some low dimensional root invariants of v_1-periodic elements at the prime 2. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/rootpub/rootpub Title: Root invariants in the Adams spectral sequence Author(s): Mark Behrens Abstract: Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the E_1 term of the E-Adams spectral sequence. The main theorems of this paper concern when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low dimensional root invariants of v_1-periodic elements at the prime 3. We also compute the root invariants of some infinite v_1-periodic families of elements at the prime 3. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/K2S/K2S Title: A modular description of the K(2)-local sphere at the prime 3 Author(s): Mark Behrens Abstract: Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This spectrum is built out of spectra related to tmf. At p=3 we show that the K(2)-local sphere is built out of Q(2) and its K(2)-local Spanier-Whitehead dual. This gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and Rezk. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Lawson/dense Title: Isogenies of elliptic curves and the Morava stabilizer group Authors: Mark Behrens and Tyler Lawson Abstract: Let MS_2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over \br{FF}_p, O the ring of endomorphisms of C, and \ell a topological generator of Z_p^x (respectively Z_2^x/{+-1} if p = 2). We show that for p > 2 the group \Gamma \subseteq O[1/\ell]^x of quasi-endomorphisms of degree a power of \ell is dense in MS_2. For p = 2, we show that \Gamma is dense in an index 2 subgroup of MS_2. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu/KS Title: Krull-Schmidt decompositions for thick subcategories Author: Sunil Chebolu AMS classifictaion numbers: Primary: 55p42; Secondary: 18E30 Address: Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7 Abstract: Following Krause, we prove Krull-Schmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. Some consequences of these decomposition results are also discussed. In particular, it is shown that all these decompositions respect $K$-theory 6. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/p1v5ams Title: Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action (Revised version) Author: Daniel Davis Address: Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067 Abstract: Let G be a closed subgroup of G_n, the extended Morava stabilizer group. Let E_n be the Lubin-Tate spectrum, let X be an arbitrary spectrum with trivial G-action, and define E^(X) to be L_K(n)(E_n ^ X). We prove that E^(X) is a continuous G-spectrum with a G-homotopy fixed point spectrum, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is the homotopy groups of the G-homotopy fixed point spectrum of E^(X). We show that the homotopy fixed points of E^(X) come from the K(n)-localization of the homotopy fixed points of the spectrum (F_n ^ X). 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/freyd On Freyd's generating hypothesis Mark Hovey We revisit Freyd's generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that $I$, the Brown-Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesis, that the homotopy of a finite spectrum that is not a wedge of spheres can never be finitely generated as a module over $\pi_{*}S$, is in fact true for finite torsion spectra. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_tkcsr Title: Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups Author: Wolfgang Lueck AMS Classification Numbers: 55N15 Address: Wolfgang Lueck Mathematisches Institut der Westfaelischen Wilhelms-Universitaet Einsteinstr. 62 48149 Muenster Germany xxx-archive: KT/0507237 Abstract: We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Rosendal Title: Toward a fundamental groupoid for tensor triangulated categories Author: Jack Morava AMS classification: 11G, 19F, 57R, 81T Abstract: Notes for a talk at the conference on arithmetic of structured ring spectra; Rosendal, Norway, August 19 - 28 2005: This very speculative talk suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/survey Degree bounds--an invitation to postmodern invariant theory Mara D. Neusel Abstract: This is a survey article on degree bounds in invariant theory of finite groups. A finite subgroup $G$ of the general linear group $\GL(n\, \F)$ over some field $\F$ acts via matrix multiplication on the vector space $V=\F^n$. This induces an action of $G$ on the polynomials $\F[x_1\commadots x_n]$ in $n$ variables. The polynomials $\F[x_1\commadots x_n]^G\subseteq \F[x_1\commadots x_n]$ invariant under this action form a subring. This ring is our center of study. In particular we will discuss how to calculate this ring. In this context degree bounds are central, and we want to present the known results. We also sketch the techniques that are used to obtain good bounds and describe open questions. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Wisniewski/piotr Connected Hopf algebras with Dixmier bases and infinite primary decomposition Mara D. Neusel, Piotr Wisniewski Abstract: In this paper we show the existence of invariant primary decompositions in the categories of modules and rings over a Hopf algebra of Dixmier type. --------------- ------------------------------------ There are 6 new papers this time, from Bergner (2), Chebolu, Hornbostel-Naumann, Lueck-Reich, and Notbohm. Mark Hovey New papers appearing on hopf between 8/4/05 and 9/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/MultiSort Title: Rigidification of algebras over multi-sorted algebraic theories Author: Julia E. Bergner AMS Classification: 18C10, 18G30, 18E35, 55P48 arXiv submission number: math.AT/0508152 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different ``sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/SimplicialMonoids Title: Simplicial monoids and Segal categories Author: Julia E. Bergner AMS Classification: 18G30, 18E35, 18C10, 55U40 arXiv submission number: math.AT/0508416 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model category structure on the category of simplicial monoids is Quillen equivalent to an appropriate model category structure on the category of simplicial spaces with a single point in degree zero. In this second model structure, the fibrant objects are reduced Segal categories. We then generalize the proof to relate simplicial categories with a fixed object set to Segal categories with the same fixed set in degree zero. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu/chromatic Title: Refining thick subcategory theorems Author: Sunil Chebolu AMS classification numbers: Primary: 55P42, 18G55, 19A99 Address: Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7 Abstract: We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification of the triangulated subcategories of perfect complexes over some noetherian rings. In the homotopy category of spectra we obtain only a partial classification of the triangulated subcategories of the finite $p$-local spectra. We use this partial classification to study the lattice of triangulated subcategories. This study gives some new evidence to a conjecture of Adams that the thick subcategory $\C_2$ can be generated by iterated cofiberings of the Smith-Toda complex. We also discuss various consequences of these classifications theorems. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel-Naumann/f-invofbeta Title: Beta-elements and divided congruences Authors: Jens Hornbostel, Niko Naumann Abstract: The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of beta-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck-Reich/lueck+reich0805 Title of Paper: Detecting K-theory by cyclic homology Author(s): Wolfgang Lueck and Holger Reich AMS Classification number: 19D55 xxx_archive: math.KT/0509002 Addresses of Authors: Mathematisches Institut Westfaelische Wilhelms-Universitaet Einsteinstr. 62 48149 Muenster Germany Text of Abstract (try for 20 lines or less) We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm/cmcomplex Title: Cohen-Macaulay and Gorenstein complexes from a topological point of view Author: Dietrich Notbohm AMS Classification numbers: 13F55, 55R35 Address of Author: Dept. of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England Abstract: The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face ring are reflected by geometric and/or combinatoric properties of the simplicial complex. We give a new proof for these result by homotopy theoretic methods and constructions. Our approach is based on ideas used very successfully in the analysis of the homotopy theory of classifying spaces. ------------------ --------------------------------- There are 8 new papers this time, from Arkowitz-Lupton, DavisDaniel(2), DavisD-Sun, Felix-Lupton, Henn, Kreck-Lueck, and Lupton-Phillips-Schochet-SmithSB. Mark Hovey New papers appearing on hopf between 9/5/05 and 10/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Lupton/ArkLupActions Homotopy Actions, Cyclic Maps and their Duals Martin Arkowitz and Gregory Lupton MSC 2000 55Q05, 55M30, 55P30 Abstract: An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1. Note: Appeared in Homology, Homotopy and Applications, vol. 7(1) (2005), 169-184. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/galois Title: Rognes's theory of Galois extensions and the continuous action of G_n on E_n Author: Daniel G. Davis Address: Purdue University Abstract: Let us take for granted that $L_{K(n)}S^0 \rightarrow E_n$ is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that "E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This inactive manuscript arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available. The author thanks Paul Goerss for his encouragement. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg Title: Attempting to construct homotopy orbits for profinite groups Author: Daniel G. Davis Address: Purdue University Abstract: This note gives a heuristic argument for how one might like to define X_{hG}, for G profinite; it represents a first step in attempting to do this. The argument is not shown to work, and though the heuristic seems plausible, the author does not know how to complete the critical Definition 4.2. Also, the proof of Theorem 5.2 is incomplete. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Sun/DavisSun A number-theoretic approach to homotopy exponents of SU(n) Donald M. Davis and Zhi-Wei Sun AMS Classifications: 55Q52, 57T20, 11A07, 11B65, 11S05 Abstract: We use methods of combinatorial number theory to prove that, for all n and p, some homotopy group pi_i(SU(n)) contains an element of order p^{n-1+ord_p([n/p]!)}, where ord_p(m) denotes the exponent of p in m. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Felix-Lupton/FelLupEval Title: Evaluation Maps in Rational Homotopy Authors: Yves Felix and Gregory Lupton AMS MSC2000: 55P62, 55Q05 arXiv: math.AT/0509632 Abstract: Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X, X;1) -> X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational homotopy groups is injective. This result has strong consequences: if the image in rational homotopy groups of ev is trivial, then the generalized evaluation map is null-homotopic after rationalization; unless X satisfies a very strong splitting condition, any generalized evaluation map induces the trivial homomorphism in rational cohomology; the map Gamma_ev is rationally a homotopy monomorphism and a generalized evaluation map may be written as a composition of a homotopy epimorphism and this homotopy monomorphism. We include illustrative examples and prove numerous subsidiary results of interest. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn/kn-res-ded Title: On finite resolutions of K(n)-local spheres Author: Hans-Werner Henn Abstract: For odd primes p we construct finite resolutions of the trivial module Z_p for the n-th Morava stabilizer group by (direct summands of) permutation modules with respect to finite p-subgroups. Furthermore we discuss the problem of realizing these resolutions by finite resolutions of the K(n)-local sphere via spectra which are (direct summands of) wedges of homotopy fixed point spectra for the action of these finite p-subgroups on the Lubin-Tate spectrum E_n. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kreck-Lueck/kreck+lueck0905 Title of Paper: Topological rigidity for non-aspherical manifolds Author(s): Matthias Kreck and Wolfgang Lueck AMS Classification number: 57N99, 57R67. xxx_archive: math.GT/0509238 The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N ---> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N ---> M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for S^k x S^d, for sphere bundles over aspherical closed manifolds of dimension less or equal to 3 and for 3-manifolds with torsionfree fundamental groups. We show that this rigidity is inherited under connected sums in dimensions greater or equal to 5. We also classify manifolds of dimension 5 or 6 whose fundamental group is the one of a surface and whose second homotopy group is trivial. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-Phillips-Schochet-SmithSB/RationalTaylor Title: Banach Algebras and Rational Homotopy Theory Authors: Gregory Lupton, N.Christopher Phillips, Claude L.~Schochet and Samuel B. Smith AMS MSC (2000): 46J05, 46L85, 55P62, 54C35, 55P15, 55P45 arXiv number: math.AT/0509269 Abstract: Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of $GL_n (A)$, the group of invertible $n \times n$ matrices with coefficients in A in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of ``last columns'' of $GL_n (A).$ We construct a natural isomorphism \[ {\check{H}}^s (Max(A); Q) \cong \pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q \] for $n > (1/2) s + 1$ which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F(X,G) for G a Lie group or, more generally, a rational H-space. ----------------- --------------------------------------- There are 7 new papers this time, from Bartels-Reich, Bousfield, Fausk-Isaksen (2), Neusel, Neusel-Sezer, and Siebenmann. Mark Hovey New papers appearing on hopf between 10/1/05 and 11/11/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Reich/erb Title: Coefficients for the Farrell-Jones Conjecture Authors: Arthur Bartels, Holger Reich Author's e-mail address: bartelsa@math.uni-muenster.de, reichh@math.uni-muenster.de Abstract: We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/kunneth Title: Kunneth theorems and unstable operations in 2-adic KO-cohomology Author: A.K. Bousfield E-mail: bous@uic.edu AMS classifications: 55N15,55S25,55U25 Abstract: We develop Kunneth theorems and obtain detailed results on unstable operations in 2-adic KO-cohomology and, more generally, in united 2-adic K-cohomology. These results are needed for work on the K-localizations of H-spaces at the prime 2 and should be of independent interest. Our proofs of relations for unstable operations rely on Atiyah's Real K-theory and on a convenient mod 2 simplification of 2-adic KO-cohomology. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Isaksen/filtered Title: Model structures on pro-categories Authors: Halvard Fausk, Daniel C. Isaksen E-mail: fausk@math.uio.no, isaksen@math.wayne.edu AMS Classification: 55U35 Primary ; Secondary 55P91, 18G55 Abstract: We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for $G$-spaces, where $G$ is a pro-finite group. The class of weak equivalences is an approximation to the class of underlying weak equivalences. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Isaksen/t-model Title: T-model structures Authors: Halvard Fausk and Daniel C. Isaksen E-mail: fausk@math.uio.no, isaksen@math.wayne.edu AMS Classification: Primary 55P42; Secondary 18E30, 55U35 Abstract: For every stable model category $\mathcal{M}$ with a certain extra structure, we produce an associated model structure on the pro-category pro-$\mathcal{M}$ and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-$\mathcal{M}$. Our motivating example is the category of pro-spectra. The extra structure referred to above is a t-model structure. This is a rigidification of the usual notion of a t-structure on a triangulated category. A t-model structure is a proper simplicial stable model category $\mathcal{M}$ with a t-structure on its homotopy category together with an additional factorization axiom. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/piotr Connected Hopf algebras with Dixmier bases and infinite primary decomposition Mara D. Neusel Mara.D.Neusel@ttu.edu Abstract: In this paper we study the existence of invariant primary decompositions for algebras and modules over Hopf algebras. This is an update of the previous preprint of Neusel-Wisniewski of the same title. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether The Noether map AUTHORS: Mara D. Neusel (Texas Tech University), M\"ufit Sezer (Bo\u gazici \"Universitesi) EMAILS: mara.d.neusel@ttu.edu mufit.sezer@boun.edu.tr ABSTRACT: Let $\rho: G\hra GL(n\/,\ \F)$ be a faithful representation of a finite group $G$. In this paper we study the image of the associated Noether map \[ \eta_G^G: \F[V(G)]^G \longrightarrow \F[V]^G\/. \] It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure $\overline{\Im(\eta_G^G)} =\F[V]^G$. This is true without any restrictions on the group, representation, or ground field. Furthermore, we show that the Noether map is surjective, i.e., its image integrally closed, if $V=\F^n$ is a projective $\F G$-module. Moreover, we show that the converse of this statement is true if $G$ is a $p$-group and $\F$ has characteristic $p$, or if $\rho$ is a permutation representation. We apply these results and obtain upper bounds on the Noether number and the Cohen-Macaulay defect of $\F[V]^G$. We illustrate our results with several examples. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Siebenmann/Schoen-02Sept2005 The Osgood-Schoenflies Theorem Revisited by Laurent Siebenmann Math'ematique, B^at. 425, Universit'e de Paris-Sud, 91405-Orsay, France http://topo.math.u-psud.fr/~lcs/contact This retrospective article presents an elementary, and hopefully direct and clear, geo- metric proof of what is usually called the (classical planar) Schoenflies Theorem; it is stated as (ST) in x4 below _ with mention of its early history, including W.F. Osgood's rarely cited contributions. This (ST) is essentially the fact _ surprising in view of known fractal curves _ that every compact subset of the cartesian plane R2 that is homeomorphic to the circle S1, is necessarily the frontier in R2 of a set homeomorphic to the 2-disk. Beware that the `Generalized Schoenflies theorem' of B. Mazur [Maz] and M. Brown [Brow1] _ proved five decades later and valid in all dimensions _ does not imply (ST) since it assumes a condition of flatness (or local flatness [Brow2]). ------------------ There are 8 new papers this time, from Arone-Lesh, BrownR, Gutierrez, Inoue-Yagita, Klein-Williams, Korbas, Lockridge, and Ziemianski Mark Hovey New papers appearing on hopf between 11/11/05 and 1/4/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lesh/arone-lesh-press Title: Filtered spectra arising from permutative categories Authors: Gregory Arone University of Virginia Kathryn Lesh Union College Abstract: Given a special Gamma-category C satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to C and the Eilenberg-Mac Lane spectrum HZ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of HZ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, A_{m}, that interpolate between bu and HZ. We think of A_{m} as a ``bu-analogue'' of the m'th symmetric power of the sphere and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the m'th subquotient is contractible unless m is a power of a prime, and in v_{k}-periodic homotopy the filtration has only k+2 nontrivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {A_{m}}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of $bu$. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture. This is a revision of a preprint previously submitted to Hopf. The paper has been accepted for publication in Journal für die reine und angewandte Mathematik (Crelle's Journal). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/PB-jordan Title: Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem Author: Ronald Brown Author's e-mail address: r.brown at bangor.ac.uk Author's mailing address: Mathematics Department, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK Author's web site: www.bangor.ac.uk/r.brown Preprint: University of Wales Math Preprint 06.01 Abstract: We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gutierrez/hlems Homological localizations of Eilenberg-Mac Lane spectra Javier J. Guti\'errez We discuss the Bousfield localization $L_E X$ for any spectrum $E$ and any $HR$-module $X$, where $R$ is a ring with unit. Due to the splitting property of $HR$-modules, it is enough to study the localization of Eilenberg--Mac\,Lane spectra. Using general results about stable $f$-localizations, we give a method to compute the localization of an Eilenberg--Mac\,Lane spectrum $L_E HG$ for any spectrum $E$ and any abelian group $G$. We describe $L_E HG$ explicitly when $G$ is one of the following: finitely generated abelian groups, $p$-adic integers, Pr\"ufer groups, and subrings of the rationals. The results depend basically on the $E$-acyclicity patterns of the spectrum $H\Q$ and the spectrum $H\Z/p$ for each prime $p$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Inoue-Yagita/bpso Title: The complex cobordism of BSOn Authors: K.Inoue and N.Yagita Abstract: In this paper, we compute MU(BSO(2n)) and show that it is generated as an MU-algebra by Conner-Floyd Chern classes and one 2n-dimensional element. For the case BO(m) are still studied by W.S.Wilson. We get the result by using (equivariant) stratification methods introduced to compute Chow rings by Guillot, Molina, Vessozi and Vistoli. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Klein-Williams/int-theoryI Homotopical intersection theory, I. by John R. Klein and E. Bruce Williams Abstract: We give a new approach to intersection theory. Our ``cycles'' are closed manifolds mapping into compact manifolds and our ``intersections'' are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Korbas/cuplength Title: Bounds for the cup-length of Poincare spaces and their applications Author: Julius Korbas AMS Classification numbers: Primary: 57N65; 55M30 Secondary: 53C30 Author's addresses: Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics, Comenius University, Mlynska dolina, SK-842 48 Bratislava 4, Slovakia or Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, SK-814 73 Bratislava 1, Slovakia Abstract: Our main result offers a new (quite systematic) way of deriving bounds for the cup-length of Poincare spaces over fields; we outline a general research program based on this result. For the oriented Grassmann manifolds, already a limited realization of the program leads, in many cases, to the exact values of the cup-length and to interesting information on the Lyusternik-Shnirel'man category. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lockridge/gh Title: The generating hypothesis in the derived category of R-modules. Author: Keir H. Lockridge Abstract: In this paper, we prove a version of Freyd's generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S is an object in D whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Ziemianski/SpinHtpReps TITLE: Homotopy representations of Spin(7) and SO(7) at prime 2 AUTHOR: Krzysztof Ziemianski ADDRESS: Faculty of Mathematics, Informatics and Mechanics Warsaw University Banacha 2 02-097 Warszawa POLAND ABSTRACT: A homotopy (complex) representation of a compact connected Lie group L at prime p is a map from BL into the p-completion of the classifying space of the unitary group. In this paper we give a partial classification of homotopy representations of SO(7) and Spin(7) at prime 2. Motiviation for considering this problem is twofold: first, one may hope that it would help to understand maps between classifying spaces. Secondly, construction of a homotopy representation of Spin(7) is a crucial step in the construction of a faithful representation of the 2-compact group DI(4). ------------------ There are 4 new papers this time, from Biedermann-Chorny-Roendigs, Bubenik-Worytkiewicz, DavisD-Theriault, and Fresse. Mark Hovey New papers appearing on hopf between 1/4/06 and 2/8/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann-Chorny-Roendigs/biedermann-chorny-roendigs Title: Goodwillie's calculus and model categories Author(s): Georg Biedermann, Boris Chorny, Oliver Roendigs Abstract: The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for classification of polynomial and homogeneous functors. Finally we show that the $n$-th derivative induces a Quillen map between the $n$-homogeneous model structure on small functors from pointed simplicial sets to spectra and the category of spectra with $\Sigma_n$-action. We consider also a finitary version of the $n$-homogeneous model structure and the $n$-homogeneous model structure on functors from pointed finite simplicial sets to spectra. In these two cases the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik-Worytkiewicz/lps title: A model category for local po-spaces author: Peter Bubenik author: Krzysztof Worytkiewicz to appear in: Homology, Homotopy and Applications abstract: Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by constructing a model category containing the category of local po-spaces. We show the category of simplicial presheaves on local po-spaces can be given Jardine's model structure, in which we identify the weak equivalences between local po-spaces. In the process we give an equivalence between the category of sheaves on a local po-space and the category of {\'e}tale bundles over a local po-space. Finally we describe a localization that should provide a good framework for studying concurrent systems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Theriault/theri6 Odd-primary homotopy exponents of compact simple Lie groups Donald M. Davis and Stephen D. Theriault We note that a recent result of the second author yields upper bounds for odd-primary homotopy exponents of compact simple Lie groups which are often quite close to the lower bounds obtained from v1-periodic homotopy theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Fresse/Bar-StructureUniqueness Title: The bar construction of an $E$-infinity algebra Author: Benoit Fresse Abstract: We consider the classical reduced bar construction of associative algebras B(A). If the product of A is commutative, then B(A) can be equipped with the classical shuffle product, so that B(A) is still a commutative algebra. This assertion can be generalized for algebras which are commutative up to homotopy. Namely, one observes that the bar construction of an E-infinite algebra B(A) can be endowed with the structure of an E-infinite algebra. The purpose of this article is to give an existence and uniqueness theorem for this claim. We would like to insist on the uniqueness property: our statement makes the construction of $E$-infinite structures easier and more flexible. Therefore, the proof of our existence theorem differs from other constructions of the literature. In addition, the uniqueness property allows to give easily a homotopy interpretation of the bar construction. ----------------- There are 4 new papers this time, from BrownR, DavisDaniel, DavisD, and Hovey. Mark Hovey New papers appearing on hopf between 2/8/06 and 3/1/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/bedlewo Title: Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions. Author: Ronald Brown AMS classification number: 01A60,53C29,81Q70,22A22,55P15 Expansion of an invited talk given to the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland). Abstract: This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/cplx2 Title: The E_2-term of the descent spectral sequence for continuous G-spectra Author: Daniel G. Davis Author's address: Purdue University Abstract: Let {X_i} be a tower of discrete G-spectra, each of which is fibrant as a spectrum, so that X=holim_i X_i is a continuous G-spectrum, with homotopy fixed point spectrum X^{hG}. The E_2-term of the descent spectral sequence for \pi_*(X^{hG}) cannot always be expressed as continuous cohomology. However, we show that the E_2-term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in lim_i M_i, where {M_i} is a tower of discrete G-modules. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/CPcrabb4 Some new immersion results for complex projective space Donald M. Davis Lehigh University, Bethlehem, PA 18015 Abstract: We prove the following two new optimal immersion results for complex projective space. First, if n equiv 3 mod 8 but n not equiv 3 mod 64, and alpha(n)=7, then CP^n can be immersed in R^{4n-14}. Second, if n is even and alpha(n)=3, then CP^n can be immersed in R^{4n-4}. Here alpha(n) denotes the number of 1's in the binary expansion of n. The first contradicts a result of Crabb, who said that such an immersion does not exist, apparently due to an arithmetic mistake. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/injective-comod Injective comodules and Landweber exact homology theories Mark Hovey Wesleyan University Middletown, CT We classify the indecomposable injective E(n)_{*}E(n)-comodules, where $E(n)$ is the Johnson-Wilson homology theory. They are suspensions of the J_{n,r}, where J_{n,r} is the E(n)-homology of the rth monochromatic piece M_{r} E(r) of E(r) and $0\leq r\leq n$. The endomorphism ring of J_{n,r} is the ring of operations in the completed E(r) theory; this ring of operations is not really known so far as I know, though it is closely related to the stabilizer group S_r. An interesting byproduct of this study is the isomorphism E^{*}(X) = \Hom_{E(n)_{*}} (E(n)_{*}M_{n}X, K) where E is completed E(n) theory and K is the n-fold desuspension of E(n)_{*}/I_{n}^{\infty}). ----------------- There are 4 new papers this time, from Blanc-Johnson-Turner, Clarke-Crossley-Whitehouse, Muro-Tonks, Ziemianski. I also wanted to say that in my paper of last time, there is an isomorphism between the completed E(n)-cohomology of X and Hom from the E(n)-homology of M_n X to an appropriate module. This isomorphism was known before to Greenlees, Hopkins, Sadofsky, and others, though it does not appear to be in print. The version of the paper now on the archive reflects that. Mark Hovey New papers appearing on hopf between 3/1/06 and 4/7/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/rdpa Title: On Realizing Diagrams of Pi-algebras Authors: David Blanc, Mark W. Johnson, and James M. Turner Abstract: Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun by Dwyer, Kan, and Stover to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Clarke-Crossley-Whitehouse/ccwDiscrete The discrete module category for the ring of K-theory operations Francis Clarke, Martin Crossley, Sarah Whitehouse We study the category of discrete modules over the ring of degree zero stable operations in p-local complex K-theory. We show that the p-local K-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K-local stable homotopy category (Amer. J. Math., 1985). We also provide an alternative characterisation of discrete modules as locally finitely generated modules. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Tonks/1tK3 Title:The 1-type of a Waldhausen K-theory spectrum Authors: Fernando Muro and Andrew Tonks Abstract: We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Ziemianski/DI4Rep TITLE: A faithful unitary representation of the 2-compact group DI(4) AUTHOR: Krzysztof Ziemianski ABSTRACT: We construct a monomorphism from the $2$-compact group $DI(4)$ into a $2$-compact unitary group. ---------------- ------------------------------ There are 7 new papers this time, from Kuhn, Neusel-Sezer (2), Pengelley-Williams, RadulescuBanu, and SanchezGarcia (2) Mark Hovey New papers appearing on hopf between 4/7/06 and 5/3/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/autgrp Title: The nilpotent filtration and the action of automorphisms on the cohomology of finite $p$--groups Author: Nicholas J. Kuhn AMS classification number: 20J06 Abstract: We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p=2, but is reduced to a question about 2--central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2--groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less. Featured is the nilpotent filtration of the category of unstable A--modules. Also featured are unstable algebras of cohomology primitives associated to central group extensions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-I The Noether Map I Mara D Neusel and M"ufit Sezer Abstract: Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a finite group G. In this paper we study the image of the associated Noether map $ \eta_G^G: F[V(G)]^G \longrightarrow F[V]^G. $ It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure $\overline{Im(\eta_G^G)} =F[V]^G$. This is true without any restrictions on the group, representation, or ground field. Moreover, we show that the extension $Im (\eta_G^G) \subseteq F[V]^G$ is a finite $p$-root extension. Furthermore, we show that the Noether map is surjective, i.e., its image integrally closed, if $V=F^n$ is a projective $FG$-module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of $F[V]^G$ and the Cohen-Macaulay defect of $F[V]^G$. We illustrate our results with several examples. Note that this paper together with noether-map-II contain stronger results than the authors' previous paper Neusel-Sezer/noether. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-II The Noether Map II Mara D Neusel and M"ufit Sezer Abstract: Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map \[ \eta_G^G: F[V(G)]^G \longrightarrow F[V]^G. \] In [Noether Map I] it has been shown that the Noether map is surjective if $V$ is a projective $FG$-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for $p$-groups (where $p$ is the characteristic of the ground field $F$) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of $V$. Note that this paper together with noether-map-I contain stronger results than the authors' previous paper Neusel-Sezer/noether. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/peng-will-oddkam The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations, and invariant theory David J. Pengelley and Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 16W22; Secondary 16W30, 16W50, 55S10, 55S12, 55S99, 57T05. We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for p=2 , in particular in the nature and consequences of the defining Adem relations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/RadulescuBanu/cofib-cat Title: Cofibrations in Homotopy Theory Author: Andrei Radulescu-Banu Author's mailing address: 86 Cedar St, Lexington, MA 02421 Abstract: We define Anderson-Brown-Cisisnski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibraction categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a right derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for left derivators. These constructions provide a natural framework for 'doing homotopy theory' in ABC (co)fibration categories. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/bredon Title: Bredon homology and equivariant K-homology of SL(3,Z) Author: Ruben Sanchez-Garcia Author's address: Department of Pure Maths, Hicks Building University of Sheffield Sheffield S3 7RH, United Kingdom Included ps or eps files: SouleFundamentalDomainLabelled.eps TruncatedCube.eps AMS classification number: 19L47, 55N91 (Primary); 19K99, 46L80 (Secondary) Other useful information: arXiv:math.KT/0601587 Abstract: We obtain the equivariant K-homology of the classifying space \underline{E}SL(3,Z) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for GL(3,Z). Our calculations give therefore the topological side of the Baum-Connes conjecture for these groups. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/coxeter Title: Equivariant K-homology for some Coxeter groups Author: Ruben Sanchez-Garcia Author's address: Department of Pure Maths, Hicks Building University of Sheffield Sheffield S3 7RH, United Kingdom Included eps files: hexagon3.eps hexagon4.eps interval.eps tessellation0.eps trianglesd2.eps AMS classification number: 19L47, 55N91 (Primary); 19K99, 46L80 (Secondary) Other useful information: arXiv:math.KT/0604402 Abstract: We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly map. --------------- There are 7 new papers this time, from Bartels-Rosenthal, Chermak-Oliver-Shpectorov, Dwyer-Wilkerson, Jardine (2), Stacey-Whitehouse, and Wilkerson. Mark Hovey New papers appearing on hopf between 5/3/06 and 6/5/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Rosenthal/asymptotic Authors: Arthur Bartels, David Rosenthal arXiv submission number: math.KT/0605088 Abstract: It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the classifying space for proper actions. The result also applies to certain groups that admit only a finite dimensional model for this space. In particular, it applies to discrete subgroups of virtually connected Lie groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chermak-Oliver-Shpectorov/fundsol The simple connectivity of $B\Sol(q)$ by Andrew Chermak, Bob Oliver, and Sergey Shpectorov Andrew Chermak Kansas State University Bob Oliver LAGA, Institut Galil\'ee Sergey Shpectorov University of Birmingham Abstract: A $p$-local finite group is an algebraic structure which includes two categories, a fusion system and a linking system, which mimic the fusion and linking categories of a finite group over one of its Sylow subgroups. The $p$-completion of the geometric realization of the linking system is the classifying space of the finite group. In this paper, we study the geometric realization, \emph{without} completion, of linking systems of certain exotic 2-local finite groups whose existence was predicted by Solomon and Benson, and prove that they are all simply connected. The file "Co3graph.eps" must be included with the dvi file. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS W. G. DWYER AND C. W. WILKERSON In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case. 1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra on V # the k-dual of V, and R = S^W the ring of invariants of under the natural action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators. Steinberg [9] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S\tensor_R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4] to other fields of characteristic zero, and by T.-C. Lin [5] to the case in which k is a finite field of characteristic prime to the order of W . The current proof is independent of previous methods. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/coc-cat3 Title: Cocycle categories Author: J.F. Jardine arXive submission number: math.AT/0605198 Abstract: A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and satisfies the extra condition that weak equivalences are closed under finite products. Various applications of this result are displayed, including the homotopy classification of torsors, abelian cohomology groups, group extensions and gerbes. The older classification results have simple new proofs involving canonically defined cocycles. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/gerbes6 Title: Homotopy classification of gerbes Author: J.F. Jardine arXive submission number: math.AT/0605200 Abstract: Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined on BF and taking values in the classifying space for the stack completion of the fundamental groupoid of F. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Other useful information: math.AT/0605471 Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/newfred Loop Spaces and Finiteness Clarence W. Wilkerson Purdue University This expository note began as comments on a shorter note of F.R. Cohen \cite{cohen}. Cohen's paper is an elegant application of powerful recent results in unstable homotopy theory to a problem of interest to analysts. {\sl {\bf Theorem :} (F.\,R. Cohen,\cite{cohen}) Let $X$ be a simply connected finite complex which is not contractible and let $\Omega^j_0X$ be the component of the constant map in the $j$-th pointed loop space of $X$. If $j \geq 2$, then the Lusternik-Schnirlman category of $\Omega^j_0X$ is not finite. }\\ This note includes a rederivation of the above theorem using H-space methods of W. Browder from the 60's, \cite{Browder-loop}, \cite{Browder-Torsion}. The aim is to reduce the prerequisites for Cohen's theorem to those available after a second course in algebraic topology. We end with a discussion of recent work of Lannes-Schwartz on various notions of finiteness properties and the behavior under looping. The common theme is extensive use of the action of the Steenrod algebra on the cohomology of a topological space. --------------- ----- There are 7 new papers this time, from Biedermann, Blanc, Oliver-Ventura, Shipley, Stacey-Whitehouse, Wuethrich, and YauD. Mark Hovey New papers appearing on hopf between 6/5/06 and 7/8/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann/presh-n-types Title: On the homotopy theory of n-types Author: Georg Biedermann Mail address: Dep. of Mathematics, Middlesex College, UWO, London, Ontario, N5X 2W8, Canada Abstract: We achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of groupoids enriched in simplicial sets. This can be viewed as a different description of the homotopy theory of higher hyperstacks. As a special case we obtain a good substitute for the homotopy theory of (weak) higher groupoids. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/comp Title: Comparing homotopy categories Author: David Blanc Address: Department of Mathematics University of Haifa 31905 Haifa Israel Abstract: Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C. Examples include finding spaces with given homology or homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov1 Extensions of linking systems with $p$-group kernel Bob Oliver and Joana Ventura LAGA Departamento de Matem\'atica Institut Galil\'ee Instituto Superior T\'ecnico Av. J-B Cl\'ement Av. Rovisco Pais 93430 Villetaneuse, France 1049--001 Lisboa, Portugal bobol@math.univ-paris13.fr jventura@math.ist.utl.pt Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: Classifying space, $p$-completion, finite groups, fusion. Abstract: We study extensions of $p$-local finite groups where the kernel is a $p$-group. In particular, we construct examples of saturated fusion systems $\calf$ which do not come from finite groups, but which have normal $p$-subgroups $A\nsg\calf$ such that $\calf/A$ is the fusion system of a finite group. One of the tools used to do this is the concept of a ``transporter system'', which is modelled on the transporter category of a finite group, and is more general than a linking system. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Shipley/zdga17 Title: HZ-algebra spectra are differential graded algebras Author: Brooke Shipley Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. Resubmitted to correct font generation problem with the conversion to postscript and PDF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings Title: Infinitesimal thickenings of Morava K-theories Author: Samuel Wuethrich Abstract: A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/GD2 Title: Gerstenhaber structure and Deligne's conjecture for Loday algebras Author: Donald Yau Abstract: A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure-Smith to prove an analogue of Deligne's conjecture for Loday algebras. ----------------- ----------------------------------- There are 7 new papers this time, from Arone-Lambrechts-Volic, Broto-Levi-Oliver, Dwyer-Wilkerson, Gillespie, Naumann, Ulrich-Wilkerson, and YauD. Mark Hovey New papers appearing on hopf between 7/8/06 and 8/4/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lambrechts-Volic/CalculusFormalityEmbeddings Title: Calculus of functors, operad formality, and rational homology of embedding spaces Authors: Gregory Arone, Department of Mathematics, University of Virginia, Charlottesville, VA, USA. Pascal Lambrechts Institut Math\'{e}matique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium Ismar Voli\'c Department of Mathematics, University of Virginia, Charlottesville, VA, USA Abstract: Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orthogonal tower'') of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E^1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor HQ /\Ebar(M,V)_+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that under the above assumption on codimension, the rational homology groups of Ebar(M,V) are determined by the rational homotopy type of M. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo3 Title of paper: Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups Authors: Carles Broto, Ran Levi, Bob Oliver AMS Classification: Primary 55R35. Secondary 55R40, 57T10 Addresses of authors: Departament de Matem\`atiques Universitat Aut\`onoma de Barcelona E--08193 Bellaterra, Spain Department of Mathematical Sciences University of Aberdeen, Meston Building 339 Aberdeen AB24 3UE, U.K. LAGA, Institut Galil\'ee Av. J-B Cl\'ement 93430 Villetaneuse, France Abstract: We define and study a certain class of spaces which includes $p$-completed classifying spaces of compact Lie groups, classifying spaces of $p$-compact groups, and $p$-completed classifying spaces of certain locally finite discrete groups. These spaces are determined by fusion and linking systems over ``discrete $p$-toral groups'' --- extensions of $(\Z/p^\infty)^r$ by finite $p$-groups --- in the same way that classifying spaces of $p$-local finite groups as defined in \cite{BLO2} are determined by fusion and linking systems over finite $p$-groups. We call these structures ``$p$-local compact groups''. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/PiOneHopf The fundamental group of a $p$-compact group W. G. Dwyer and C. W. Wilkerson The notion of a $p$-compact group is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of $p$-compact groups, one for each prime number~$p$. A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form. There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus. For some time there has in fact been a corresponding formula for the center of $p$-compact groups, but in general the fundamental group has eluded analysis. The purpose of the present paper is to remedy this deficit. For any space $Y$, we let $\HZp_i(Y)$ denotes $\lim{}_n\HH_i(Y;\Z/p^n)$. Suppose that $X$ is a connected $p$-compact group, with maximal torus $T$ and torus normalizer $\NT$. It is known that the map $\pi_1(T)\to\pi_1(X)$ is surjective or equivalently that the map $\HZp_2(\BB T)\to\HZp_2(\BB X)$ is surjective. We prove the following statement. Main Theorem: If $X$ is a connected $p$-compact group, then the kernel of the map $\HZp_2 \BB T\to \HZp_2\BB\NT$ is the same as the kernel of the map $\HZp_2 \BB T\to \HZp_2\BB X$. Equivalently, the image of the map $\HZp_2\BB T\to \HZp_2(\BB\NT)$ is (naturally) isomorphic to $\pi_1X$. There is a proof of the corresponding statement for compact Lie groups which relies on the Feshbach double coset formula Our proof of the MainTheorem uses a transfer calculation that in practice amounts to a weak homological reflection of the double coset formula; we can get away with this because we have a splitting of $\HZp_2(\BB\NT)$. It is possible to derive from the MainTheorem a more explicit formula for $\pi_1X$; this formula is known for $p$~odd as a consequence of the classification theorem for $p$ odd. Our demonstration does not use the classification theorem. Let $W$ denote the Weyl group of $X$. If $p$ is odd, then $\pi_1X$ is naturally isomorphic to the module of coinvariants $\HH_0(W;\HZp_2(\BB T))$ . If $p=2$, then up to factors which do not contribute to $\pi_1X$, the normalizer of the torus in $X$ is derived by $\Ftwo$-completion from the normalizer $\NT_G$ of a maximal torus $T_G$ in a connected compact Lie group~$G$ . The image of the map $\HH_2(\BB T_G;\Z)\to\HH_2(\BB\NT_G;\Z)$ is isomorphic to $\pi_1G$ , and so by the MainTheorem the tensor product of this image with $\Ztwo$ is $\pi_1X$. This image can be computed from the marked reflection lattice $(\pi_1T_G, \{b_\sigma,\beta_\sigma\})$ corresponding to the root system of $G$ or, after tensoring with $\Ztwo$, from the marked complete reflection lattice $(\pi_1T,\{b_\sigma,\beta_\sigma\})$ associated to $X$ The upshot is that $\pi_1X$ is the quotient of $\pi_1T=\pi_2\BB T=\HZtwo_2\BB T$ by the $\Ztwo$--submodule generated by the elements $\{b_\sigma\}$. Another way to describe this calculation is the following. For each reflection $s_\alpha$ in the Weyl group~$W$, let $u_\alpha$ be a generator over $\Zp$ of the rank~1 submodule of $\pi_1T$ given by the image of $(1-s_\alpha)$. If $p$ is odd let $v_\alpha=u_\alpha$; if $p=2$, let $v_\alpha=u_\alpha$ or $u_\alpha/2$, according to whether the marking of $s_\alpha$ is trivial or non-trivial. Then $\pi_1X$ is the quotient of $\pi_1T$ by the $\Zp$-span of the elements~$v_\alpha$. See the upcoming even classification by Andersen and Grodal for more details. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie/quasi-coherent Title: A Quillen Approach to Derived Categories and Tensor Products Author: James Gillespie AMS Classification numbers: 55U35, 18G15, 18E30 4000 University Drive Penn State McKeesport McKeesport, PA 15132 Abstract: We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies methods used by the author to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category G, a nice enough class of objects, which we call a Kaplansky class, induces a model structure on the category Ch(G) of chain complexes. We also find simple conditions to put on the Kaplansky class which will guarantee that our model structure in monoidal. We see that the common model structures used in practice are all induced by such Kaplansky classes. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Naumann/qisoneu Niko Naumann Quasi-isogenies and Morava stabilizer groups For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent interest. For example, it gives a precise solution to the problem of extending automorphisms of the $p$-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ulrich-Wilkerson/uw06rev1 Field degrees and multiplicities for non-integral extensions Bernd Ulrich Clarence W. Wilkerson Department of Mathematics, Purdue University, West Lafayette, IN 47907 Department of Mathematics, Purdue University, West Lafayette, IN 47907 ulrich@math.purdue.edu cwilkers@purdue.edu Let $k$ be a field and $S = k[t_1,\hdots,t_d]$ a polynomial ring with variables $t_i$ of degree one. Consider a $k$-subalgebra $R$ generated by $m$ homogeneous elements $\{x_1,\hdots,x_m\}$. In general, if $x$ is a homogeneous element in a graded object, we denote its degree by $|x|$. {\bf Problem.} {\it Let $[S:R]$ denote the degree of the underlying fraction field extension. If $S$ is algebraic over $R$, calculate $[S:R]$ from the $\{|x_i|\}$ }. First, one has a form of Bezout's Theorem: \begin{thm}\label{BezoutsThm} If $S$ is integral over $R$, the following hold: \begin{enumerate} \item $[S:R]$ divides $\prod{|x_i|}$. \item If $m=d$, then $[S:R] = \prod{|x_i|}$. \end{enumerate} \end{thm} In this paper, we consider the case that $m = d$ and obtain a converse to part (b) above: \begin{thm}\label{MainTheorem} If $S$ is algebraic over $R$, $m=d$, and $[S:R] = \prod{|x_i|}$, then $S$ is integral over $R$ $($equivalently, $S$ is finitely generated as an $R$-module$)$. \end{thm} We also note that if $S$ is not integral over $R$, then $[S:R]$ need not even divide $\prod{|x_i|}$. Our proofs rely on reduction to the case of standard graded $k$-algebras. An interesting application of Theorem 1.2 is in the study of rings of invariants of finite groups acting on a polynomial ring: \begin{thm}\label{Invariants} Let $V$ be a $d$-dimensional vector space over the field $k$, $V^\#$ its $k$-dual, and $S = S[V^\#] = k[t_1,\hdots,t_d]$ the algebra of polynomial functions on $V$. Let $W \subset GL(V)$ be a finite group. There is an induced action on $S$. Then $S^W = R$ is a polynomial algebra over $k$ if and only if there exist homogeneous elements $\{x_1, \hdots,x_d\}$ of $R$ such that \begin{enumerate} \item $S$ is algebraic over $k[x_1,\hdots,x_d]$, and \item $|W| = \prod{|x_i|}$. \end{enumerate} \end{thm} 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/module_alg Title: Cohomology and deformation of module-algebras Author: Donald Yau Email: dyau@math.ohio-state.edu Abstract: An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered. --------------- ----------------------------- I took a month off; sorry about the delay. There are 9 new papers this time, from Bergner, Chebolu-Christensen-Minac, DavisDaniel, Dugger-Isaksen, Fausk (2), GrayB, Hovey-Lockridge-Puninski, and Wuethrich. Mark Hovey New papers appearing on hopf between 8/4/06 and 10/6/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/ReedyFib Title: A characterization of fibrant Segal categories Author: Julia E. Bergner AMS Classification: 55U35, 18G30 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure Secat_c. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories which are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure Secat_f. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghost [Note: I had trouble with the dvi file of this paper. I expect to have it up by 10/7/06--Mark] TITLE: Ghosts in modular representation theory AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42 ABSTRACT: We study ghosts in the stable module category of a finite group. That is, we study maps between modular representations of finite groups which are invisible in Tate cohomology. We establish various sets of conditions which guarantee the existence of a non-trivial ghost out of a given representation. We then investigate the generating hypothesis which is the statement that there are no non-trivial ghosts between finite-dimensional representations. This is done by focusing on three quintessential examples: the cyclic $p$-groups (finite representation type), the Klein four group (domestic representation type), and the quaternion groups (tame representation type). In the examples where the generating hypothesis fails, we obtain bounds on the ghost number: the smallest integer $l$ such that the composition of any $l$ ghosts between finite-dimensional representations is trivial. In particular, we obtain bounds on the ghost numbers for all $2$-groups which have a cyclic subgroup of index $2$. Projective classes in the stable module category play a key role in getting these bounds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg3 Title: The homotopy orbit spectrum for profinite groups Author: Daniel G. Davis Abstract: Let G be a profinite group. We define an S[[G]]-module to be a G-spectrum X that satisfies certain conditions, and, given an S[[G]]-module X, we define the homotopy orbit spectrum X_{hG}. When G is countably based and X satisfies a certain finiteness condition, we construct a homotopy orbit spectral sequence whose E_2-term is the continuous homology of G with coefficients in the graded profinite Z[[G]]-module pi_*X. Let G_n be the extended Morava stabilizer group and let E_n be the Lubin-Tate spectrum. As an application of our theory, we show that the function spectrum F(E_n,L_{K(n)}(S^0)) is an S[[G_n]]-module with an associated homotopy orbit spectral sequence. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/etdq Title: Etale homotopy and sums-of-squares formulas Authors: Daniel Dugger, Daniel C. Isaksen AMS classification number: 55P60, 15A63 Abstract: This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p > 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized etale cohomology theory called etale BP2. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk/ArtinBrauer Generalized Artin and Brauer induction for compact Lie groups Halvard Fausk Abstract: Let $G$ be a compact Lie group. We present two induction theorems for certain generalized $G$-equivariant cohomology theories. The theory applies to $G$-equivariant $K$-theory $K_G$, and to the Borel cohomology associated to any complex oriented cohomology theory. The coefficient ring of $K_G$ is the representation ring $R(G)$ of $G$. When $G$ is a finite group the induction theorems for $K_G$ coincide with the classical Artin and Brauer induction theorems for $R(G)$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk/Gspectra-Fausk Title: Equivariant homotopy theory for pro--spectra Author: Halvard Fausk Abstract. We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The $G-$homotopy theory is ``pieced together'' from the $G/U-$homotopy theories for suitable quotient groups $G/U$ of $G$; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In this category Postnikov towers are studied from a general perspective. We introduce pro$-G-$spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We give a careful discussion of two version of a model structure with ``underlying weak equivalences''. One of the versions only make sense for pro$-$spectra. In the end we use the theory to study homotopy fixed points of pro$-G-$spectra. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/fiber Filtering the fiber of the pinch map Brayton Gray This paper develops the similarity between the loops on an odd dimensional sphere and the fiber F of the pinch map from an odd dimensional mod p^r Moore space to the sphere, for p odd. In particular, a Hopf invariant map is defined and there is an EHP sequence up to a factor which is the loops on a bouquet of higher dimensiona Moore spaces. As a consequence we have two technical results about the mysterious connecting map from the double loops on the sphere to the loops on F. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge-Puninski/derived-gen-hyp Title: The generating hypothesis in the derived category of a ring. Authors: Mark Hovey, Keir Lockridge, and Gena Puninski Abstract: We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular, and therefore does not satisfy the strong form of the generating hypothesis. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings_rev Title: Infinitesimal thickenings of Morava K-theories (revised version) Author: Samuel Wuethrich AMS classification number: 55P42, 55P43; 55U20, 55N22 Abstract: This is a revised version. A few points have been clarified and some typos have been corrected. A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well. ----------------- ------------------------ There are 4 new papers this time, from Chebolu-Christensen-Minac, DavisDaniel, Stacey-Whitehouse, and Yagita. Mark Hovey New papers appearing on hopf between 10/6/06 and 11/5/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/GH-StMod TITLE: Groups which do not admit ghosts AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42 ABSTRACT: A ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups of order 2 and 3. We compare this to the situation in the derived category of a commutative ring. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/siterplusg Title: The site R^+_G for a profinite group G Author: Daniel G. Davis AMS classification number: 55P42, 55U35, 18B25 Abstract: Let G be a non-finite profinite group and let G-Sets_{df} be the canonical site of finite discrete G-sets. Then the category R^+_G, defined by Devinatz and Hopkins, is the category obtained by considering G-Sets_{df} together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R^+_G is a site when equipped with the pretopology of epimorphic covers. Also, we explain why the associated topology on R^+_G is not subcanonical, and hence, not canonical. We note that, since R^+_G is a site, there is automatically a model category structure on the category of presheaves of spectra on the site. Finally, we point out that such presheaves of spectra are a nice way of organizing the data that is obtained by taking the homotopy fixed points of a continuous G-spectrum with respect to the open subgroups of G. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse AMS classification number: 55S25, 55P47 Other useful information: math.AT/0605471 Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the Bousfield-Kuhn functor. This is an updated version of an earlier submission. The proof of proposition 3.2 has been corrected; other minor improvements have been made. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/abp Algebraic BP-theory and norm varieties Nobuaki Yagita Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan Primary 14C15, 57T25; Secondary 55R35, 57T05 Let X be a smooth variety over a field k of characteristic zero. For a fixed prime p, the algebraic BP-theory ABP(X) is the algebraic version of the topological BP-theory. Given a nonzero symbol a in K_{n+1}^M (k)/p, the norm variety V_a is a variety such that a=0 in K_{n+1}^M (k(V_a))/p and V_a(C)=v_n. In this paper, we mainly study ABP(V_a) for p an odd prime. -------------There are 8 new papers this time, from Andersen-Grodal (the completion of the classification theorem for p-compact groups!), Benson-Chebolu-Christensen-Minac, BrownR-Sivera, Bousfield, Kadzisa-Mimura, Kuhn, Morel, and Snaith. Mark Hovey New papers appearing on hopf between 11/5/06 and 12/7/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Andersen-Grodal/2classification Title: The classification of 2-compact groups Authors: Kasper K. S. Andersen and Jesper Grodal Abstract: We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group. Combined with our earlier work with Moeller and Viruel for p odd, this establishes the full classification of p-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over the p-adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson-Chebolu-Christensen-Minac/GH-pgroup-new Title: Freyd's generating hypothesis for the stable module category of a $p$-group Authors: David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac. Abstract: Freyd's generating hypothesis, interpreted in the stable module category of a finite $p$-group $G$, is the statement that a map between finite-dimensional $kG$-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial $p$-group $G$ if and only if $G$ is either $\mathbb{Z}/2$ or $\mathbb{Z}/3$. We also give various conditions which are equivalent to the generating hypothesis. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/normalisation Title: Normalisation for the fundamental crossed complex of a simplicial set Author(s): Ronald Brown, Rafael Sivera R. Brown University of Wales, Bangor, Dean St., Bangor, Gwynedd LL57 1UT, U.K. R. Sivera, Departamento de Geometria y Topologia, Universitat de Valencia, 46100 Burjassot, Valencia, Spain Abstract: Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes, such as the monoidal closed structure. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/Klocal On the 2-adic K-localizations of H-spaces A.K. Bousfield Department of Mathematics University of Illinois at Chicago Chicago, IL 60607 We determine the 2-adic K-localizations for a large class of H-spaces and related spaces. As in the odd-primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary v1-periodic homotopy groups of our spaces. The present v1-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic K-cohomology algebras (including the 2-adic KO-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic K-cohomology algebras with specified operations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kadzisa-Mimura/mbflsc1 Authors: Hiroyuki Kadzisa, Mamoru Mimura Title: Morse-Bott functions and the Lusternik-Schnirelmann category, I The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used to give an upper bound for Lusternik-Schnirelmann categories of topological spaces. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C^1 and their gradient flows, and to apply the result to some homogeneous spaces to determine their Lusternik-Schnirelmann categories. In particular, the Morse-Bott functions on the Stiefel manifolds considered by Frankel are effectively used for constructing all the cone-decompositions in this paper. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/primitives2 Title: Primitives and central detection numbers in group cohomology Author: Nicholas J. Kuhn Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 abstract: Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we show that d_0(G) is bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity Conjecture holds. In particular, this holds for all groups such that the p--rank of G minus the depth of H^*(G) is at most 2. When we look at examples with p=2, we learn that d_0(G) is at most 7 for all groups with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and d_0(G)=14). Enroute we recover and strengthen theorems of Adem and Karagueuzian on essential cohomology, and Green on depth essential cohomology, and prove theorems about the structure of cohomology primitives associated to central extensions. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morel/A1homotopy A^1-algebraic topology over a field Fabien Morel Mathematisches Institut der Universität München Theresienstr. 39 D-80333 München Text of Abstract: In this work we prove some basic results in the context of A1-homotopy theory of smooth schemes over a field k : the analogue of the Brouwer degree, the Hurewicz theorem, the theory of A1-coverings and its relationship to the fundamental A1-homotopy sheaf, and some fundamental computations involving unramified Milnor-Witt K-theory like the fundamental A1-homotopy sheaves of P^n and SL_n . 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Snaith/UTTArf Title: Upper triangular technology and the Arf-Kervaire invariant Author: Victor Snaith address: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, England Abstract. This paper introduces the upper triangular technology (UTT) into classical homotopy theory. This is a new and easy to use method to calculate the effect of the left unit map in 2-adic connective K-theory; the map which is the basis for operations in bu-theory. By way of application, UTT is used to give a new, very simple proof of a conjecture of Barratt- Jones-Mahowald, which rephrases K-theoretically the existence of framed manifolds of Arf-Kervaire invariant one. -------------------------------------------------------------- There are 7 new papers this time, from Castellana-Crespo-Scherer, Clement-Scherer, Colman, Maltsiniotis, Matthey-Pitsch-Scherer, Nakagawa, and Ruiz. Mark Hovey New papers appearing on hopf between 12/7/06 and 1/1/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/covers Title: On the cohomology of highly connected covers of finite complexes Authors: Natalia Castellana, Juan A. Crespo, and Jerome Scherer Abstract: Relying on the computation of the Andre-Quillen homology groups for unstable Hopf algebras, we prove that the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Clement-Scherer/exponent Title: Homology exponents for H-spaces Authors: Alain Clement and Jerome Scherer Abstract: We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of the integral homology. Our main result states that if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2^r, S^1, K(Z,2), and K(Z,3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the same methods that simply connected H-spaces whose mod 2 cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod 2 finite H-spaces with copies of K(Z,2) and K(Z,3). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Colman/ColmanHTG Title of Paper: On the homotopy type of Lie groupoids Author: Hellen Colman Address of Author: Department of Mathematics, Wilbur Wright College, 4300 N. Narragansett Avenue, Chicago, IL 60634 USA Text of Abstract: We propose a notion of groupoid homotopy for generalized maps. This notion of groupoid homotopy generalizes the notions of natural transformation and strict homotopy for functors. The groupoid homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the orbifold homotopy between orbifold maps induced by the groupoid homotopy. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Maltsiniotis/adjquill Title: Le theoreme de Quillen, d'adjonction des foncteurs derives, revisite Author: Georges MALTSINIOTIS English Translation: math.AT/0611952 Address: Université Paris 7 Denis Diderot Case Postale 7012 2, place Jussieu F-75251 PARIS CEDEX 05 Abstract: The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for "absolute" derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Matthey-Pitsch-Scherer/Blochlift Title: Generalized orientations and the Bloch invariant Authors: Michel Matthey, Wolfgang Pitsch, and Jerome Scherer Abstract: For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K_3(C). Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We also discuss the non-compact case, in which there appears a Z/2-ambiguity. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Nakagawa/cohomologyE8 Title: The integral cohomology ring of E_8/T^1 E_7 Author: Masaki Nakagawa Address of author: Department of General Education, Takamatsu National College of Technology, 355 Chokushi-cho, Takamatsu, 761-8058, Japan Abstract: The generalized flag manifolds are homogeneous spaces of the form G/C, where G is a compact connected Lie group and C is the centralizer of a torus in G. These homogeneous spaces play an important role in algebraic topology, algebraic geometry and differential geometry. In this paper, using the Borel presentation and a method due to Toda, we determine the integral cohomology ring of a certain generalized flag manifold which is a quotient space of the exceptional Lie group E8. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ruiz/ar-gl-rv Title: Exotic normal fusion subsystems of General Linear Groups. Author: Albert Ruiz Institution: Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Cerdanyola del Valles, Spain. Abstract: We classify the saturated fusion subsystems of index prime to $p$ of the general linear group over $F_q$ over a Sylow $p$-subgroup, where $q$ is a prime power prime to an odd prime $p$. In this classification we get some of the exotic $p$-local finite groups discovered by C. Broto and J. Moller as saturated fusion subsystems of the general linear group. --------------------------------------------------------------------- There are 4 new papers this time, from Bartels-Lueck-Reich, Davis-Dula-Mahowald, and Vespa (2). In adddition, there are 3 updates of papers recently posted to Hopf; I will just list these rather than including the abstracts again. They are Benson-Chebolu-Christensen-Minac/GH-pgroup-new Chebolu-Christensen-Minac/GH-Stmod Kuhn/primitives Mark Hovey New papers appearing on hopf between 1/1/07 and 2/3/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Lueck-Reich/blr-hyperbolic Title: The K-theoretic Farrell-Jones Conjecture for hyperbolic groups Authors: Arthur Bartels, Wolfgang Lueck, Holger Reich Abstract: We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Davis-Dula-Mahowald/imms2 Immersions of RP^{2^e-1} Donald M. Davis, Giora Dula, and Mark Mahowald Abstract: We prove that RP^{2^e-1} cannot be immersed in R^{2^{e+1}-e-8} provided e>6. If e>13, this is 2 better than previously known immersions. Our method is primarily an induction on geometric dimension, incorporating also sections obtained from the Radon-Hurwitz theorem. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Vespa/Fquad Generic representations of orthogonal groups: the functor category Fquad Christine Vespa In this paper, we define the functor category Fquad associated to vector spaces over the field with two elements equipped with a quadratic form. We show the existence of a fully-faithful, exact functor from F to Fquad, which preserves simple objects, where F is the category of functors from the category of finite dimensional vector spaces over the field with two elements to the category of all vector spaces. We define a subcategory Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully-faithful functor which preserves simple objects. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Vespa/mixtes Generic representations of orthogonal groups: the mixed functors Christine Vespa In previous work, we defined the category of functors Fquad, associated to vector spaces over the field with two elements equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category Fquad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category Fquad. --------------- I am running late this month. There are 3 new papers this time, fron BrownR, Muro, and SmithL. Mark Hovey New papers appearing on hopf between 2/3/07 and 3/19/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/glob-gpds2 AUTHOR: Ronald Brown AUTHOR ADDRESS: School of Computer Science, University of Wales, Dean St., Bangor, Gwynedd, LL57 1UT, UK; TITLE: A new higher homotopy groupoid: the fundamental globular omega-groupoid of a filtered space MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25. KEY WORDS: filtered space, higher homotopy van Kampen theorem, cubical singular complex, free globular groupoid xxxLANL archive: math.AT/0702677 2 eps files, 19 pages ABSTRACT: We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of globular omega--groupoid. The proofs use an analogous fundamental cubical omega--groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular omega-groupoid on one element of dimension n is the fundamental crossed complex of the n-globe. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro/tcwm6 Title: A triangulated category without models Author(s): Fernando Muro Author's mailing address: Universitat de Barcelona, Facultat de Matemŕtiques, Departament d'Ŕlgebra i Geometria, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain AMS classification number: 18E30, 55P42, 16E40 Abstract: We exhibit a triangulated category which is neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/13a50 (This abstract was written by Mark) Reference list on Invariant Theory Author: Larry Smith AMS Code: 13A50 Invariant Theory Address: Mathematisches Institut Bunsenstrasse 3--5 D 37073 Goettingen Federal republic of germany Abstract: This is a list of references in invariant theory. The .tex file is included so that one can import references into a document. The journals.tex file includes macros for journals. -------------------------------------------------- 6 new papers this month, from Benson, Chebolu-Christensen-Minac, DavisD-Mahowald, Muro-Schwede-Strickland, Oliver-Ventura, and Panin-Pimenov-Roendigs. Mark Hovey New papers appearing on hopf between 3/19/07 and 4/19/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson/loops An algebraic model for chains on $\Omega BG\phat$ Dave Benson Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE Abstract: We provide an interpretation of the homology of the loop space on the $p$-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If $f$ is an idempotent in $kG$ such that $f.kG$ is the projective cover of the trivial module $k$, and $e=1-f$, then we exhibit isomorphisms for $n\ge 2$: H_n(\Omega BG\phat;k) \cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG) H^n(\Omega BG\phat;k) \cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG). Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghostnumber TITLE: Ghosts in modular representation theory AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42 ABSTRACT: A \emph{ghost} over a finite group $G$ is a map between modular representations of $G$ which is invisible in Tate cohomology. Motivated by the failure of the \emph{generating hypothesis}---the statement that ghosts between finite-dimensional $G$-representations factor through a projective---we define the \emph{compact ghost number} of $kG$ to be the smallest integer $l$ such that the composition of any $l$ ghosts between finite-dimensional $G$-representations factors through a projective. In this paper we study ghosts and the compact ghost numbers of $p$-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation $k$, holds for all $p$-groups. We do this by proving that a map between finite-dimensional $G$-representations is a ghost if and only if it is a \emph{dual ghost}. We then compute the compact ghost numbers of all cyclic $p$-groups and all abelian $2$-groups with $C_2$ as a summand. We obtain bounds on the compact ghost numbers for abelian $p$-groups and for all $2$-groups which have a cyclic subgroup of index $2$. Using these bounds we determine the finite abelian groups which have compact ghost number at most $2$. %Finally, using universal ghosts, we establish various sets of conditions which %guarantee the existence of a non-trivial ghost out of a $G$-representation. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory. COMMENTS: This version replaces an earlier one with file name ghost.tex. This is a substantial improvement with many new results and major reorganisation of the paper. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Mahowald/overlook Nonimmersions of RP^n implied by tmf, revisited Donald M. Davis and Mark Mahowald In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Schwede-Strickland/tcwm17 Author(s): Fernando Muro, Stefan Schwede, Neil Strickland Abstract: We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov2 Saturated fusion systems over $2$-groups Bob Oliver & Joana Ventura AMS classification: Primary 20D20. Secondary 20D45, 20D08 Abstract: We develop methods for listing, for a given 2-group $S$, all nonconstrained centerfree saturated fusion systems over $S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $2^7$ and two of order $2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Panin-Pimenov-Roendigs/BGL-post Author: Ivan Panin Author2: Konstantin Pimenov Author3: Oliver Roendigs Title: On Voevodsky's algebraic K-theory spectrum BGL Under a certain normalization assumption we prove that the Voevodsky's spectrum BGL which represents algebraic $K$-theory is unique over the integers. Following an idea of Voevodsky, we equip the spectrum BGL with the structure of a commutative ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over the integers We pull this structure back to get a distinguished monoidal structure on BGL for an arbitrary Noetherian base scheme. ------------------------------------------------------------ 4 new papers this month, from Barge-Lannes, Biedermann, Bubenik, and Devinatz. Mark Hovey New papers appearing on hopf between 4/19/07 and 5/14/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Barge-Lannes/SMB Title: Suites de Sturm, indice de Maslov et p\'e riodicit\'e de Bott Authors: Jean Barge and Jean Lannes Abstract: This memoir presents a reworking of a very classical subject; it is related to works of many people, especially: Richard W. Sharpe, Max Karoubi, Andrew Ranicki, Fran\c{c}ois Latour... We explain in particular how the usual theory of Sturm sequences is linked to the fundamental theorem of hermitian K-theory (due to Karoubi) and to Bott periodicity. Keywords: Sturm sequences, Maslov index, Bott periodicity, hermitian K-theory. AMS classification: 19G38, 19C99. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann/L-stable L-stable functors by Georg Biedermann We generalize and greatly simplify the approach of Lydakis and Dundas-R\"ondigs-{\O}stv{\ae}r to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V=M=S_* pointed simplicial sets and L=S^1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik/sep Author: Peter Bubenik Title: Separated Lie models and the homotopy Lie algebra AMS classification number: Primary 55P62; Secondary 17B55 to appear in the Journal of Pure and Applied Algebra Abstract: The homotopy Lie algebra of a simply connected topological space, X, is given by the rational homotopy groups on the loop space of X. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property we call separated. The homology of a separated dgL has a particular form which lends itself to calculations. We give connections to the radical of the homotopy Lie algebra and the Avramov-Felix conjecture. Examples that are worked out in detail include wedges of spheres on any "thickness" and connected sums of products of spheres. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/towardsfiniteness Title: Towards the finiteness of the homotopy groups of the K(n)-localization of S^0. Author: Ethan S. Devinatz Abstract: Let G be a closed subgroup of the nth Morava stabilizer group S_n, n>1, and let E_n^{hG} denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. If G=, the subgroup topologically generated by an element z in the p-Sylow subgroup S_n^0 of S_n, and z is non-torsion in the quotient of S_n^0 by its center, we prove that the E_n^{h}-homology of any K(n-2)-acyclic finite spectrum annihilated by p is of essentially finite rank. (The definition of essentially finite rank is given in the paper.) We also show that the units in the coefficient ring of E_n which are fixed by z are just the units in the Witt vectors with coefficients in the field of p^n elements. If n=2 and p>3, we show that, if G is a closed subgroup of S_n^0 not contained in the center, then G contains an open subnormal subgroup U such that the mod(p) homotopy of E_n^{hV} is of essentially finite rank, where V is the product of U with the units in the field of p elements. --------------------------------------------------------------------------------------- 4 new papers this month, from Arone-Dwyer-Lesh, Bendersky-DavisD, Karoubi, and Wuethrich. Mark Hovey New papers appearing on hopf between 5/14/07 and 6/8/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Dwyer-Lesh/LoopStructuresTaylorTowers Title Loop structures in Taylor towers Authors G. Z. Arone, W. G. Dwyer, K. Lesh Kerchof Hall, U. of Virginia, P.O. Box 400137, Charlottesville VA 22904 USA Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 Department of Mathematics, Union College, Schenectady, NY 12308 Abstract We study spaces of natural transformations between homogeneous functors in Goodwillie's calculus of homotopy functors and in Weiss's orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU(V). The interest in such deloopings stems from conjectures made by the first and the third author in a 2007 paper that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/DW2 v1-periodic homotopy groups of the Dwyer-Wilkerson space Martin Bendersky Donald M. Davis Abstract: The Dwyer-Wilkerson space DI(4) is the only exotic 2-compact group. We compute its v1-periodic homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Karoubi/Karoubi Cochaines quasi-commutatives en Topologie Algebrique Max Karoubi Abstract : We describe a new category of "quasi-commutative" DGA's , called D*, where the product is "almost" commutative : it is commutative on a subcomplex of C = D* tensor D* (with some axioms). To each simplicial set (or even ringed space) we associate a quasi-commutative DGA, from which we recover the homotopy type and are able to describe an explicit procedure to "compute" homotopy groups and cohomology operations. The basic idea of the construction is to use difference calculus, instead of differential calculus as in Sullivan's theory. This paper is an extension of ideas posted in the Archives a few years ago under the title "Methodes quantiques en Topologie Algebrique". However, the point of view is simpler and the proofs are now complete. It is going to appear in the Quarterly Journal of Pure and Applied Math. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings_final Title: Infinitesimal thickenings of Morava K-theories (final version) Author: Samuel Wuethrich AMS classification number: 55P42, 55P43; 55U20, 55N22 arXive submission number: math.AT/0607110 Comments: 25 pages. Final version, to appear in J. Pure Appl. Algebra. Contents of former section 5 mostly rewritten and reorganized into two sections; some minor corrections and changes Abstract: A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well. ------------------------------------------------------------- 5 new papers this month, from Bisson-Tsemo, ChornyB, and Neusel(3). Mark Hovey New papers appearing on hopf between 6/8/07 and 8/13/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bisson-Tsemo/AlgGeomOps Title: Extended powers and Steenrod operations in algebraic geometry (Preliminary Draft, July 2007 version) Authors: Terrence Bisson and Aristide Tsemo Abstract: Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/BrownRep Title: Brown representability for space-valued functors Author(s): Boris Chorny Abstract: In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weakly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hanoi Inseparable Extensions of Algebras over the Steenrod Algebra with Applications to Modular Invariant Theory of Finite Groups II author: Mara D. Neusel abstract: We continue our study of the homological properties of the purely inseparable extensions of integrally closed unstable Noetherian integral domains over the Steenrod algebra. It turns out that the projective dimension of an algebra is a lower bound for the projective dimension of its inseparable closure. Furthermore, its depth is an upper bound for the depth of its inseparable closure. Moreover, both algebras have the same global dimension. We apply these results to invariant theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid Degree bounds and the regular representation author: Mara D. Neusel abstract: Let rho : G --> GL(n , F) be a faithful representation of a finite group G. Denote by beta(F[V]^G) the maximal degree of an F-algebra generator of the ring of polynomial invariants F[V]^G in a minimal generating set. We prove the old conjecture that in the nonmodular case beta(F[V]^G)<= beta(F[FG]^G), where FG is the regular representation. Along the way we show that rings of permutation invariants that are Cohen-Macaulay always satisfy Noether's bound. Furthermore, we show that rings of invariants of sums of permutation representations that are Cohen-Macaulay are generated by polarizations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/unstable The Unstable Parts Functor and Injective Objects author: Mara D. Neusel abstract: The unstable part functor Un assigns to an arbitrary module over the Steenrod algebra the largest unstable submodule. We start by showing some general properties of this functor. Then we study the functor Un S^{-1} obtained from Un by precomposition with a localization. We show that Un S^{-1} is an exact functor from the category of unstable noetherian modules over some unstable noetherian algebra to itself. Along the lines we describe the injective objects in this category. ---------------- ----------------------------------------------- 7 new papers this month, from Ausoni-Rognes, DavisD, Gonzalez-Wilson, Kitchloo-Wilson(2), Neusel, and Neusel-Sezer. Mark Hovey New papers appearing on hopf between 8/13/07 and 9/24/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ausoni-Rognes/ausoni-rognes-kku Title: Rational algebraic K-theory of topological K-theory. Authors: Christian Ausoni and John Rognes. MSC-class: 19D55; 55N99 arXiv:0708.2160v1 [math.KT] Christian Ausoni Mathematical Institute University of Bonn John Rognes Department of Mathematics University of Oslo Abstract: We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/pcpt Homotopy type and v1-periodic homotopy groups of p-compact groups Donald M. Davis Lehigh University, Bethlehem, PA 18015 Abstract: We determine the v1-periodic homotopy groups of all irreducible p-compact groups. In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, with several exceptions, every irreducible p-compact group is a product of explicit spherically-resolved spaces which occur also as factors of p-completed Lie groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Wilson/products The ${BP}$-theory of two-fold products of projective spaces Jes\'us Gonz\'alez Departamento de Matem\'aticas Centro de Investigaci\'on y de Estudios Avanzados del IPN W. Stephen Wilson Department of Mathematics Johns Hopkins University We compute the BP (co)homology of the product of two (stunted) projective spaces. The behavior under maps (particularly of the Tor term) is studied. This is used extensively by Kitchloo and Wilson in their work on non-immersions. Additional work with lens spaces is also included. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions1 The second real Johnson-Wilson theory and non-immersions of $RP^n$ Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) W. Stephen Wilson Department of Mathematics Johns Hopkins University Hu and Kriz construct the real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$ periodic, from the $2(2^n-1)$ periodic spectrum $E(n)$. $ER(1)$ is just $KO_{(2)}$ and $E(1)$ is just $KU_{(2)}$. We compute $ER(n)^*(RP^\infty)$ and set up a Bockstein spectral sequence to compute $ER(n)^*(-)$ from $E(n)^*(-)$. We combine these to compute $ER(2)^*(RP^{2n})$ and use this to get new non-immersions for real projective spaces. Our lowest dimensional new example is an improvement of 2 for $RP^{48}$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions2 The second real Johnson-Wilson theory and non-immersions of $RP^n$, Part II. Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) W. Stephen Wilson Department of Mathematics Johns Hopkins University This paper is a continuation of the study begun in the paper with the same name. We analyze $ER(2)^{16*+8}(RP^{2n})$ and compute $ER(2)^*(RP^{16K+1})$ and use these to prove more non-immersion theorems for $RP^n$ including many in fairly low dimensions. In particular, we get 12 new non-immersion results for $RP^n$ where $n \le 192$, the range included in Don Davis's tables. These complement the 10 already found in part I. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmidappendix title: Degree Bounds and the Regular Representation: Appendix author: Mara D. Neusel subjclass[2000]: Primary 13A50 keywords: Invariant Theory of Finite Permutation Groups, Permutation Representation, Cohen-Macaulay, Gorenstein, Complete Intersection, Hypersurface, Polynomial Algebra, Pseudo-Reflection abstract: Let G be a matrix group consisting of permutation matrices. Let F and K be two different fields. We show that if the polynomial invariants F[V]^G and K[V]^G are both Cohen-Macaulay, then they are simultaneously Gorenstein, complete intersections, hypersurfaces, resp. polynomial. Thus Cohen-Macaulay rings of permutation invariants are polynomial exactly when G is generated by pseudo-reflections. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/psquare title: the invariants of modular indecomposable representations of Z_{p^2} authors: Mara D. Neusel and M\"ufit Sezer abstract: We consider the invariant ring for an indecomposable representation of a cyclic group of order p^2 over a field F of characteristic p. We describe a set of F-algebra generators of this ring of invariants, and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants. This bound, as a polynomial in p, is of degree two. -------------- ------------------------------------- 6 new papers this time, from Chebolu-Christensen-Minac, Elmendorf-Mandell, Flores-Foote, Gray-Theriault, Kahn-Maltsiniotis, and Stacey-Whitehouse. Mark Hovey New papers appearing on hopf between 9/24/07 and 11/29/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/GH-periodic TITLE: Freyd's generating hypothesis for groups with periodic cohomology. AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada ABSTRACT: Let $G$ be a finite group and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/EM2v5 A. D. Elmendorf and M. A. Mandell Multiplicative structure in infinite loop space theory We extend the K-theory functor constructed in our previous paper (Rings, modules, and algebras in infinite loop space theory, Advances in Mathematics 205 (2006) 163-228) to the bicomplete symmetric monoidal closed category of based (symmetric) multicategories, to which our previous source category of permutative categories and lax morphisms maps fully and faithfully. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores-Foote/Flores-Foote Title: Strongly closed subgroups and the cellular structure of classifying spaces Authors: Ram\'on J. Flores and Richard M. Foote Abstract: In this paper we give a complete classification of the finite groups that contain a strongly closed $p$-subgroup, generalizing previous work of the second author to the case of an odd prime. We use this result to also obtain a description of the BZ/p-cellularization (in the sense of Dror-Farjoun) of all the classifying spaces of finite groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gray-Theriault/Gray-Theriault An elementary construction of Anick's fibration Brayton Gray Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045 Stephen Theriault Department of Mathematical Sciences University of Aberdeen Aberdeen, AB24 3UE, United Kingdom Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> \Omega S^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>=5 and r>=1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>=1. We go on to establish several properties of the space T. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kahn-Maltsiniotis/StrDer Structures de dérivabilité Bruno KAHN & Georges MALTSINIOTIS Institut de Mathématiques de Jussieu We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of "homotopical categories", and by Radulescu-Banu in the context of Cisinski's "derivable categories". 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/hopf Title: The Hunting of the Hopf Ring Authors: Andrew Stacey and Sarah Whitehouse Addresses of authors: Andrew Stacey Institutt for matematiske fag NTNU 7491 Trondheim Norway Sarah Whitehouse Department of Pure Mathematics University of Sheffield Sheffield S3 7RH UK Abstract: We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another. ----------------- Happy New Year! 10 new papers this time, from Anton, DavisDaniel, Harper (2) (that is John E. Harper of Notre Dame, not John Harper of Rochester), Hovey-Lockridge (2), Neusel (2), and Yagita (2). Mark Hovey New papers appearing on hopf between 11/29/07 and 1/18/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/homologicalSymbols Title: Homological symbols and the Quillen Conjecture Author(s): Marian F. Anton Abstract: We formulate a "correct" version of the Quillen conjecture on the cohomology of linear groups by defining an unstable form of Milnor K-theory and show that this version can be solved by a finite process. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/fibrantmodel4 Title: Explicit fibrant replacement for discrete G-spectra Author: Daniel G. Davis Abstract: If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-operads-monoidal Title: Homotopy theory of modules over operads and non-Sigma operads in monoidal model categories Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 49 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper studies the existence of model category structures on modules and algebras over operads in monoidal model categories. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-spectra Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 21 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/gen-gen-hyp The ghost dimension of a ring Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We introduce the concept of the ghost dimension of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that the ghost dimension of R is less than or equal to the weak dimension of R, with equality if R is coherent or has weak dimension 1. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/triproj Triangulations of Projective Modules Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modules over a symmetric ring spectrum or $S$-algebra, where semisimple and von Neumann regular ring spectra are defined and discussed. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hilbert Title: On the Hilbert Ideal Author: Mara D. Neusel Abstract: We prove the Hilbert number conjecture. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid Title: Degree Bounds and the Regular Representation Author: Mara D. Neusel Abstract: This is a revised version of the paper with the same name posted during last summer. We prove Schmid's inequality in the general case, and Killius' conjecture for permutation representations. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/coniveaufilt Title: Coniveau filtration of cohomology of group Author: Nobuaki Yagita Abstract: We consider natural filtrations of mod p cohomology of a classifying space BG for a compact Lie group G, such that the reduced power operation preserves the filtration but the Bockstein opration descends the filtration degree one. An example of such filtrations is defined by the image from the motivic cohomology. For example, when BG=BO(n), this filtration coincides the coniveau filtration defined by Grothendieck. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/torsorEE Title: Note on Chow rings of nontrivial G-torsors over a field. Author: Nobuaki Yagita Abstract: Let G(k) be a split reductive group over a field k corresponding to a compact Lie group G. Let E be a nontrivial G(k)-torsor over a field k. In this paper we study the Chow ring of nontrivial G(k)-torsors E. For example when (G,p)=(F_4,3), we see that the positive degree of the mod 3 Chow ring of E is zero. ---------------- ---------------------------------------------- 4 new papers this time, from Blanc, Harper (again, this is John E. Harper of Notre Dame, not John Harper of Rochester), Kuhn, and Sati-Schreiber-Stasheff. Mark Hovey New papers appearing on hopf between 1/18/07 and 3/3/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/quil Title: Generalized Andre-Quillen Cohomology Author: David Blanc Address: Dept. of Mathematics, U. Haifa, Haifa, Israel Abstract: We explain how the approach of Andre and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them. As a side benefit, we clarify exactly what assumptions on an (algebraic) category are needed in order for the approach of Beck and Andre-Quillen to work. We also show how the description may be applied to construct universal coefficient and reverse Adams spectral sequences. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/QuillenHomology Title: Bar constructions and Quillen homology of modules over operads Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 33 pages, uses xy-pic; compiled the .tex file without using the dvips,ps options in xy-pic, to ensure .dvi is device independent, but diagrams may now appear jagged, etc. Abstract: This paper shows that Quillen derived homology of modules and algebras over an operad, for symmetric sequences of symmetric spectra and unbounded chain complexes, can be calculated using simplicial bar constructions, modulo cofibrancy conditions. Working with several model category structures, a homotopical proof is given, after showing that certain homotopy colimits in modules and algebras over an operad can be easily understood. The key result here, which is at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/telescopic Title: A guide to telescopic functors Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 Abstract: In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Sati-Schreiber-Stasheff/LCon Title: L-infinity algebra connections and applications to String- and Chern-Simons n-transport Authors: Hisham Sati, Urs Schreiber and Jim Stasheff Abstract: We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity -algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) -> U(H) -> PU(H) to higher categorical central extensions, like the String-extension B U(1) -> String(G) -> G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String- extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures'' whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class. -------------------- ---------------------------------- My semester has ended, my daughter has chosen a college, and I finally have some time to deal with Hopf. Sorry for the long delay. 7 new papers this time, from Blanc-Johnson-Turner, Broto-Moller-Oliver, Carlson-Chebolu-Minac, Neusel-Sezer, Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov, Yagita (2). Mark Hovey New papers appearing on hopf between 3/3/07 and 5/15/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/lgss Title: Local-to-global spectral sequences for the cohomology of diagrams Authors: David Blanc, Mark W. Johnson, and James M. Turner Address: Department of Mathematics, University of Haifa, 31905 Haifa, Israel Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA Abstract: The cohomology of diagrams arises in various areas of mathematics, such as deformation theory, classifying diagrams of groups, and in homotopy theory, in the context of the rectification of homotopy-commutative diagrams, and thus in the study of higher homotopy and cohomology operations. For this purpose we construct ``local-to-global'' spectral sequences for the cohomology of a diagram, which can be used to compute the cohomology of the full diagram in terms of smaller pieces. We also explain why such a local-to-global approach is relevant to higher operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Moller-Oliver/bmo1 Authors: C. Broto, J. M. M\o{}ller, and B. Oliver Title: Equivalences between fusion systems of finite groups of Lie type Subject class: Primary 20D06, Secondary 55R37, 20D20 keywords: groups of Lie type, fusion systems, classifying spaces, p-completion Abstract: We prove, for certain pairs $G,G'$ of finite groups of Lie type, that the $p$-fusion systems $F_p(G)$ and $F_p(G')$ are equivalent. In other words, there is an isomorphism between a Sylow $p$-subgroup of $G$ and one of $G'$ which preserves $p$-fusion. This occurs, for example, when $G=\Gamma(q)$ and $G'=\Gamma(q')$ for a simple Lie ``type'' $\Gamma$, and $q$ and $q'$ are prime powers, both prime to $p$, which generate the same closed subgroup of $p$-adic units. Our proof uses homotopy theoretic properties of the $p$-completed classifying spaces of $G$ and $G'$, and we know of no purely algebraic proof of this result. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Carlson-Chebolu-Minac/fgt Finite generation of Tate cohomology Jon F. Carlson Department of Mathematics University of Georgia Athens, GA 30602, USA Sunil K. Chebolu Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Abstract: Let G be a finite group and let k be a field of characteristic p. If M is a finitely generated indecomposable non-projective kG-module, we conjecture that the Tate cohomology of G with coefficients in M is finitely generated over the Tate cohomology ring of G if and only if the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results all of which support this conjecture. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/separating TITLE: Characterizing Separating Invariants AUTHORS: Mara D.~Neusel and M\"uf\.it Sezer ABSTRACT: We study separating algebras for rings of invariants of finite groups. We give an algebraic characterization for these. Furthermore, we describe a particularly nice separating subalgebra for rings of invariants of p-groups in characteristic p. This leads to a characterization of subalgebras such that their p-root and integral closure is equal to the ring of invariants. Finally, we present separating sets for invariants rings of nonmodular representations of abelian groups whose size depends only on the degree of the representation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov/flow [Your moderator was in a quandary over this paper, which is clearly not remotely algebraic topology, but decided to err on the side of openness] Integrable isotropic geometrical flows and Heisenberg ferromagnets N.S.Serikbaev, Zh.M.Bitibaeva, K.K.Yerzhanov, R.Myrzakulov* Department of General and Theoretical Physics, Eurasian National University, Astana, 010008, Kazakhstan Abstract Geometrical Flows (GF) play an important role in modern mathematics and physics. In this letter we have considered some integrable isotropic GF Ricci Flows (RF) and mean curvature flows (MCF) ~ which are related with integrable Heisenberg ferromagnets. In 2+1 dimensions, these GF have a singularity at t = t0. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/motsplitG Title: Note on the mod p motivic cohomology of algebraic groups. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: Let G_k be a split reductive group over a field k of ch(k)=0 corresponding to a compact Lie group G. In this paper, we show that the mod p motivic cohomology is isomorphic to the tensor product of the usual mod p cohomology H^*(G;Z/p) and the motivic cohomology H^{*,*'}(Spec(k);Z/p), when G=SO_n,G_2,F_4,E_6. We also give an example of nonsplit case (G=G_2,p=2,k=R) which does not hold the above isomorphism. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/realchow Title: Note on motivic cohomology of anisotropic real quadrics. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: In this paper, we compute the mod 2 motivic cohomology H^{*,*'}(X;Z/2) for the anisotropic quadric X over R the field of real numbers. ----------- ------------------------------------------------------------ Hmm, I seem to be losing momentum on these Hopf announcements. 9 new papers this time, from Behrens-Davis, Gillespie-Hovey, Gonzalez-Landweber, Hovey (2), Hovey-Lockridge, Kashiwabara, Kuhn, and Monico-Neusel. Mark Hovey New papers appearing on hopf between 5/15/08 and 9/6/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Davis/funcspec15 Title of Paper: The homotopy fixed point spectra of profinite Galois extensions Authors: Mark Behrens, Daniel G. Davis AMS Classification numbers: 55N20, 55P43 ArXiv ID: math.AT/0808.1092 Abstract: Let E be a k-local profinite G-Galois extension of an E_\infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie-Hovey/gorenstein Gorenstein model structures and generalized derived categories James Gillespie and Mark Hovey In a previous paper, the second author introduced the Gorenstein projective and Gorenstein injective model structures on $R$-Mod, the category of $R$-modules, where $R$ is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable homotopy category of $R$. Here we show that if such a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$ is Gorenstein and the three associated Gorenstein model structures on $R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-modules, are nothing more than the usual projective, injective and flat model structures on Ch($R$), the category of chain complexes of $R$-modules. Although these correspondences only recover these model structures on Ch($R$) when $R$ has finite global dimension, we can set $R = \Z$ and use general techniques from model category theory to lift the projective model structure from Ch($\Z$) to Ch($R$) for an arbitrary ring $R$. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when $\Z[x]/(x^2)$ is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring $R$ as well as the derived category of $R$ and we give some examples of such generalizations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Landweber/symmotion Title: Symmetric topological complexity of projective and lens spaces Authors: Jesus Gonzalez and Peter Landweber Adresses: Departamento de Matematicas, CINVESTAV-IPN, Mexico City 07000, MEXICO Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. This paper describes the corresponding relationship between the symmetrized versions of (b) and (c) to the Euclidean embedding dimension of projective spaces. Extensions to the case of lens spaces and complex projective spaces are discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/bpbp1 The homotopy of MString and MU<6> at large primes Mark Hovey We use Hopf rings to compute the homotopy rings $\pi_{*}\MO{8}$ and $\pi_{*}\MU{6}$ at primes $>3$. In this case, the additive structure is well-known, but the ring structure is not polynomial. Instead, these rings are quotients of polynomial rings by infinite regular sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/intersection-homology Intersection homological algebra Mark Hovey We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space $X$, we get intersection homology groups $I^{\perversity{p}}H_{n}X$ depending on the choice of an $n$-perversity $\perversity{p}$. The $n$-perversities form a lattice, and we can think of $IH_{n}X$ as a functor from this lattice to abelian groups, or more generally $R$-modules. Such perverse $R$-modules form a closed symmetric monoidal abelian category. We study this category and its associated homological algebra. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/ssrs Semisimple ring spectra Mark Hovey and Keir Lockridge Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension 0, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/hqs2008 The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE In this paper we give a generator-relation description of mod $p$ Bockstein-nil homology of $QS^n$ for odd prime $p$. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/loopSS Title: Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 AMS classification number: 55S10 arXiv:0806.3281 Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Monico-Neusel/counting TITLE: Counting Special Monomials AUTHORS: Chris Monico and Mara D. Neusel Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409 ABSTRACT: In this paper we study the number of orbits of special monomials of G acting by permutations on the polynomials in n variables. We give formulae for several crucial families of groups, for direct sums of representations, as well as for vector invariants. In addition we give two algorithms for arbitrary permutation groups, one relying on the geometry of G acting on the underlying vector space, the other relying on the representation theory of the symmetric groups. ----------------- BABYL OPTIONS: -*- rmail -*- Version: 5 Labels: Note: This is the header of an rmail file. Note: If you are seeing it in rmail, Note: it means the file has no messages in it.  1, edited,, Mail-from: From dmd1@lehigh.edu Sat Jan 17 18:19:15 1998 Received: from nss4.cc.Lehigh.EDU (root@nss4.CC.Lehigh.EDU [128.180.1.13]) by mail.wesleyan.edu (8.8.6/8.7.3) with ESMTP id SAA10234 for ; Sat, 17 Jan 1998 18:20:52 -0500 (EST) Received: from ns4-1.CC.Lehigh.EDU (root@ns4-1.CC.Lehigh.EDU [128.180.1.42]) by nss4.cc.Lehigh.EDU (8.8.8/8.8.5) with ESMTP id SAA119110; Sat, 17 Jan 1998 18:23:02 -0500 Received: (from dmd1@localhost) by ns4-1.CC.Lehigh.EDU (8.8.5/8.8.5) id SAA39528; Sat, 17 Jan 1998 18:19:17 -0500 Message-Id: <199801172319.SAA39528@ns4-1.CC.Lehigh.EDU> Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) X-Mailer: SENDM [Version 2.0.17] Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) Content-Type: text X-UIDL: aacd4710beaa4a6483935a131ded8f1b Lines: 256 Xref: picard.math.wesleyan.edu davis:291 X-Gnus-Newsgroup: davis:291 Sun Jan 18 06:29:52 1998 *** EOOH *** Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) Content-Type: text Xref: picard.math.wesleyan.edu davis:291 *** EOOH *** Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) X-UIDL: aacd4710beaa4a6483935a131ded8f1b Xref: picard.math.wesleyan.edu davis:291 X-Gnus-Newsgroup: davis:291 Sun Jan 18 06:29:52 1998 Hmm, I seem to be losing momentum on these Hopf announcements. 9 new papers this time, from Behrens-Davis, Gillespie-Hovey, Gonzalez-Landweber, Hovey (2), Hovey-Lockridge, Kashiwabara, Kuhn, and Monico-Neusel. Mark Hovey New papers appearing on hopf between 5/15/08 and 9/6/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Davis/funcspec15 Title of Paper: The homotopy fixed point spectra of profinite Galois extensions Authors: Mark Behrens, Daniel G. Davis AMS Classification numbers: 55N20, 55P43 ArXiv ID: math.AT/0808.1092 Abstract: Let E be a k-local profinite G-Galois extension of an E_\infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie-Hovey/gorenstein Gorenstein model structures and generalized derived categories James Gillespie and Mark Hovey In a previous paper, the second author introduced the Gorenstein projective and Gorenstein injective model structures on $R$-Mod, the category of $R$-modules, where $R$ is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable homotopy category of $R$. Here we show that if such a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$ is Gorenstein and the three associated Gorenstein model structures on $R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-modules, are nothing more than the usual projective, injective and flat model structures on Ch($R$), the category of chain complexes of $R$-modules. Although these correspondences only recover these model structures on Ch($R$) when $R$ has finite global dimension, we can set $R = \Z$ and use general techniques from model category theory to lift the projective model structure from Ch($\Z$) to Ch($R$) for an arbitrary ring $R$. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when $\Z[x]/(x^2)$ is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring $R$ as well as the derived category of $R$ and we give some examples of such generalizations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Landweber/symmotion Title: Symmetric topological complexity of projective and lens spaces Authors: Jesus Gonzalez and Peter Landweber Adresses: Departamento de Matematicas, CINVESTAV-IPN, Mexico City 07000, MEXICO Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. This paper describes the corresponding relationship between the symmetrized versions of (b) and (c) to the Euclidean embedding dimension of projective spaces. Extensions to the case of lens spaces and complex projective spaces are discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/bpbp1 The homotopy of MString and MU<6> at large primes Mark Hovey We use Hopf rings to compute the homotopy rings $\pi_{*}\MO{8}$ and $\pi_{*}\MU{6}$ at primes $>3$. In this case, the additive structure is well-known, but the ring structure is not polynomial. Instead, these rings are quotients of polynomial rings by infinite regular sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/intersection-homology Intersection homological algebra Mark Hovey We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space $X$, we get intersection homology groups $I^{\perversity{p}}H_{n}X$ depending on the choice of an $n$-perversity $\perversity{p}$. The $n$-perversities form a lattice, and we can think of $IH_{n}X$ as a functor from this lattice to abelian groups, or more generally $R$-modules. Such perverse $R$-modules form a closed symmetric monoidal abelian category. We study this category and its associated homological algebra. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/ssrs Semisimple ring spectra Mark Hovey and Keir Lockridge Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension 0, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/hqs2008 The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE In this paper we give a generator-relation description of mod $p$ Bockstein-nil homology of $QS^n$ for odd prime $p$. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/loopSS Title: Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 AMS classification number: 55S10 arXiv:0806.3281 Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Monico-Neusel/counting TITLE: Counting Special Monomials AUTHORS: Chris Monico and Mara D. Neusel Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409 ABSTRACT: In this paper we study the number of orbits of special monomials of G acting by permutations on the polynomials in n variables. We give formulae for several crucial families of groups, for direct sums of representations, as well as for vector invariants. In addition we give two algorithms for arbitrary permutation groups, one relying on the geometry of G acting on the underlying vector space, the other relying on the representation theory of the symmetric groups. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this listserv, go to https://lists.lehigh.edu/mailman/listinfo/algtop-l. To see past issues of new submissions to Hopf, go to http://math.wesleyan.edu/~mhovey/archive/ To get the papers listed above, go to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There is a web form for submitting papers to Hopf on this site as well. You should submit an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/new-html/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker at math.purdue.edu telling him what you have uploaded. The largest archive of math preprints is at http://arxiv.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at the arXiv, send e-mail to math@arxiv.org with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.  -------------------------------------------- Sorry once again for the long delay. 12 new papers this time, from BrownR, BrownR-Sivera, Glover-Henn, Harper, Henn-Karamanov-Mahowald, LinJP, SmithL (2 papers), SmithL-Stong (4 papers). Mark Hovey New papers appearing on hopf between 9/6/08 and 11/20/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-art Author: Ronald Brown Title: Crossed complexes and higher homotopy groupoids as non commutative tools for higher dimensional local-to-global problems Abstract: We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types in non simply connected situations. Web page: www.bangor.ac.uk/r.brown Futher information: This is a revised version (2008) of a paper published in Fields Institute Communications 43 (2004) 101-130, which was an extended account of a lecture given at the meeting on `Categorical Structures for Descent, Galois Theory, Hopf algebras and semiabelian categories', Fields Institute, September 23-28, 2002. The author is grateful for support from the Fields Institute and a Leverhulme Emeritus Research Fellowship, 2002-2004, and to M. Hazewinkel for helpful comments on a draft. This paper is to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/fibcat Title: Algebraic colimit calculations in homotopy theory using fibred and cofibred categories Author(s): Ronald Brown and Rafael Sivera Abstract: Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several base points, but we also describe briefly the situation for triads. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Glover-Henn/oct28-2008 Title: On the mod-p cohomology of Out(F_{2(p-1)}) Authors: Henry Glover Hans-Werner Henn Abstract: We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/ModulesSpectra14 Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Institute of geometry, algebra and topology, EPFL, CH-1015 Lausanne, Switzerland Comments: 33 pages, uses xy-pic. Significant revision. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. *** Please note: this is not a new submission to the Hopf arxiv, but a revision of an earlier manuscript with the same title. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn-Karamanov-Mahowald/hkm Title: The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited Authors: Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Abstract: In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/LinJP/Lin08 Homology Rings of Homotopy Associative $H$--spaces James P. Lin Let $X$ be a homotopy associative mod $p$ $H$--space for $p$ an odd prime. The homology $H_*(X; \mathbb{F}_p)$ is an associative ring, but not necessarily commutative. We study conditions when $[\overline{x}, \overline{y}] \neq 0$ for $\overline{x}, \overline{y}$ elements of $H_*(X; \mathbb{F}_p)$. Under certain conditions $[\overline{x}, \overline{y}] \neq 0$ imply $ad^l (\overline{x},\overline{y}) \neq 0$ for $l=p-2$ or $p-1$. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action (Hamanaka et. al., 1996), (Kono et. al., 1993), (Kono et. al., 2003). We also generalize results of (Kane, 2006) to nonfinite mod $p$ homotopy associative $H$--spaces. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/stable Title: Stable Invariants of Finite General Linear Groups and Symmetric Groups in Odd Characteristic Author: Larry Smith (AG-Invariantentheorie) We show that the stable invariants of the finite general linear group $\GL(n, \F_q)$ over a Galois field $\F_q$ with an odd characteristic coincide with the Hilbert ideal. The same argument applies to the tautological representation of the symmetric group in odd characteristic. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/steintri Title: On R. Steinberg's Theorem on Algebras of Coinvariants Author Larry Smith (AG-Invariantentheorie} Steinberg's Theorem on the coinvariant algebra $\C[V]_G$ of a complex representation $\rho : G \hra \GL(n, \C)$ of a finite group $G$ says that $\C[V]_G$ is a Poincar\'e duality algebra if and only if the invariant algebra $\C[V]^G$ is a polynomial algebra. The extension of this to the nonmodular case has been achieved in stages, the final result being obtained by W.G. Dwyer and C.W. Wilkerson. We show that the main module theoretic tool they use extends to the following characteristic free result: If $\F[V]_G$ is a Poincar\'e duality algebra of formal dimension $d$\/, then $\F[V]^G$ is a polynomial algebra if and only if $\Hom_{\F[V]^G} (\F[V], \F[V])$ contains a nonzero element of degree $-d$\/. In the nonmodular case an easy transfer argument then recovers their extension of Steinberg's Theorem by means of some representation theory. Combined with some new results concerning the $\Delta$ operators of Demazure, our characteristic free result yields the following for reflection groups: A reflection group $G$ for which $\F[V]_G$ is a Poincar\'e duality algebra in which the trivial $G$-representation $1_G$ occurs only once as a subrepresentation has a polynomial algebra for its invariant algebra $\F[V]^G$\/. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/binary Title: Invariants of Binary Forms Modulo Two Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) We examine the invariant theory of binary bilinear forms over the field $\F_2$ of two elements that arises in the classification of (standardly graded) Poincar\'e duality algebras with two algebra generators over the field $\F_2$ of two elements. We compute the corresponding ring of invariants and find seperating invariants for the orbit space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pbi Title : Projective Bundle Ideals : Construction of Maximal Primary Irreducible Ideals in Polynomial Algebras Authors: Larry Smith and R. E. Stong Summary: We formalize the algebra of the Projective Bundle Theorem and use it to construct and study maximal primary irreducible ideals in polynomial algebras. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pda_quos Title: Poincar\'e Duality Algebras Modulo Two and Macaulay's Inverse Systems Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) If $H$ is a Poincar\'e duality algebra generated by its homogeneous component of degree $1$ it is called {\bf standardly graded} and the dimension of its homogeneous component $H_1$ of degree one is called its {\bf rank}\/. Standardly graded Poincar\'e duality algebras occur as quotient algebras of a (standardly graded) polynomial algebra by a maximal primary irreducible ideal Such ideals were studied in the work of F. S. Macaulay at the start of the last century who developed an elegant means of constructing them. The fact that these quotients are Poincar\'e duality algebras is a special case of a result of W. Gr\"obner. In this note we study the classification of Poincar\'e duality algebras over the field $\F_2$ of two elements. We obtain a complete classification of surfaces, i.e., Poincar\'e duality algebras of formal dimension two. To do so we determine the Grothendieck group of standardly graded surface algebras over an arbitrary field under the operation of connected sum. This group turns out to be $\Z$\/, hence finitely generated, and mirrors faithfully the topological classification of closed surfaces. By contrast, for Poincar\'e duality algebras (standardly graded or not) of formal dimension strictly greater than two the Grothendieck group fails to be finitely generated. We make a systematic study of standardly graded threefolds, i.e., Poincar\'e duality algebras of formal dimension three that are generated by their elements of degree one. The isomorphism classes of threefolds of rank at most three are in bijective correspondence with the orbits of the action of $\GL(3, \F_2)$ on a $10$-dimensional vector space, the space of catalecticant matrices. To determine the number of isomorphism classes we count the number of orbits using invariant theory. As a byproduct we obtain a classification of arbitrary bilinear forms in up to three variables. We determine explicitly all the standardly graded threefolds of rank at most three. There are 21 isomorphism classes. Twelve of these admit an unstable Steenrod algebra action, so could in theory be realized as the mod $2$ cohomology of a closed manifold. We exhibit for each such example a corresponding manifold; most of these are obvious, but there is one example of a slightly exotic $3$-manifold that is a torus bundle over a circle to which we devote some space. For threefolds of higher rank we explain one of several ways to construct such algebras that are not connected sums using Macaulay's theory of inverse systems. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/rank_two Title: On Maximal Primary Irreducible Ideals in $\F[x, y]$ Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) At the beginning of the last century F.~S.~Macaulay developed an elegant theory describing homogeneous ideals in polynomial rings. This theory makes the maximal-primary irriducible ideals $I \subset \F[z_1\commadots z_n]$ correspond to a single homogeneous inverse polynomial $\theta_I \in \F[z_1^{-1} \commadots z_n^{-1}]$\/. Macaulay's theory has recently attracted attention in connection with problems arising in invariant theory and algebraic topology. In this note we show how given an inverse binary form $\theta \in \F[x^{-1}, y^{-1}]$ one may explicitly write down generators of the corresponding maximal-primary irreducible ideal $I(\theta) \subset \F[x, y]$\/. As a bonus we obtain an elementary proof of a theorem of Vasconcelos that such an ideal is always generated by a regular sequence. ----------------- -------------------------------------------------------- 4 new papers this time, from Bailey, Chebolu-Minac, Nguyen-Schwartz-Tran, and Ostvaer. Mark Hovey New papers appearing on hopf between 11/20/08 and 3/9/09 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bailey/bailey-bosmtmf Title: On the spectrum bo \wedge tmf Author(s): Scott M. Bailey AMS classification number: 55P10 (Primary) 55P42, 55Q51 (Secondary) Abstract: M. Mahowald in his work on bo-resolutions, constructed a bo-module splitting of the spectrum bo ^ bo into a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar splitting of bo ^ tmf is constructed. This splitting is then used to understand the bo_*-algebra structure of bo_* tmf and allows for a description of bo^* tmf. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Minac/Auslander-Reiten Title: Auslander-Reiten sequences for homotopists and arithmeticians Authors: Sunil Chebolu, Jan Minac Comments: 16 pages, to appear in "Annales des sciences math'ematiques du Quebec" Abstract: We introduce Auslander-Reiten sequences for group algebras and give several recent applications. The first part of the paper is devoted to some fundamental problems in Tate cohomology which are motivated by homotopy theory. In the second part of the paper we interpret Auslander-Reiten sequences in the context of Galois theory and connect them to some important arithmetic objects. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Nguyen-Schwartz-Tran/nguyen-schwartz-tran Title: La fonction generatrice de Minc et une "conjecture de Segal" pour certains spectres de Thom Authors: Nguyen Dang Ho Hai, Lionel Schwartz, Tran Ngoc Nam Abstract On construit dans cet article une r'esolution injective minimale dans la cat'egorie U des modules instables sur l'alg`ebre de Steenrod modulo 2, de la cohomologie de certains spectres obtenus `a partir de l'espace de Thom du fibr'e, associ'e `a la repr'esentation r'eguli`ere r'eduite du groupe ab'elien 'el'ementaire (Z=2)n, au dessus de l'espace B(Z=2)n. Les termes de la r'esolution sont des produits tensoriels de modules de Brown-Gitler J(k) et de modules de Steinberg Ln introduits par S. Mitchell et S. Priddy. Ces modules sont injectifs d'apr`es J. Lannes et S. Zarati, de plus ils sont ind'ecomposables. L'existence de cette r'esolution avait 'et'e conjectur'ee par Jean Lannes et le deuxi`eme auteur. La principale indication soutenant cette conjecture 'etait un r'esultat combinatoire de G. Andrews : la somme altern'ee des s'eries de Poincar'e des modules consid'er'ees est nulle. Ce r'esultat a des cons'equences homotopiques et permet de d'emontrer pour ces spectres un r'esultat du type de la conjecture de Segal pour les classifiants des 2-groupes ab'eliens 'el'ementaires. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Ostvaer/cstar Title: Homotopy theory of C*-algebras Author: Paul Arne Ostvaer MSC classes: 46L99; 55P99 Abstract: In this work we construct from ground up a homotopy theory of C*-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C*-algebras. The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C*-homotopy theory. The stable homotopy category of C*-algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C*-algebras. In addition, we employ homotopy theory to define a new type of K-theory of C*-algebras. ------------------ 5 new papers this time, from Blanc-Johnson-Turner, BrownR, Chebolu-Minac, Hovey-Lockridge, and Huber-Kings-Naumann Mark Hovey New papers appearing on hopf between 3/9/09 and 5/28/09 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/hoc19 Title: Higher Homotopy Operations and Cohomology Authors: David Blanc, Mark W. Johnson, and James M. Turner Comments: 28 pages, to appear in the Journal of K-theory Abstract: The question of whether a homotopy-commutative diagram is rectifiable can be addressed via a cohomological obstruction theory developed by Dwyer-Kan-Smith. In this paper, the authors study a general notion of pointed homotopy operations which generalize, for example, Toda brackets. These topologically defined operations are constructed as homotopy-commutative diagrams and it is shown that they may be may be identified, under mild assumptions, with (the last of) the Dwyer-Kan-Smith cohomological obstructions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/dbgpdnew09 TITLE: Double modules, double categories and groupoids, and a new homotopical double groupoid AUTHOR: Ronald Brown ABSTRACT: We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups. AUTHOR'S ADDRESS: School of Computer Science, Bangor University, Bangor, Gwynedd, LL57 1UT, UK web site: www.bangor.ac.uk/r.brown 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Minac/BIRS-Survey Title: Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture Authors: Sunil K. Chebolu and Jan Minac Abstract: In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal pro-p-quotients of absolute Galois groups. Comments: To appear in the proceedings of the BIRS workshop "New Topological Contexts for Galois Theory and Algebraic Geometry" in Topology and Geometry monographs. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/weak-dim-2 The ghost and weak dimensions of rings and ring spectra Mark Hovey and Keir Lockridge The primary object of this paper is to prove the conjecture of the authors from a previous paper, explaining how to recover the weak dimension of a ring from its derived category. In the process, we develop a theory of weak dimension, which we call ghost dimension, for the generalized rings, known as ring spectra, that arise in algebraic topology. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Huber-Kings-Naumann/Lazard-complements Title: Some complements to the Lazard isomorphism Authors: Annette Huber, Guido Kings and Niko Naumann Abstract: Lazard showed in his seminal work that for rational coefficients continuous group cohomology of $p$-adic Lie-groups is isomorphic to Lie-algebra cohomology. We refine this result in two directions: firstly we extend his isomorphism under certain conditions to integral coefficients and secondly, we show that for algebraic groups, his isomorphism can be realized by differentiating locally analytic cochains. ------------------- ---------------------------------------------------------------------- It is obvious that I am no longer the best person for this job, given that I have not sent out a message since May 28. I would be happy if someone volunteered to take it over. One could, no doubt, mechanize the process, but I have obviously not done that. Anyway, there are 11 new papers this time, including some pretty interesting ones (by which I do not mean the two from me!). They are from Aldrovandi-Noohi, Breen-Mikhailov, Chebolu-Efrat-Minac, Colman, Hovey (2), Kashiwabara, Laures-McClure, Levi-Ragnarsson, Ramras, and Yagita. Mark Hovey New papers appearing on hopf between 5/28/09 and 11/30/09 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aldrovandi-Noohi/torsors Title: Butterflies II: Torsors for 2-group stacks Authors: Ettore Aldrovandi and Behrang Noohi Abstract: We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the ``fraction'' (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Breen-Mikhailov/derived-homotopy Title: Derived functors of non-additive functors and homotopy theory Authors: Lawrence Breen and Roman Mikhailov Abstract: We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces M(A,n), based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors. As an illustration of this method, we retrieve in a purely algebraic manner the 3-torsion component of the homotopy groups of the 2-sphere up to degree 14, and give a unified presentation of the low degree homotopy groups of M(A,n), for small values of n. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Efrat-Minac/decomposables Title: Quotients of absolute Galois groups which determine the entire Galois cohomology Authors: Sunil Chebolu, Ido Efrat, and Jan Minac Abstract: For prime power q=p^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F, Z/q) is determined by a quotient G_F^{[3]} of the absolute Galois group G_F related to its descending q-central sequence. Conversely, we show that G_F^{[3]} is determined by the lower cohomology of G_F. This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Colman/Colman Title of Paper: On the homotopy type of Lie groupoids Author: Hellen Colman Address of Author: Department of Mathematics, Wilbur Wright College, 4300 N. Narragansett Avenue, Chicago, IL 60634 USA Text of Abstract: We propose a notion of groupoid homotopy for generalized maps. This notion of groupoid homotopy generalizes the notions of natural transformation and strict homotopy for functors. The groupoid homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the orbifold homotopy between orbifold maps induced by the groupoid homotopy. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/symmetric-monoidal Additive closed symmetric monoidal structures on $R$-modules Mark Hovey Wesleyan University In this paper, we classify additive closed symmetric monoidal structures on the category of left R-modules by using Watts' theorem. An additive closed symmetric monoidal structure is equivalent to an R-module \Lambda _{A,B} equipped with two commuting right R-module structures represented by the symbols A and B, an R-module K to serve as the unit, and certain isomorphisms. We use this result to look at simple cases. We find rings R for which there are no additive closed symmetric monoidal structures on R-modules, for which there is exactly one (up to isomorphism), for which there are exactly seven, and for which there are a proper class of isomorphism classes of such structures. We also prove some general structual results; for example, we prove that the unit K must always be a finitely generated R-module. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/watts The Eilenberg-Watts theorem in homotopical algebra Mark Hovey The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or smash product) with a fixed object is in fact such a tensor product, up to homotopy. The right half says any functor that looks like it could be Hom into a fixed object is so, up to homotopy. More precisely, suppose we have a closed symmetric monoidal category (resp. Quillen model category) M. Then the functor T_{B} that takes A to A tensor B is an M-functor and a left adjoint. The same is true if B is an E-E'-bimodule, where E and E' are monoids in M, and T_{B} takes an E-module A to A tensored over E with B. Define a closed symmetric monoidal category (resp. model category) to be left self-contained (resp. homotopically left self-contained) if every functor F from E-modules to E'-modules that is an M-functor and a left adjoint (resp. and a left Quillen functor) is naturally isomorphic (resp. naturally weakly equivalent) to T_{B} for some B. The classical Eilenberg-Watts theorem in algebra then just says that the category of abelian groups is left self-contained, so we are generalizing that theorem. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/bpinfla Title: BP infinite loop algebras Author Takuji Kashiwabara In this paper we define the notion of BP infinite loop algebras, a sort of BP -analogue of A- R-allowable Hopf algebras, and show that under some conditions, BP -cohomology of infinite loop spaces has such a structure. Furthermore we show that the BP infinite loop algebra structure gives a serious restriction on underlying unstable BP -algebra structure. Key words: Brown-Peterson cohomology, Infinite loop spaces, Generalized cohomology operations, Unstable Algebras, Dyer-Lashof operations, Nishida relations, Landweber-Novikov algebra, Steenrod algebra, Hopf ring 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Laures-McClure/quinn Title: Multiplicative properties of Quinn spectra. Authors: Gerd Laures and J.E. McClure Abstract: We give a simple sufficient condition for Quinn's ``bordism-type spectra'' to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's functor from bordism-type theories to spectra lifts to the category of symmetric spectra. We also give a new account of the foundations. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Ragnarsson/plfgCoh p-Local Finite Group Cohomology R. Levi and K. Ragnarsson We study cohomology for p-local finite groups with non-constant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Ramras/stable-moduli Title: The stable moduli space of flat connections over a surface Author: Daniel A. Ramras Abstract: We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawson's work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/ChowD8 Title: Chow rings of nonabelian p-groups of order p cubed Author: Nobuaki Yagita Abstract. Let G be a nonabelian p group of order p3 (i.e., extraspecial p-group), and BG its classifying space. Then CH*(BG) = H2*(BG) where CH*(-) is the Chow ring over the field k = C. We also compute mod(2) motivic cohomology and motivic cobordism of BQ8 and BD8. ------------------ -------------------------------------------------------- Happy New Year! If you have a suspicious mind like me, you may think that the fact that I am suddenly more efficient at getting out this announcement is because I am one of the authors this month. But I can honestly say that I was (helpfully!) prodded into this by Brayton Gray, another of this month's authors. There are 2 new papers this time, from GrayB and Hovey-Lockridge. Mark Hovey New papers appearing on hopf between 11/30/09 and 1/7/10 1. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/Gray-Whitehead On Generalized Whitehead Products Brayton Gray A symmetric monoidal product G o H is defined on the category of co-H spaces that are either suspensions or simply connected, together with a Whitehead product map G o H ---> G v H whose mapping cone is the product G x H. In particular, G o SX is equivalent to G ^ X. This generalizes some work of Theriault and allows one to analyze the Whitehead product structure of co-H spaces. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/finite-dim Homological dimensions of ring spectra by Mark Hovey and Keir Lockridge We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5. ------------------- -------------------------------------------------- Back to my old inefficiency. There are 7 new papers this time, from BrownR, Dwyer-Wilkerson, Kuhn, Levi-Seeliger, and Pengelley-Williams (3). Mark Hovey New papers appearing on hopf between 1/7/10 and 6/1/10 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/freeloopspace Title: Crossed modules and the homotopy 2-type of the free loop space Author: Ronald Brown Author's home page: www.bangor.ac.uk/r.brown AMS classification number: 18D15,55Q05,55Q52 arXive submission number: arXiv:1003.5617 Key words: free loop space, crossed module, crossed complex, closed category, classifying space, higher homotopies Abstract: The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We show how this depends on the homotopy 2-type of X by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX; that is we describe crossed modules representing the 2-type of each component of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS W. G. DWYER AND C. W. WILKERSON In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case. 1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra on V # the k-dual of V, and R = S^W the ring of invariants of under the natural action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators. Steinberg [9] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S\tensor_R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4] to other fields of characteristic zero, and by T.-C. Lin [5] to the case in which k is a finite field of characteristic prime to the order of W . The current proof is independent of previous methods. (Revised Jan. 25, 2010 to correct typos and to incorporate some remarks by R. J. Shank .) 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/kuhnnilpotence Title: Nilpotence in Group Cohomology Author: Nicholas J. Kuhn AMS classification number: 20J06, 55R40 arXiv:1002.4662 Subject class: math.GR; math.AT Abstract: We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p--groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p--group P. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Seeliger/LoopSol TITLE: Loop space homology associated to the mod 2 Dickson invariants} AUTHORS: Ran Levi, Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, U.K.} Nora Seeliger LAGA, Institut Galilee, Av. J-B Clement, 93430 Villetaneuse, France AMS CLASSIFICATION: Primary 55R35. Secondary 55R40, 20D20 arXiv:1001.3581v1 ABSTRACT The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop space homology of the 2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/beyond Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to CP(infinity) and BU. David J. Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 55R40; Secondary 55R45, 55S05, 55S10 We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of `digits') and is zero in other degrees. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/FiniteRealProj46 Unstable module presentations for the cohomology of real projective spaces David J. Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 55R40; Secondary 55R45, 55S05, 55S10 There is much we still do not know about projective spaces. We describe here how the mod two cohomology of each real projective space is built as an unstable module over the Steenrod algebra A, or equivalently, over K, the algebra of inherently unstable mod two "lower operations" originally introduced by Steenrod. In particular, to produce the cohomology of projective space of each dimension we consider the well-known minimal set of unstable module generators and construct a minimal set of unstable relations. Three new perspectives we blend for this purpose are: 1. to focus solely on the two-power Steenrod squares that generate A to understand the A-action in a process we call "shoveling ones"; 2. to describe every element in a canonical way from a particular unstable generator by composing operations from the algebra K; 3. to shift attention when studying an unstable A-module to consid- ering and analyzing it directly as an equivalent K-module. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/symmetric-hit-problem-hopf The hit problem for symmetric algebras over the Steenrod algebra David Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 The hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebra over the field with p elements, for p an arbitrary prime, and treat the equivalent problem of determing the set of A-primitive elements in its dual. We produce a method for generating new A-primitives from known ones via a new action of the Kudo-Araki-May algebra, K, and consider the K-module structure of the A-primitives, which form a sub K-algebra of the dual of the symmetric algebra over the Steenrod algebra. ---------------- Hello everybody, This is the first installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Feedback is welcomed. Instructions for subscribing or unsubscribing to this list, as well as getting the papers on the list, are at the end of this message. Please tell me if I am using an out-of-date e-mail address for you. Papers uploaded to Hopf between Dec. 20, 1994 and Jan. 1, 1995: 1. Hopkins-Ravenel-Wilson/moravaktheory Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov systems, by Michael J. Hopkins, Douglas C. Ravenel, and W. Stephen Wilson. We have three somewhat independent sets of results. Our first results are a mixed blessing. We show that Morava $K$-theories don't see $k$-invariants for homotopy commutative $H$-spaces which are finite Postnikov systems, i.e. for those with only a finite number of homotopy groups. Since $k$-invariants are what holds the space together, this suggests that Morava $K$-theories will not be of much use around such spaces. On the other hand, this gives us the Morava $K$-theory of a wide class of spaces which is bound to be useful. In particular, this work allows the recent work in \cite{RWY} to be applied to compute the Brown-Peterson cohomology of all such spaces. Their Brown-Peterson cohomology turns out to be all in even degrees (as is their Morava $K$-theory) and flat as a $BP^{*}$ module for the category of finitely presented $BP^{*}(BP)$ modules. Thus these examples have extremely nice Brown-Peterson cohomology which is as good as a Hopf algebra. Our second set of results produces a large family of spaces which behave as if they were finite Postnikov systems from the point of view of Morava $K$-theory even though they are not. This allows us to apply the above results to an even wider class of spaces than finite Postnikov systems. These examples come from spaces in omega spectra with certain properties. There are many well known examples with these properties. In particular, we compute the $K(n)$ homology of all the spaces in the $\Omega$-spectra for $P(q)$ and $k(q)$ where $q > n$. In order to prove our results on finite Postnikov systems we need our third set of results; a beginning of an analysis of bicommutative Hopf algebras over $K(n)_*$. 2. JPMay/completions Completions in algebra and topology by J.P.C. Greenlees and J.P. May Abstract We discuss algebraic completions at ideals and localizations away from ideals in a commutative ring, and we use the framework of ``Modern foundations for stable homotopy theory'' to show how this algebra can be mimicked topologically for ideals in the coefficient ring of an E infinity ring spectrum. The algebraic fact that completion is not exact forces us to work with the derived functors of completion, and we explain how topological completions of spectra mimic an algebraic description of these derived functors in terms of ``local homology groups''. These constructs are relevant to cohomology theories. The dual constructs relevant to homology theories involve Grothendieck's ``local cohomology groups''. There are concomitant notions of ``\v{C}ech homology and cohomology groups'', which fit into algebraic fibre sequences that we mimic by fibre sequences of spectra. These lead to a new theory of localizations of spectra away from ideals. When specialized to MU-modules, these localizations shed light on the chromatic filtration and the chromatic convergence theorem. Contents: Algebraic definitions: local and \v{C}ech cohomology and homology Connections with derived functors; calculational tools Topological analogues of the algebraic definitions Completion at ideals and Bousfield localization Localization away from ideals and Bousfield localization The specialization to ideals in $MU_*$ This paper is to appear in ``The handbook of Algebraic Topology'', edited by Ioan James. 3. JPMay/derived_categories Derived categories in algebra and topology by J.P. May Abstract An analogy between the derived category of modules over a commutative ring and the stable homotopy category of spectra is elaborated to a much closer analogy between the derived category of E infinity modules over an E infinity algebra and the derived category of E infinity module spectra over an E infinity ring spectrum. In both the algebraic and topological contexts, these new derived categories allow one to study ``modules up to homotopy'' over ``commutative algebras up to homotopy'' in much the same way that one studies ordinary modules in classical homological algebra. There are many applications in algebraic topology, algebraic K-theory, and algebraic geometry. This expository note explains the ideas and gives a brief summary of the relevant definitions in both contexts. This paper will appear in the proceedings of the Eleventh International Conference on Topology, Trieste, 1993. 4. JPMay/equivariant_theory Equivariant stable homotopy theory by J.P.C. Greenlees and J.P. May Abstract After sketching the basic concepts of space level equivariant homotopy theory, we introduce the basic ideas and constructions of spectrum level equivariant homotopy theory, combining earlier work of Lewis and May with the framework of ``Modern foundations for stable homotopy theory''. We then illustrate ideas by explaining the fundamental localization and completion theorems that relate equivariant to nonequivariant homology and cohomology. A key idea is that ``completion theorems'' in cohomology are sometimes consequences of results that deserve to be called ``localization theorems'' in homology. For example, for finite groups G, the Atiyah-Segal completion theorem that computes the K cohomology of BG is a consequence of a localization theorem that computes the K homology of BG. We describe a recent result that gives the same kind of localization and completion theorems for the spectrum MU(G) that represents a stabilized version of equivariant complex cobordism and for all module spectra over MU(G). For example, this applies to equivariant versions of Brown-Peterson and Morava homology and cohomology theories. We also discuss equivariant cohomotopy, a theory for which the cohomological completion theorem is true, by Carlsson's proof of the Segal conjecture, but the homological localization theorem is false. Contents: Equivariant homotopy The equivariant stable homotopy category Homology and cohomology theories and fixed point spectra Change of groups and duality theory Mackey functors, $K(M,n)'s$ and $RO(G)$-graded cohomology Philosophy of localization and completion theorems How to prove localization and completion theorems Examples of localization and completion theorems This paper is to appear in ``The handbook of Algebraic Topology'', edited by Ioan James. 5. JPMay/modern_foundations Modern foundations for stable homotopy theory by A.D. Elmendorf, I. Kriz, and J.P. May} Abstract We describe the foundations of stable homotopy theory to be established in our monograph ``Rings, algebras, and modules in stable homotopy theory'', in preparation, which will have Michael Mandell as a fourth author. Contents: Spectra and the stable homotopy category Smash products and twisted half-smash products The category of S-modules and its derived category The smash product of S-modules A infinity and E infinity ring spectra and their modules The smash product of R-modules and function R-modules Tor and Ext in topology and algebra Universal coefficient and Kunneth spectral sequences Algebraic constructions in the derived category of R-modules Algebra structures on localizations and on quotients by ideals The specialization to MU-modules and algebras The paper is already slightly obsolete, in that the definitive treatment will be based on a modified category of S-modules with a smash product that is not only commutative and associative but also unital. This paper is to appear in ``The handbook of Algebraic Topology'', edited by Ioan James. 6. JPMay/operads_motives Operads, algebras, modules, and motives by Igor Kriz and J.P. May Abstract With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent Parts: Definitions and examples of operads and their actions Partial algebraic structures and conversion theorems Derived categories from a topological point of view Rational derived categories and mixed Tate motives Derived categories of modules over $E_{\infty}$ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras ``up to homotopy", for example commutative algebras, n-Lie algebras, n-braid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA's and derived categories of modules up to homotopy over DGA's up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral. This monograph will appear in Asterisque. -------------------------- Hello everybody, This is the second installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Messages: 1. If you got this message from me (hovey---math.mit.edu) then you are currently subscribed. I am sure that not every one who wants to be on this list is on it, so I encourage subscribers to forward this to others who are not on it and are interested. Current graduate students in particular should know about it. 2. Last time the papers in the JPMay directory were not actually ready to be downloaded. They are now. I will try to prevent this kind of mistake from happening in the future. 3. The WWW web interface for the archive has improved. Open the URL http://hopf.math.purdue.edu/pub/hopf.html using Lynx, Mosaic, or Netscape. Mark Hovey Papers uploaded to Hopf between Jan. 2 and Jan. 18, 1995: 1. /pub/Nakano-Palmieri/nakano-palmieri.dvi D.K. Nakano and J.H. Palmieri, Support varieties for the Steenrod algebra In this paper we study the cohomological varieties associated to the finite-dimensional sub-Hopf algebras of the Steenrod algebra. A stratification theorem like the Quillen/Avrunin-Scott stratification theorem for finite groups is proven. With this stratification one can then invoke results from restricted Lie algebra cohomology to study these cohomological varieties. As a result, we get a description of the cohomology of these Hopf algebras, modulo nilpotence; we also prove a conjecture of Margolis about $P^{s}_{t}$-homology of a tensor product of modules. 2. /pub/Palmieri/palmieri-quasi.dvi A note on the cohomology of finite dimensional cocommutative Hopf algebras John H. Palmieri In the context of finite dimensional cocommutative Hopf algebras, we prove versions of various group cohomology results: the Quillen-Venkov theorem on detecting nilpotence in group cohomology, Chouinard's theorem on determining whether a $kG$-module is projective by restricting to elementary abelian $p$-subgroups of $G$, and Quillen's theorem which identifies the cohomology of $G$, ``modulo nilpotent elements.'' This last result is only proved for graded connected Hopf algebras. ----------------- Hello everybody, This is the third installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. New Message: Those with a World-Wide Web browser will notice some changes at http://hopf.math.purdue.edu/pub/hopf.html . In particular, there is now a link to a home page for the Purdue math department, which list seminars and colloquia happening there. There are also some conference announcements on the hopf.html page, including the Great Lakes K-theory conference in March, the Midwest topology seminar in February, the Lehigh conference in June, and the SUNY Geometry festival in April. There is also a link to the K-theory calendar of events, maintained by Dan Grayson at the University of Illinois at Chicago Circle. 2. If you have a Mac, you may be interested in /pub/macweb1.00A3.sea.hqx, which a BinHexed Stuffit archive of MacWeb, a WWW-browser similar to Mosaic and Netscape. Old message: If you got this message from me (hovey---math.mit.edu) then you are currently subscribed. I am sure that not every one who wants to be on this list is on it, so I encourage subscribers to forward this to others who are not on it and are interested. Current graduate students in particular should know about it. Mark Hovey Papers uploaded to Hopf between Jan. 18 and Feb. 2, 1995: 1. /pub/Ravenel-Wilson/pnhopfring.dvi \title{The Hopf ring for $P(n)$} \author{Douglas C. Ravenel \thanks{Partially supported by the National Science Foundation} \\University of Rochester\\Rochester, New York 14627\\ {\small drav---troi.cc.rochester.edu} \and W. Stephen Wilson \\Johns Hopkins University\\Baltimore, Maryland 21218\\ {\small wsw---math.jhu.edu}} \maketitle \begin{abstract} We show that $E_*(\pn{n}{*})$, the $E$-homology of the $\Omega$-spectrum for $P(n)$, is an $E_*$ free Hopf ring for $E$ a complex oriented theory with $I_n$ sent to $0$. This covers the cases $P(q)_*(\pn{n}{*})$ and $K(q)_*(\pn{n}{*})$, $q \geq n$. The generators of the Hopf ring are those necessary for the stable result. The motivation for this paper is to show that $P(n)$ satisfies all of the conditions for the machinery of unstable cohomology operations set up in Boardman-Johnson-Wilson. This can then be used to produce splittings analogous to those for $BP$. \end{abstract} 2. /pub/Jardine/README (Note from Hovey: Jardine has set up his own home page. I will not be announcing which papers he has there unless he uploads them to Hopf. If you are using a WWW-browser, go to /pub/Jardine and you will see a link to his home page.) If you've got a web browser like mosaic or lynx, go to Jardine's subdirectory on the UWO Math. Dept. WWW server, which is right here.

Alternatively, Jardine's preprints are available by anonymous ftp at jardine.math.uwo.ca in the subdirectory /pub/papers/jardine. ------------------ This is the fourth installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. It seems to me that everyone will have their own home page on the WWW within five years or so. However, not everyone has WWW access at the moment, so please continue putting papers on the Hopf archive even if you do set up your own home page. It would be helpful as well, if you develop your own home page, to put abstracts as well as dvi files there and to make sure to date the papers by time of last modification. Clarence is willing to put a simple html document on the archive with a pointer to your home page, like the one Jardine put on hopf (/pub/Jardine/jardine.html). Don Davis has his own home page now: http://www.lehigh.edu/dmd1/public/www-data/dmd1.html Mark Hovey Papers uploaded to Hopf between Feb. 3 and Feb. 16, 1995: 1. /pub/Bendersky-DDavis-Mahowald/spn.dvi v1-periodic homotopy groups of Sp(n) Martin Bendersky Donald M. Davis Mark Mahowald In this paper we calculate the 2-primary v1-periodic homotopy groups of the symplectic groups Sp(n). The proof utilizes new methods of calculating the unstable Novikov spectral sequence. One corollary is that some homotopy group of Sp(n) contains an element of order 2^{2n-1} . 2. /pub/Bousfield-DDavis/bous.dvi The unstable Adams spectral sequence of $SO$ and $U$, and a splitting of unstable Ext groups by A.K. Bousfield and Donald M. Davis We construct algebraic spectral sequences which are conjectured to agree with the unstable Adams spectral sequences for the infinite unitary and special orthogonal groups $U$ and $SO$. A closely related conjecture is that the unstable Ext groups of $H^*(\Sigma CP^\infty)$ and $H^*(RP^\infty)$ split as direct sums of the unstable Ext groups for their subquotients consisting of classes whose degrees have a fixed number of 1's in their binary expansions. 3. /pub/DDavis/harp.dvi Equivalences of some v1-telescopes DONALD M. DAVIS Abstract.Certain naturally occurring spaces have isomorphic v1-periodic homotopy groups. To each is associated a mapping telescope whose ordi- nary homotopy groups equal the v1-periodic homotopy groups of the space. It is proved that the mapping telescopes of the spaces are homotopy equivalent. Lehigh University, Bethlehem, Pennsylvania 18015 4. /pub/DDavis/survey.dvi Computing v1 -periodic homotopy groups of spheres and some compact Lie groups Donald M. Davis Contents 1.Introduction 2.Definition of v1-periodic homotopy groups 3.The isomorphism v11ss (S2n+1) ss v11sss2n1 (Bqn) 4.J-homology 5.The v1-periodic homotopy groups of spectra 6.The v1-periodic UNSS for spheres 7.v1-periodic homotopy groups of SU(n) 8.v1-periodic homotopy groups of some Lie groups References HANDBOOK OF ALGEBRAIC TOPOLOGY Edited by I.M. James 1995 Elsevier Science B.V. All rights reserved 5. /pub/DDavis-Yang/huaj.dvi Tractable formulas for $v_{1}$-periodic homotopy groups of $SU(n)$ when $n \leq p^{2}-p+1$. by Donald M. Davis and Huajian Yang Let $p$ be a fixed odd prime. In \cite{Davis}, it was proved that for $\epsilon=0$ and 1, $v_1^{-1}\pi_{2k-\epsilon}(SU(n))$ has order $p^{e(k,n)}$, where $e(k,n)=\min\{\nu_p(j!S(k,j)):n\le j\le k\}$, with $S(k,j)$ the Stirling number of the second kind. In this paper, we give a more tractable formula for $e(k,n)$ when $n\le p^2-p+1$ by calculating the unstable Novikov spectral sequence. We also determine the abelian group structure when $\epsilon=1$; it was known to be cyclic when $\epsilon=0$. 6. /pub/Hovey-Sadofsky/tate-bousfield-class.dvi Tate Cohomology Lowers Chromatic Bousfield Classes By Mark Hovey and Hal Sadofsky Let $G$ be a finite group. We use the results of \cite{greenlees-sadofsky} to show that the Tate homology of $E(n)$-local spectra with respect to $G$ produces $E(n-1)$ local spectra. We also show that the Bousfield class of the Tate homology of $L_{n}X$ (for $X$ finite) is the same as that of $L_{n-1}X$. To be precise, recall that Tate homology is a functor from $G$-spectra to $G$-spectra. To produce a functor $P_{G}$ from spectra to spectra, we look at a spectrum as a naive $G$-spectrum on which $G$ acts trivially, apply Tate homology, and take $G$-fixed points. This composite is the functor we shall actually study, and we'll prove that $\langle P_{G}(L_{n}X)\rangle = \langle L_{n-1}X \rangle$ when $X$ is finite. When $G=\Sigma_{p}$, the symmetric group on $p$ letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald's functor $RP_{-\infty}(-)).$ ------------------------ This is the fifth installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between Feb. 17 and Mar. 9, 1995: 1. /pub/Aguade-Broto-Notbohm/cookemod2.abstract A mod two analogue of a conjecture of Cooke by J. Aguad\'e, C. Broto and D. Notbohm Abstract: We study spaces whose mod 2 cohomology has the form: Poly(x)\otimes Exterior(Sq^1x). We prove: Theorem: There is a space X with this cohomology if and only if x has degree 2, 4 or 8. The 'only if' part can be considered as the mod 2 version of a conjecture of Cooke. Its proof is similar to the proof for old primes (contained in Cooke.dvi) but one should be slightly more careful in small degrees. The most interesting goal of this paper is probably the construction of what we think to be remarkable space X with H*(X;F_2) = F_2[x_8] \otimes E(Sq^1x_8) 2. /pub/Boardman/stabop.abstract DVI FILE: stabop.dvi TITLE: Stable operations in generalized cohomology AUTHOR: J. Michael Boardman TO APPEAR: Handbook of Algebraic Topology, ed. I.M.James, Elsevier (Amsterdam, 1995). We describe the structure of the stable operations on E-cohomology following Adams, in a manner that generalizes to unstable operations. The appropriate context is the language of comonads and coalgebras over a comonad. The necessary category theory is developed in detail. Five examples are presented: ordinary mod p cohomology, unitary cobordism MU, Brown- Peterson cohomology BP, complex K-theory KU, and Morava K-theory K(n). 3. /pub/Boardman-Johnson-Wilson/bjw.abs DVI FILE: bjw.dvi TITLE: Unstable operations in generalized cohomology AUTHORS: J. Michael Boardman, David Copeland Johnson, W. Stephen Wilson TO APPEAR: Handbook of Algebraic Topology, ed. I.M.James, Elsevier (Amsterdam, 1995). We describe the structure of the unstable operations on E-cohomology in terms of comonads, in the style of the companion paper on stable operations. There are two variants, depending on whether we consider only the additive operations, or all unstable operations. For practical use, we unpack the comonad information and express it in terms of Hopf rings. Five examples are discussed: ordinary mod p cohomology, unitary cobordism MU, Brown-Peterson BP-cohomology, complex K-theory KU, and Morava K-theory K(n). We give two applications to BP-cohomology. The first shows that the presence of unstable operations imposes dimensional restrictions on the Landweber filtration of the BP-cohomology of a finite complex. The second constructs idempotent operations in degree k that recover the known unstable splittings of BP-cohomology. 4. /pub/Shipley/convergence.new (This is a significantly revised version of a paper already on the archive. I reproduce here an abstract followed by a brief description of the changes--Mark.) We produce new convergence conditions for the homology spectral sequence of a cosimplicial space by requiring that each codegree of the cosimplicial space has finite type mod $p$ homology. Specifically, we find conditions which ensure strong convergence if and only if the total space has $p$-good components. We also find exotic convergence conditions for cosimplicial spaces not covered by the strong convergence conditions. These results give new convergence conditions, for example, for the Eilenberg-Moore spectral sequence and for mapping spaces. This new version contains several generalizations of the old results. Specifically, the requirement of a non-empty total space is no longer needed. Also, Corollary 10.3 is a new strong convergence result requiring p-complete codegrees instead of p-nilpotent codegrees. Of course, there have been other minor changes and corrections. 5. /pub/MWeiss/betticurv.abstract Curvature and Finite Domination, by Michael Weiss. Abstract. Gromov obtained an upper bound on the Betti numbers of a closed Riemannian manifold in terms of a lower bound on the sectional curvature. It is shown that Gromov's upper bound is an upper bound on the minimum number of cells in CW-spaces dominating the manifold. 6. /pub/MWeiss/embed.abstract Calculus of Embeddings, by Michael Weiss Abstract. This is a study of spaces of smooth embeddings emb(M,N) in the spirit of immersion theory, and in the spirit of "Calculus". It leads to very efficient calculations of emb(M,N) when dim(M) is small compared to dim(N). Immersion theory appears as the "first derivative" of embedding theory, and the game is to find the higher derivatives, i.e. the "Taylor Series". The Taylor series converges when the codimension, dim(N)-dim(M), is at least 3. This follows from a multiple disjunction lemma proved recently by Goodwillie (not in his thesis). It's an announcement - no proofs. 7. /pub/MWeiss/ortho.abstract Orthogonal Calculus, by Michael Weiss Abstract. Orthogonal calculus is a way to explore spaces equipped with a filtration indexed by the finite dimensional linear subspaces V of an infinite dimensional euclidean space. Example: BO, filtered by subspaces BO(V), or BTOP, filtered by subspaces BTOP(V). Those who like to split big spaces may be interested, and the hardy ones who still like surgery theory may also be interested, since many of the moduli spaces in surgery theory come with such a filtration. Orthogonal calculus is modelled on Goodwillie calculus: Among the spaces equipped with a filtration of the type above, some are "polynomial of degree n", and the game is to approximate arbitrary ones by polynomial ones (Taylor approximation). First order approximations in orthogonal calculus have been used heavily by Bruce Williams and me in papers related to surgery. They look like generalized total Stiefel-Whitney classes, and second order approximations look like generalized total Pontryagin classes plus generalized total Stiefel-Whitney classes. ------------ This is the sixth installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Note that there is a new directory, /pub/pictures, containing pictures of topologists in gif and jpeg format. You need some kind of viewing program to see these, like xv (Unix), lview.exe (Windows), or JPEGView (Mac). With Netscape or Mosaic you can just click on them by first going to the usual spot, http://hopf.math.purdue.edu/pub/hopf.html . The file INDEX gives some information about who is in the pictures. Also note that the papers by J. P. May and coauthors announced in the first installment of this list are now in more appropriate directories. That is, there are now directories like Kriz-May and Greenlees-May, whereas before they were all in JPMay. Mark Hovey Papers uploaded to Hopf between Mar. 10 and May 3, 1995: 1. /pub/RBruner/newQ8.abstract Real connective $K$-theory and the quaternion group Dilip Bayen and Robert R. Bruner Mathematics Department Wayne State University Detroit, Michigan, 48202 dbayen---math.wayne.edu rrb---math.wayne.edu April, 1995 Let ko be the real connective K theory spectrum. We compute ko_*BG and ko^*BG for groups G whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients t(ko)^G_* of the G fixed points of the Tate spectrum t(ko) for G = Sl_2(3) and G = Q_8. The results provide a counterexample to the optimistic conjecture of Greenlees and May [Generalized Tate Cohomology, Conj 13.4], by showing, in particular, that t(ko)^G is not a wedge of Eilenberg-Maclane spectra, as occurs for groups of prime order. --------- This is the seventh installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between May 3 and May 13, 1995: 1. /pub/Blanc/Blanc_hspace.abstract % % Homotopy operations and the obstructions to being an H-space % David Blanc % % November 14, 1994 % The question of whether a given space X possesses such an H-space structure has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations. This is done by reformulating the question in terms of the realizability of a certain morphism of abelian \Pi-algebras, which translates in turn (using the author's obstruction theory for realization of such morphisms) into the requirement that a certain sequence of higher homotopy operations, taking values in \pi_{\star} X, vanishes coherently. We illustrate the theory by a couple of examples: it can be used to calculate the obstruction to CP^2 being an H-space rationally; we also show that the "torsion Whitehead product" (which we define) may be thought of as ``the first higher order obstruction'' to being an H-space, and give another example. 2. /pub/Blanc/Blanc_loop.abstract % % Loop spaces and homotopy operations % David Blanc % % April 27, 1995 % We describe two obstruction theories for a given topological space X to be a loop space, both defined in terms of higher homotopy operations: First, we explain how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{\star-1} X (which are isomorphic to \pi_{\star} Y if X\simeq\Omega Y). \ This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be stated in terms of the author's obstruction theories for realizing Pi-algebras and their morphisms. A more concrete approach, which does not require a given H-space structure on X, yields the following: Theorem A: If X is a CW complex such that all Whitehead products vanish in \pi_{\star} X, then X is homotopy equivalent to a loop space if and only if a certain collection of higher homotopy operations vanish coherently. The higher homotopy operations in question depend only on maps between wedges of spheres, and take value in homotopy groups of spheres. They are constructed by means of a certain collection of convex polyhedra which may be of independent interest. 3. /pub/Blanc/model.abstract % % New Model Categories from Old % David Blanc % % (revision: January 25, 1995) % Model categories, first introduced by Quillen, have proved useful in a number of areas - most notably in his treatment of rational homotopy, and in defining homology and other derived functors in non-abelian categories. From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E^2-term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E^2-term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra. The original purpose of this note was to provide an element in this identification which appears to be missing from the literature: namely, an explicit model category structure for the category cCA of cosimplicial coalgebras as above. What one would really like is a model category for arbitrary categories of cosimplicial universal coalgebras, analogous to Quillen's treatment of simplicial universal algebras, which is based on Quillen's ``small object argument'', and implicitly uses a procedure for transfering model category structures by means of adjoint functors (in the direction of the left adjoint; the procedure is made explicit in the paper). Unfortunately, Quillen's procedure cannot be dualized, in the categorical sense. The reason is essentially set-theoretic: more can be said about maps into a sequential colimit of sets than about maps out of a sequential limit (and thus, for example, colim is exact, for R-modules, while lim is not). Therefore, for our purposes we describe alternative (and less elegant) conditions for using adjoint functors to create new model category structures. The dual version then allows us to define model category structures for certain categories of cosimplicial universal coalgebras - including cCA. ======================== 4. /pub/Blanc/towers.abstract % % Colimits for the Pro category of towers of simplicial sets % David Blanc % % January 18, 1995 % The Pro category of towers of spaces (and of other categories) has been studied in several contexts, and used for a variety of applications in homotopy theory, shape theory, geometric topology, and algebraic geometry - as well as in the study of v_n-periodicity in unstable homotopy theory. One problem in the usual version of the Pro category of towers is that, while finite limits and colimits exist, and may be constructed in a straightforward (levelwise) manner, the same does not hold for infinite colimits; and these were needed for the application to v_n-periodicity. The construction presented here embeds a suitable subcategory of the Pro category Tow of towers of simplicial sets in a certain category Net of strict Ind-towers, in which we have explicit constructions for all colimits, as well as finite limits. This category Net can thus be thought of as a cocompletion of the Pro category of towers of spaces. There are other cocomplete categories in which Tow may be embedded - for example, the category of all pro-simplicial sets, or the full category of all inductive systems of towers. One advantage of the approach described here is that one obtains a smaller, and more mangeable, cocompletion, in this special case, and the construction of the colimits may be made quite explicitly. A side effect of our approach is the elimination of certain ``phantom phenomena'' from the Pro category of towers. 5./pub/Elmendorf-Kriz-Mandell-May/ekmm.abstract Title: Rings, modules, and algebras in stable homotopy theory Authors: A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May address: Purdue University Calumet, Hammond IN 46323 email: aelmendo---math.purdue.edu address: The University of Michigan, Ann Arbor, MI 48109-1003 email: ikriz---math.lsa.umich.edu address: The University of Chicago, Chicago, IL 60637 email: mandell---math.uchicago.edu address: The University of Chicago, Chicago, IL 60637 email: may---math.uchicago.edu Let $S$ be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category $\sM_S$ of ``$S$-modules'' whose derived category $\sD_S$ is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\sma_S R \darrow R$. These notions are essentially equivalent to the earlier notions of $A_\infty$ and $E_\infty$ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\sma_S M\darrow M$. When $R$ is commutative, the category $\sM_R$ of $R$-modules also has an associative, commutative, and unital smash product, and its derived category $\sD_R$ has properties just like the stable homotopy category. Working in the derived category $\sD_R$, we construct spectral sequences that specialize to give generalized universal coefficient and K\"{u}nneth spectral sequences. Classical torsion products and Ext groups are obtained by specializing our constructions to Eilenberg-Mac~Lane spectra and passing to homotopy groups, and the derived category of a discrete ring $R$ is equivalent to the derived category of its associated Eilenberg-Mac~Lane $S$-algebra. We also develop a homotopical theory of $R$-ring spectra in $\sD_R$, analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as $MU$-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise. Working in the module category $\sM_R$, we show that the category of finite cell modules over an $S$-algebra $R$ gives rise to an associated algebraic $K$-theory spectrum $KR$. Specialized to the Eilenberg-Mac~Lane spectra of discrete rings, this recovers Quillen's algebraic $K$-theory of rings. Specialized to suspension spectra $\Sigma^{\infty}(\Omega X)_+$ of loop spaces, it recovers Waldhausen's algebraic $K$-theory of spaces. Replacing our ground ring $S$ by a commutative $S$-algebra $R$, we define $R$-algebras and commutative $R$-algebras in terms of maps $A\sma_R A\darrow A$, and we show that the categories of $R$-modules, $R$-algebras, and commutative $R$-algebras are all topological model categories. We use the model structures to study Bousfield localizations of $R$-modules and $R$-algebras. In particular, we prove that $KO$ and $KU$ are commutative $ko$ and $ku$-algebras and therefore commutative $S$-algebras. We define the topological Hochschild homology $R$-module $THH^R(A;M)$ of $A$ with coefficients in an $(A,A)$-bimodule $M$ and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac~Lane spectra and passing to homotopy groups. 6. /pub/Green-Leary/extra.abstract Chern classes and extraspecial groups David J. Green and Ian J. Leary djg---math.uchicago.edu leary---mpim-bonn.mpg.de Abstract: The mod-$p$ cohomology ring of the extraspecial $p$-group of exponent~$p$ is studied for odd~$p$. We investigate the subquotient~$ch(G)$ generated by Chern classes modulo the nilradical. The subring of~$ch(G)$ generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them. ---------- This is the eighth installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. This list is maintained by Mark Hovey (hovey---math.mit.edu). Instructions at the end. Perhaps I should mention that generally I find out about new papers on hopf the day or day after Clarence makes them available. The erratic schedule of this list is due to variability in when people put new papers on the archive. New business: 1. I have decided to distribute this list through Don Davis' topology mailing list. The effect of this on you should be minimal, except that you will receive the other submissions to Don's list as well as my mailings. This means that you should subscribe and unsubscribe through Don, not me. The instructions at the end are suitably revised to reflect this. Let me know your reactions to this, if any. 2. I now have a home page on the world-wide web!!! It contains the back issues of these mailings, and my own papers, as well as some various computer and Emacs-related links. No graphics though. The URL is http://www.mit.edu:8001/afs/athena.mit.edu/user/h/o/hovey/Public/homepage.html Add it to your bookmarks so you don't have to type it more than once! Mark Hovey Papers uploaded to Hopf between May 13 and July 10, 1995: 1. /pub/DJGreen/m24.abstract Author: David J Green Title : The 3-local cohomology of the Mathieu group M_24 Status: To appear in Glasgow Math. J. Date : Submitted 8th August 1994. Resubmitted 11th November 1994. Abstract: The localisation at the prime 3 of the integral cohomology ring of the Mathieu group $M_{24}$ is calculated. The Chern classes of the Todd representation in $GL_{11} (F_2)$ generate the even-degree part of this ring. The mod-3 cohomology ring is also calculated. [These results have been used by C. B. Thomas to prove that the elliptic cohomology of the classifying space $BM_{24}$ is generated by Chern classes, and is therefore concentrated in even dimensions.] 1991 Mathematics Subject Classification: 20J06 (primary), 20D08 2. /pub/DJGreen/p5.abstract Author: David J Green Title : Chern classes and extraspecial groups of order $p^5$ Date : 7th June 1995 A presentation is obtained for the Chern subring modulo nilradical of both extraspecial $p$-groups of order $p^5$, for $p$ an odd prime. Moreover, it is proved that, for every extraspecial $p$-group of exponent $p$, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical. Finally, a related question about symplectic invariants is discussed, and solved for $Sp_4 (F_p)$. The main innovation in this work is to consider extraspecial groups as central products, and to partition the maximal elementary abelian subgroups of the central product into those which lift to abelian subgroups of the corresponding direct product, and those which do not. 1991 Mathematics Subject Classification: 20J06 3. /pub/Henderson/Ext_Mon_HA.abstract (I think this is an updated version of a paper that was already on the archive-- Mark) Hopf Algebra Extensions of Monogenic Hopf Algebras Gregory D. Henderson Pennsylvania State University William M. Singer has described a cohomology theory of connected Hopf algebras which classifies extensions of a cocommutative Hopf algebra by a commutative Hopf algebra in much the same way as the cohomology of groups classifies extensions of a group by an abelian group. We compute these cohomology groups for monogenic Hopf algebras, construct an action of the base ring on the cohomology groups in the case of trivial matched pairs, and use these results to further study Singer's cohomology. 4. /pub/Thomason/thomason_SymMon_equals_Spectra.abstract Symmetric monoidal categories model all connective spectra R. W. Thomason The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of -1-connective spectra. 5. /pub/Welker-Ziegler-Zivaljevic/compare.abstract Abstract : Comparison Lemmas and Applications for Diagrams of Spaces V. Welker, G.M. Ziegler, R.Zivaljevic We provide a ``toolkit'' of basic lemmas for the comparison of homotopy types of (homotopy) limits of diagrams of spaces over finite partially ordered sets, among them several new ones. In the setting of this paper, we obtain simple inductive proofs that provide explicit homotopy equivalences. (In an appendix we provide the link to the general setting of diagrams of spaces over an arbitrary small category.) We show how this toolkit of old and new diagram lemmas can be used on quite different fields of applications. In this paper we demonstrate this with respect to * the ``generalized homotopy-complementation formula'' by Bj\"orner * the topology of toric varieties (which turn out to be homeomorphic to homotopy limits, and for which the homotopy limit construction provides a suitable spectral sequence), * in the study of homotopy types of arrangements of subspaces, where we establish a new, general combinatorial formula for the homotopy types of ``Grassmannian'' arrangements, and * in the analysis of homotopy types of subgroup complexes. 6. /pub/Wolbert/current.abstract Toward an algebraic classification of module spectra by J. Wolbert Department of Mathematics, University of Chicago, Chicago, IL 60637, USA Abstract: The category of modules over an $S$-algebra (\Ai\ or \Ei\ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the $S$-algebra in question form a sufficiently nice ring, it is possible to see the deviation of the category of modules over an $S$-algebra from the corresponding algebraic module category. In particular, many algebraic modules are realized as homotopy groups of topological modules over $S$-algebras. Examples studied include real and complex $K$-theory, both connective and periodic. Further, Bousfield localization by a smashing spectrum is shown to yield a category of modules over the localized sphere. For periodic $K$-theory, these methods yield an algebraic criterion to determine when a local spectrum is a module over the $K$-theory $S$-algebra, real or complex. ------- This is the ninth installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. This list is maintained by Mark Hovey (hovey---math.mit.edu). Instructions at the end. Papers uploaded to Hopf between July 11 and July 16, 1995: 1. /pub/Hovey-Palmieri-Strickland/axiomatic.abstract Axiomatic Stable Homotopy Theory Mark Hovey, John Palmieri and Neil Strickland We define and investigate a class of categories with formal properties similar to those of the homotopy category of spectra. This class includes suitable versions of the derived category of modules over a commutative ring, or of comodules over a commutative Hopf algebra, and is closed under Bousfield localization. We study various notions of smallness, questions about representability of (co)homology functors, and various kinds of localization. We prove some theorems analogous to those of Hopkins and Smith about detection of nilpotence and classification of thick subcategories. We define the class of Noetherian stable homotopy categories, and investigate their special properties. Finally, we prove that a number of categories occurring in nature (including those mentioned above) satisfy our axioms. (Note from Mark: For a hyperlinked dvi version of this file (for use with xhdvi) see my home page, whose URL is in the instructions at the end). 2. /pub/Kashiwabara-Strickland-PTurner/dlk.abstract The Morava K-Theory Hopf Ring for BP Takuji Kashiwabara, Neil Strickland and Paul Turner Let $K$ be a $p$-local complex-oriented homology theory. The $K$-homology of the even spaces in the $\Omega$-spectrum for $BP$ form a Hopf ring. In~\cite{rawi:hrc} Ravenel and Wilson chararacterise this Hopf ring by a purely algebraic universal property, and also prove that the $K$-homology of each component of each even space is polynomial under the star product. The star-indecomposables in this Hopf ring form an algebra under the circle product. In this paper we take $K$ to be 2-periodic Morava $K$-theory, and study the resulting ring $R$ of indecomposables. In propositions~\ref{pr:pres} and~\ref{pr:iso} we give an algebraic universal property which characterises $R$, and relate this to a better-known description of the stable ring $K_*(BP)$. In theorem~\ref{th:split} we nearly provide a splitting of $R$ as a product of indecomposable factors, each of which is isomorphic modulo nilpotents to $K_*(BP)$. In the case $n=1$, there are no nilpotents and $R$ is the subring of an infinite product of copies of $K_*(BP)$ defined by a certain asymptotic condition; this is proved as theorem~\ref{th:tauiso}. We give a very simple description of the Dyer-Lashof operation on $R$ in these terms. 3. /pub/Strickland/fpfp.abstract Functorial Philosophy for Formal Phenomena Neil Strickland The purpose of this paper is to introduce the ``schematic viewpoint'' in algebraic topology. This seems to be the most natural framework in which to discuss the algebraic structures which arise from complex-oriented cohomology theories. Many of the parts which are original are joint work with Mike Hopkins and Matthew Ando. We give a definition of (formal) schemes which is well adapted to the particular technicalities which arise in the study of Morava K-theory and completed E(n)-theory. We show how to interpret the generalised (co)homology of $CP^\infty$, $Z\times BU$, $B\Sigma_{p^m}$, projective bundles and Thom spaces of complex vector bundles, and various other spaces, using the language of formal group theory. 4. /pub/Strickland/sigma4.abstract Notes on K(B\Sigma_4) at q=4 Neil Strickland In this document we describe the Morava $K$ theory (with $n=p=2$) of $\Sigma_4$ and its subgroups in excruciating detail. We use Chern classes and their transfers as generators, and describe the ring structure and all transfer and restriction maps. Much of the calculation was done using Mathematica. 5. /pub/Strickland/subgp.abstract Finite Subgroups of Formal Groups Neil Strickland In this paper we discuss various moduli problems involving the classification of finite subgroups or related structures on formal groups of finite height. Analogous problems for elliptic curves have of course been widely studied. The moduli spaces which we consider turn out to be surprisingly well-behaved. They are all Cohen-Macaulay, and most of them are smooth. The original motivation for this work came from algebraic topology, in particular the study of power operations in certain homology theories constructed by Morava. I learnt most of what I know about these questions from Mike Hopkins, and a great deal of the theory presented here was developed in discussions with him. See section~\ref{se:at} for a brief discussion of how moduli problems arise in algebraic topology. 6. /pub/Strickland/rme.abstract Rational Morava E-theory and DS^0 Neil Strickland and Paul Turner The extended-power spectrum $DS^0$ has two coproducts and two products, which interact in an intricate way. Given an $H_\infty$ ring spectrum $E$, the resulting algebraic structure on $E^*(DS^0)$ gives a framework in which to encode information about power operations. (However, we will not study power operations in this paper). Fix a prime $p$ and an integer $n>0$. We shall take $E$ to be a suitable completed and extended version of $E(n)$, which we shall call Morava $E$-theory. It is represented by a spectrum which we shall also call $E$. It is known (by unpublished work of Miller and Hopkins) that $E$ is an $E_\infty$ ring spectrum (but we shall not use this fact). In the present work, we discuss the ring $L(\ds)$ obtained from $E^0(DS^0)$ by making a certain algebraic extension and inverting $p$. Let $\Lambda$ be the group $(Q_p/Z_p)^n$, and $\Lambda^* = \Hom(\Lambda,Q/Z) = Z_p^n$ its dual. Write $\burn$ for the Burnside semiring of $\Lambda^*$, in other words the semiring of isomorphism classes of finite sets with an action of $\Lambda^*$. Write $F(\burn,L)$ for the set of functions from $\burn$ to $L$. Our central result is to give an isomorphism of $L(DS^0)$ with $F(\burn,L)$, and show that this respects all structure in sight. --------- This is the tenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Papers uploaded to Hopf between July 16 and July 28, 1995: 1. /pub/Monks/bases TITLE: Change of basis, monomial relations, and $P_t^s$ bases for the Steenrod algebra AUTHOR: Ken Monks Department of Mathematics University of Scranton Scranton, PA 18510 email: monks---uofs.edu FILENAME: BASES.DVI ABSTRACT: The relationship between several common bases for the mod 2 Steenrod algebra is explored and a new family of bases consisting of monomials in distinct $P_t^s$'s is developed. A recursive change of basis formula is produced to convert between the Milnor basis and each of the bases for which the change of basis matrix in every grading is upper triangular. In particular, it is shown that the basis of admissible monomials, the new $P_t^s$ bases, and two bases due to D. Arnon, are all bases having this property, and the corresponding change of basis formula is produced for each of them. Some monomial relations for the mod 2 Steenrod algebra are then obtained by exploring the change of basis transformations. 2. /pub/Monks/polymods TITLE: Polynomial Modules Over the Steenrod Algebra and Conjugation in the Milnor Basis AUTHOR: Ken Monks Department of Mathematics University of Scranton Scranton, PA 18510 email: monks---uofs.edu FILENAME: polymods.dvi ABSTRACT: Let $P_s=\Bbb F_2\left[ x_1,\ldots ,x_s\right]$ be the mod 2 cohomology of the $s $-fold product of $\Bbb R\text{P}^\infty $ with the usual structure as a module over the Steenrod algebra. A monomial in $P_s$ is said to be hit if it is in the image of the action $\overline{A} \otimes P_s\rightarrow P_s$ where $ \overline{A}$ is the augmentation ideal of $A$. We extend a result of Wood to determine a new family of hit monomials in $P_s$. We then use similar methods to obtain a generalization of antiautomorphism formulas of Davis and Gallant. 3. /pub/Monks/pstpaper TITLE: The nilpotence height of $P_t^s$ AUTHOR: Ken Monks Department of Mathematics University of Scranton Scranton, PA 18510 email: monks---uofs.edu FILENAME: PSTHEIGHT.DVI ABSTRACT: The method of Walker and Wood is used to completely determine the nilpotence height of the elements $\Pst$ in the Steenrod algebra at the prime 2. In particular, it is shown that $(\Pst)^{2\lfloor s/t \rfloor+2}=0$ for all $s\ge 0$, $t\ge 1$. In addition, several interesting relations in $A$ are developed in order to carry out the proof. ---------- This is the eleventh installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Papers uploaded to Hopf between July 29 and August 9, 1995: 1. /pub/Benson-Greenlees/Liegroupca.abstract \title{Commutative algebra for cohomology rings of classifying spaces of compact Lie groups.} \author{D.~J.~Benson} \address{Department of Mathematics, University of Georgia, Athens, GA 30602, USA} \email{djb---byrd.math.uga.edu} \author{J.~P.~C.~Greenlees } \address{School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} \date{} \begin{abstract} We apply the techniques of highly structured ring and module spectra to prove a duality theorem for the cohomology ring of the classifying space of a compact Lie group. This generalizes results of Benson--Carlson \cite{bc3,bc5} and Greenlees \cite{groupca} in the case of finite groups. In particular, we prove a functional equation for the Poincar\'e series in the oriented Cohen--Macaulay case. We make essential use of the theorem of Elmendorf--May \cite{em} that Borel cohomology is represented by a highly structured ring spectrum. \end{abstract} 2. /pub/Bisson-Joyal/cr_one.abstract CRone.abstract The Dyer-Lashof Algebra in Bordism Terrence Bisson and Andr\'e Joyal (June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada) We present a theory of Dyer-Lashof operations in unoriented bordism (the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where $N_*(\ )$ is unoriented bordism and $H_*(\ )$ is homology mod 2, does not respect these operations). For any finite covering space we define a ``polynomial functor'' from the category of topological spaces to itself. If the covering space is a closed manifold we obtain an operation defined on the bordism of any $E_\infty$-space. A certain sequence of operations called squaring operations are defined from two-fold coverings; they satisfy the Cartan formula and also a generalization of the Adem relations that is formulated by using Lubin's theory of isogenies of formal group laws. We call a ring equipped with such a sequence of squaring operations a $D$-{\it ring}, and observe that the bordism ring of any free $E_\infty$-space is free as a $D$-ring. In particular, the bordism ring of finite covering manifolds is the free $D$-ring on one generator. In a second compte-rendu we discuss the (Nishida) relations between the Landweber-Novikov and the Dyer-Lashof operations, and show how to represent the Dyer-Lashof operations in terms of their actions on the characteristic numbers of manifolds. 3. /pub/Bisson-Joyal/cr_two.abstract CRtwo.abstract Nishida Relations in Bordism and Homology Terrence Bisson and Andr\'e Joyal (June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada) This is the second of a series of Compte Rendus. In the first [1] we have presented a theory of Dyer-Lashof operations in unoriented bordism. Here we shall discuss the (Nishida) relations between Dyer-Lashof and Landweber-Novikov operations. They are used to represent the algebra $N_*\Sigma$ of covering manifolds in terms of their homology characteristic numbers. The proofs are based on the properties of the covering space operations and the notions of $D$-ring and $Q$-ring introduced in [1]. 4. /pub/Elmendorf-May/SGalgebras.abstract Algebras over equivariant sphere spectra by A.D.Elmendorf and J.P.May Abstract: We study algebras over the sphere spectrum S_G of a compact Lie group G. In particular, we show how to construct S_G algebras from S-algebras, where S is the nonequivariant sphere spectrum. This gives a reservoir of equivariant examples to which recently developed algebraic techniques in stable homotopy theory can be applied. A special case will be used in a companion paper of Benson and Greenlees to study the ordinary cohomology of the classifying space BG. 5. /pub/Greenlees-May/gmMU.abstract Localization and completion theorems for MU-module spectra by J.P.C.Greenlees and J.P.May Abstract: Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU; examples include all the standard homotopical MU-modules, such as the Brown-Peterson and Morava K-theory spectra. We prove localization and completion theorems for the computation of M_*(BG) and M^*(BG). The G-spectrum MU_G that represents stablilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum S_G and there is an MU_G module M_G whose underlying MU-module is M. This allows the use of topological analogues of constructions in commutative algebra. The computation of M_*(BG) and M^*(BG) is expressed in terms of spectral sequences whose respective E_2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU_*^G and its module M_*^G. The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I_*-functor with smash product. -------- This is the twelfth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Let me request that all submitters to the Hopf archive remember that your abstracts should be in human readable form. The first of the following abstracts had to be heavily edited by me to make it at all readable. My apologies for any errors thereby introduced. Mark Hovey Papers uploaded to Hopf between August 9 and August 16, 1995: 1. /pub/Lindenstrauss/thh6.abstract (This paper is a revised version of thh3.dvi previously on the archive-Mark) Topological Hochschild Homology of Extensions of Z/pZ by Polynomials, and of Z[x]/(xn) and Z[x]/(xn-1) Ayelet Lindenstrauss Princeton University Princeton, NJ 08544 Introduction The Dennis trace is a map from the algebraic K-theory of a ring into Hochschild homology. The Hochschild homology groups are easier to calculate,but they are relatively simple. The chance of finding invariants which have interesting pullbacks to K-theory increases if we succeed in factoring the Dennis trace map through intermediate groups, and obtaining trace maps which land in groups with more structure. The K-groups of a ring R are homotopy groups of BGL +(R). The Hochschild homology groups are homotopy groups of a space CH(R),the realization as a simplicial space of the standard (bar) complex used to calculate Hochschild homology. The Dennis trace map is the map on homotopy groups induced by a continuous map BGL +(R) --> CH(R) ; which can be factored through a space called THH(0)(R), which is the 0th space in the topological Hochschild homology spectrum THH(R). The existence of this intermediate step was conjectured by Goodwillie,and shown by Bokstedt in [2]. Topological Hochschild homology is constructed analogously to the bar complex CH(R), with Eilenberg-MacLane spectra K(R;m) and smash products substituted for the ring R and tensor products. The factoring of the Dennis trace (which is explained for example in [4]) consists of a map BGL +(R) --> THH (0)(R) which is the 0-level part of a spectrum version of the Dennis trace, and a map THH (0)(R) --> CH(R) induced by the component map . Bokstedt calculated the homotopy type of this spectrum (which actually is an spectrum) for R = Z and R = Z/pZ. Thus the Dennis trace map can be made to factor through ss (THH (0)(R)) = ssS(THH (R)): Bokstedt used his calculations of THH(Z) to obtain results concerning the algebraic K-theory of Z. Perhaps it is also possible to factor the Dennis trace map through an object similar both to cyclic homology and to topological Hochschild homology- a sort of `cyclic topological Hochschild homology'. Bokstedt,Hsiang and Madsen (see [4]) succeeded in creating such a theory in a different context which involves objects that can be treated using the same FSP formalism as the functor X 7! R[X] which we will use here, and indeed constructed spectra which were p-equivalent to the K-theory spectrum for any given prime p. The first part of this paper briefly describes the construction of topological Hochschild homology, following [2], and the method and results of Bokstedt's calculation of THH(Z) and THH(Z/pZ) in [3]. The second part of the paper calculates HS(THH(R);Z/pZ) for any ring R which is of the form Z/qZ[x]/(f(x)) (q prime,f(x) in Z/qZ[x]),and for the rings Z[x]/(xn) and Z[x]/(xn-1). The calculation uses the same kind of spectral sequence Bokstedt used in [2]. The result of these calculations is a splitting HS (THH(R); Z/pZ) = HH(R; Z/pZ) cross HS(THH(Z);Z/pZ) of the ring of stable homology classes,where the multiplication is induced by a shuffle-product. The third part of this paper describes the explicit form of the homotopy type of THH(R) for the rings Z/qZ[x]/(f(x)) (for any polynomial f), Z[x]/(xn), and Z[x]=(xn-1). 2. /pub/JPMay/modnew.abstract Equivariant and nonequivariant module spectra by J.P. May Abstract: Let $G$ be a compact Lie group, let $R_G$ be a commutative algebra over the sphere $G$-spectrum $S_G$, and let $R$ be its underlying nonequivariant algebra over the sphere spectrum $S$. When $R_G$ is split as an algebra, as holds for example for $R_G=MU_G$, we show how to ``extend scalars'' to construct a split $R_G$-module $R_G\sma_R M$ from an $R$-module $M$. This allows the wholesale construction of highly structured equivariant module spectra from highly structured nonequivariant module spectra. In particular, it applies to construct $MU_G$-modules from $MU$-modules and therefore gives conceptual constructions of equivariant Brown-Peterson and Morava $K$-theory spectra. ----------- This is the thirteenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between August 16 and August 30, 1995: 1. /pub/GLewis/eqv_splt_sptr SPLITTING THEOREMS FOR CERTAIN EQUIVARIANT SPECTRA L. Gaunce Lewis, Jr. Let $\Gamma $ be a compact Lie group, $\Pi $ be a normal subgroup of $\Gamma $, $G=\Gamma / \Pi $, $X$ be a $G$-space and $Y$ be a $\Gamma $-space. There are a number of results in the literature giving a direct sum decomposition of the group of $\Gamma $-equivariant stable homotopy classes of maps from the suspension spectrum of $X$ to the suspension spectrum of $Y$. Here, these results are extended to a decomposition of the group $[B,C]_\Gamma $ of equivariant stable homotopy classes of maps from an arbitrary $G$-spectrum $B$ to any $\Gamma $-spectrum $C$ carrying a cosplitting (a new type of structure introduced here). Any naive $\Gamma $-spectrum, and any spectrum derived from such by a change of universe functor, carries a cosplitting. This decomposition of $[B,C]_\Gamma $ is a consequence of the fact that, if $C$ is cosplit and $({\scr{F}}^\prime ,\scr{F})$ is any pair of families of subgroups of $\Gamma $, then there is a splitting of the cofibre sequence $$(E{\scr{F}}_+ \wedge C)^\Pi \rightarrow (E{\scr{F}}^\prime _+ \wedge C)^\Pi \rightarrow (E({\scr{F}}^\prime ,\scr{F}) \wedge C)^\Pi $$ constructed from the universal spaces for the families. Both the decomposition of the group $[B,C]_\Gamma $ and the splitting of the cofibre sequence are proven here not just for complete $\Gamma $-universes, but for arbitrary $\Gamma $-universes. Various technical results about incomplete $\Gamma $-universes that should be of independent interest are included in this paper. These include versions of the Adams and Wirthm\"uller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed point spectrum $(E({\scr{F}}^\prime ,\scr{F} ) \wedge C)^\Pi $ which gives computational force to the intuition that what really matters about a $\Gamma $-universe $U$ is which orbits $\Gamma /\Lambda $ embed as $\Gamma $-spaces in $U$. -------------- This is the fourteenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between August 30 and September 9, 1995: 1. /pub/Henn/kmod COMMUTATIVE ALGEBRA OF UNSTABLE $K$ - MODULES, LANNES' $T$ - FUNCTOR AND EQUIVARIANT MOD - P COHOMOLOGY by Hans--Werner Henn Let $p$ be a fixed prime and let $K$ be an unstable algebra over the mod - $p$ Steenrod algebra $A$ such that $K$ is finitely generated as graded $\FF_p$ - algebra. Let $K_{fg}-\Ua$ denote the abelian category of finitely generated $K$ - modules with a compatible unstable $A$ - module structure. We study various concepts of commutative algebra in this setting. The r\^ole of the prime ideal spectrum of a commutative ring is here taken by a category $\Rav (K)$ which, roughly speaking, consists of the $A$ - invariant prime ideals of $K$ together with certain ``Galois information''; sheafs will correspond to functors on this category, and the r\^ole of the sheaf associated to a module will be taken by the components of Lannes' $T$ - functor. We discuss the notions of support, of ${ \gl a}$ - torsion modules (for an invariant ideal ${ \gl a}$ of $K$) and of localization away from the Serre subcategory $\Ta ors ({ \gl a})$ of ${ \gl a}$ - torsion modules in our setting. We show that the category $K_{fg}-\Ua$ has enough injectives and use these injectives to study these localizations and their derived functors; they are closely related to the derived functors of the ${ \gl a}$ - torsion functor $F_{{ \gl a}}$. Our results are formally analogous to Grothendieck's results in the classical situation of modules over a noetherian commutative ring R [Gr]. Important for applications is the case $K=H^*BG$, the mod - $p$ cohomology of a classifying space of a compact Lie group (or a suitable discrete group), and $M=H^*_GX$ where $X$ is a (suitable) $G$ - $CW$ - complex. In these cases the category $\Rav (K)$ and the functor on $\Rav (K)$ associated to $H^*_GX$ can be described in terms of group theoretic and geometric data, and our theory yields a far-reaching generalization of a result of Jackowski and McClure [JM] resp. of Dwyer and Wilkerson [DW2]. As a concrete application of our theory we describe the size of the kernel of the restriction map from the unknown mod - $2$ cohomology of the $S$ - arithmetic group $GL(n,\Z[1/2])$ to the known cohomology of its subgroup $D_n$ of diagonal matrices. 2. /pub/Moller/normalizer J.M. Moller: Normalizers of maximal tori Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that ``normalizer'' commutes with ``centralizer''. 3. /pub/Moller/survey J.M. Moller: Homotopy Lie groups This is a survey of Dwyer and Wilkerson's p-compact groups intended for a general mathematical audience. ---------- This is the fifteenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. My home page appears to be broken. I'll try to fix it, but in any case I have been slow about updating it recently. Sorry about that. In case you are looking for more recent back issues of this list, Don Davis' home page should have them. The URL for his discussion list is http://www.lehigh.edu/dmd1/public/www-data/algtop.html . Mark Hovey Papers uploaded to Hopf between September 9 and September 23, 1995: 1. /pub/Ando/PowerOpsEll.dvi Power operations in elliptic cohomology and representations of loop groups Matthew Ando The first part of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. The second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups. The third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part. 2. /pub/Lindenstrauss/ram2.ps (Note: At present this file is only available in .ps form--Mark) Abstract of The Topological Hochschild Homology of the Gaussian Integers by Ayelet Lindenstrauss: Topological Hochschild homology is calculated explicitly for the rings ${\bf Z}[\sqrt2]$, ${\bf Z}[\sqrt{-2}]$, and ${\bf Z}[i]$. The 2-torsion of the topological Hochschild homology is calculated for the ring of integers in any quadratic extension of the rationals. 3. /pub/Rudyak/ts.dvi THE SPECTRA k AND kO ARE NOT THOM SPECTRA Yu. B. Rudyak Abstract. Here is proved that neither k nor kO are Thom spectra. This was conjectured by Mahowald in 1979. Mathematisches Institut Universitat Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg 1, Germany. ---------- This is the sixteenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between September 24 and October 24, 1995: This time we have updated versions of two papers that were already on the archive: /pub/Ando/PowerOpsEll and /pub/Lindenstrauss/ram2 . The paper by Ando is on power operations, elliptic cohomology, and representations of loop groups, and the paper by Lindenstrauss is about calculating the topological Hochschild homology of the Gaussian integers. I don't know how extensive the revisions are. We also have one new paper: 1. /pub/Henderson/Spec_Seq_Ext_HA.abstract SPECTRAL SEQUENCES FOR THE CLASSIFICATION OF EXTENSIONS OF HOPF ALGEBRAS Gregory D. Henderson October 15, 1995 We construct spectral sequences which provide a way to compute the cohomology theory that classifies extensions of graded connected Hopf algebras over a commutative ring as described by William M. Singer. Specifically, for (A,B) an abelian matched pair of graded connected R-Hopf algebras, we construct a pair of spectral sequences relating H^*(B,A) to Ext_B(R,Cotor_A(R,R)). To illustrate these spectral sequences, we examine the special case of B a monogenic graded connected Hopf algebra and also analyze an extension of Hopf algebras given by James P. Lin. ----------- This is the seventeenth installment of abstracts of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between October 24 and October 31, 1995: We have one new paper: 1. /pub/Dwyer-Mitchell/curves On the $K$-theory spectrum of a smooth curve over a finite field by W. G. Dwyer and S. A. Mitchell We study the algebraic $K$-theory spectrum associated to a smooth curve (either complete or affine) over a finite field, and determine (for instance) - the topological $K$-theory groups of the spectrum - the Bousfield localization of the spectrum with respect to topological $K$-theory - the topological $K$-theory groups of the zero space in the associated $\Omega$-spectrum This determination is done in terms of classical algebraic invariants of the curve. We also prove that the above Bousfield localization of the spectrum is a retract (in positive dimensions) of the spectrum itself. Actually, to obtain the results it is necessary to $\ell$-complete the spectrum at an odd prime $\ell$ which is different from the characteristic of the finite field; similarly, "topological $K$-theory" as used above means $\ell$-completed topological $K$-cohomology. --------- We have one new paper at Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between October 31 and November 7, 1995: 1. /pub/Ravenel-Wilson-Yagita/bpcohfrommork Brown-Peterson cohomology from Morava K-theory Douglas C. Ravenel W. Stephen Wilson and Nobuaki Yagita We give some structure to the Brown-Peterson cohomology (or its $p$-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even dimensional. We can say that the Brown-Peterson cohomology is even dimensional (concentrated in even degrees) and is flat as a $BP^*$-module for the category of finitely presented $BP^*(BP)$-modules. At first glance this would seem to be a very restricted class of spaces, but the world abounds with naturally occurring examples: Eilenberg-MacLane spaces, loops of finite Postnikov systems, classifying spaces of all finite groups whose Morava K-theory is known (including the symmetric groups), $QS^{2n}$, $BO(n)$, $MO(n)$, $BO$, $\ImJ$, etc. We finish with an explicit algebraic construction of the Brown-Peterson cohomology of a product of Eilenberg-Maclane spaces. (Note from Mark: Actually Igor Kriz has recently given a finite group whose Morava K-theory is not concentrated in even degrees.) --------- We have three new papers at Clarence Wilkerson's archive. Instructions at the end. My home page has moved, and also has been somewhat updated. The new URL is http://www-math.mit.edu/~hovey/ Mark Hovey Papers uploaded to Hopf between November 7 and November 25, 1995: 1./pub/Dwyer/decompositions Homology decompositions for classifying spaces of finite groups by W. G. Dwyer We look at ways of expressing the classifying space of a finite group G, at least up to mod p homology, as a homotopy colimit of classifying spaces of subgroups of G. What results is a general theory which includes as special cases the decompositions of Jackowski-McClure and of Jackowski-McClure-Oliver. 2. /pub/Slack/tfodd Infinite loop spaces with odd torsion free homology by Michael Slack Abstract: It is shown that an infinite loop space with no odd torsion in its integral homology also has no odd torsion in its homotopy. Combined with known results of Steve Wilson, this gives a complete classification; all such spaces are products of the Wilson spaces, which are the building blocks of the spaces in the omega spectrum for BP. Comments, questions, and corrections are all welcomed. For other papers by Slack (published, preprints, and in progress), you can visit his homepage at http://www.wmich.edu/math-stat/faculty/slack/. 3. /pub/JWu/Simplicial-group-1 On Combinatorial Descriptions of Homotopy Groups of K(ss; 1) Jie Wu November 13, 1995 Abstract We will give a combinatorial description of homotopy groups of K(ss;1). Thus, by Kan-Thurston Theorem, a combinatorial descrip- tion of the homotopy groups of a simply connected suspension space is given.In particular, all of the homotopy groups of the 3-sphere are combinatorially given. 4. /pub/JWu/Wu.copy ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS JIE WU In [?], G. Carlsson introduced a simplicial group construction which gives a generalization of Milnor's F(K) construction [?]. Roughly speaking, if we construct a simplicial group which is a free product of a simplicial group G over a pointed simplicial set X, then we get a simplicial group construction for (BG ^ X), where BG is the classifying space of G. In this article, we give a categorial view of this construction. Let C be a category. A fibrewise simplicial object over C, roughly speaking, is a diagram over C with indices in a simplicial set. This is an abstract view of fibrewise topology [?] or sheaf theory. If the category C has coproducts, then the abstract F-construction is defined to be certain coadjoint functor from the category of fibrewise simplicial objects over C to the category of simplicial objects over C. Suppose that there is a functor T from C to the category of pointed simplicial sets such that T preserves coproducts up to homotopy. Then there is an induced functor T from the category of fibrewise simplicial objects over C to the category of pointed bisimplicial sets. Theorem ?? shows that T is homotopy equivalent to T ffiF. Let C be a category of monoids. Notice that the bar-construction B preserves coproduct up to homotopy [?]. A corollary of this abstract theorem is the Carlsson theorem. An application of Carlsson's construciton to homotopy theory is to give a representation of the homotopy groups of simply connected suspension spaces to certain combinatorial groups as centers [?]. Applications of Carlsson's construction to minimal simplicial groups are given in [?]. In this paper, we pay more attention to the geometry of the Carlsson construction. The word length filtration is considered. The resulting cofibres are certain smash product pinched out certain reduced diagonal elements (Proposition ??). Our construction in the monoid case is a generalization of the James construction [?]. We construct certain natural map Hn : (Y ^X ) ! n(Y n^ (X(n)=4n )), which is similar to the James-Hopf map,for any path connected CW-complex Y and any pointed CW-complex X, where X(n)is the n-th fold self smash product of X and 4n = f(x1 ^: :^:xn) 2 X(n)j xi= xi+1for some ig (Theorem ??). A direct application of these natural maps is to give a decomposition of H(F P 1^ X) for F =R, C or H. Let F P21= F P 1=F P 1. Research at MSRI is supported in part by NSF grant DMS-9022140. Theorem 0.1. Let F = R,C or H and let X be a pointed space. Suppose that H is a multiplicative homology theory such that (1) both H (F P 1) and H (F P21) are free H (pt)-modules;and (2) the inclusion of the bottom cell Sd !F P 1 induces a monomorphism in the homology. Then there is a product filtration fFrH (F P 1^ X)gr0 of H(F P 1 ^X) such that F0 =H (pt) and Fr=Fr1 = (d1)rH (X (r)=4r); where d = dimR F and is the suspension. Furthermore, this filtration is natural for X. ----------- We have four new papers at Clarence Wilkerson's archive. This is a good time to remind submitters of papers: Abstracts must be readable by humans!! I do some editing to improve readability, but I can only do so much, and this time I didn't even feel like doing that. Instructions at the end. Mark Hovey Papers uploaded to Hopf between November 25 and December 8, 1995: 1./pub/Chacholski/B-M A Generalization of The Triad Theorem of Blakers-Massey. Wojciech Chacholski Let F-->A-->X and H-->A-->E be fibration sequences. Take the homotopy push-out B=hocolim(X<--A-->E), and then take the homotopy pull-back Y=holim(X-->B<--E). There is a natural map q:A-->Y. The main statement of the paper is that if F and H are connected, then the homotopy fiber of the suspenson of q is built from the smash of F and H using homotopy push-outs, wedges and telescopes. 2. /pub/Chacholski/barc CLOSED CLASSES WOJCIECH CHACHOLSKI 1. Introduction A non empty class C of connected spaces is said to be a closed class if it is closed under weak equivalences and pointed homotopy colimits. Aclosed class can be characterized as a non empty class of connected spaces which is closed under weak equivalences and is closed under certain simple operations: arbitrary wedges, homotopy push-outs and homotopy sequential colimits. The notion of a closed class was introduced by E. Dror Farjoun [6]. Two important constructions give rise to examples ofclosed classes. The first one is the Bousfield-Dror periodization functor PA [2]. The class ofthose spaces X, such that PA X is weakly contractible, forms a closed class. By looki* *ng just at the properties of this class wecan prove, for example, that PA X is weakly equivalent toPA X (see [2], [4]). The second construction is E. Dror Farjoun's colocalization functor C WA. The class of those spaces X, for which there exists a space Y, such that X is weakly equivalent toC WAY , forms a closed class. This class is denoted by C(A) and is called the class of A-cellul* *ar spaces. By looking just at the properties ofthe class C (A) we can prove, for example, that CWAX is weakly equivalent to CWA X (see [4], [6]). We say that a closed class C is closed under extensions by fibrations, if for every fibration sequence (Z! E ! B), such that Z andB belong to C, E belongs to C. A closed class C is closed under extensions by fibrations if and only if for every diagram F : I! C ,such that the classifying space BI belongs to C, the unpointed homotopy colimit hocolimIF belongs to C. The purpose of this paper is to understand to what extent a closedclass is closed under extensions by fibrations and under taking unpointed homotopy colimits. We start with proving a theorem that, in particular, implies: fflLet F : I ! Spaces? be a pointed diagram, such that the classifying space BI belongs to C. If for every i 2 I, F (i)b elongs to C , thenso does the unpointed homotopy colimit hocolimIF. fflLet (Z ! E ! B) be a fibration sequence with a section. If Z and B belong to C, then so does E. fflLet F : I ! C andG : I ! C be diagrams and : F ! G be a natural transformation. If hocolimIF belongs to C, then so does hocolimIG. 2 WOJCIECH CHACHOLSKI Surprisingly these and many other results are the consequences of just one statement, see theorem 5.1. We continue with investigating the properties of a base space B (respec- tively of the classifying space BI),which will guarantee that a closed class C * *is closed under extensions by fibrations with base B (respectively C is closed un- der taking the unpointed homotopy colimit of diagrams F : I ! C ). We study the following class: D(C ) =fB I jifF : I ! C isa diagram, then hocolimIF 2 Cg The main result of this paper is: Theorem. The class D(C) is a closed class and it is closedunder extensions by fibrations. 3. /pub/Chacholski/thesis On The Functors CW_{A} and P_{A}. Wojciech Chacholski I am looking at the relation between Bousfield's localization functor P_{A} and Dror Frajoun's colocalization functor CW_{A}. I am studying the question: to what extent the following sequence is exact: Spaces --P_{A}--> Spaces --CW_{A}--> Spaces --P_{A}--> Spaces The image of P_{A} is equal to the kernel of CW_{A}. The correlation between the kernel of P_{A} and the image of CW_{A} is more complicated. I proved that The kernel of P_{A} is the closure of the image of CW_{A} under taking extensions by fibrations. In the paper I am giving algorithms to construct the functor CW_{A} out of P_{A} and vice versa. I am using these algorithms to show that S^{n} is in the image of CW_{\Omega S^{n+1}} if and only if n=1,3,7 and that for every n S^{n} is in the kernel of P_{\Omega S^{n+1}}. 4. /pub/JWu/Min_Simpl_Set \title{On products in the minimal simplicial sets} \author{Jie Wu} It is well known that every fibrant simplicial set~$X$ is homotopy equivalent to a minimal simplicial set~$Y$. If~$X$ is a loop space, then there is an induced multiplication on~$Y$ given by the composite $$Y\times Y\to X\times X\to X\to Y.$$ In this case, $Y$ is a minimal simplicial set together with a multiplication. It is an old story in topology to look for a product in the minimal simplicial sets. J.F.~Adams showed that the two-stage Postnikov system~$X$ with $\pi_n(X)\ne0$ and $\pi_{n+1}(X)\ne0$ is homotopy equivalent to a minimal simplicial group~\cite{A}. J.~Milnor gave a counterexample that $\Omega(S^{n+1}\la n+1,n+2,n+3\ra)$ is not homotopy equivalent to a minimal simplicial group, where $S^{n+1}\la n+1,n+2,n+3\ra$ is the $3$-stage Postnikov system by taking the first three homotopy groups of $S^{n+1}$~\cite{Wu1}. G.~Whitehead gave some non-associative minimal simplicial abelian groupoids. In this paper, we study minimal simplicial $H$-sets, \ie, minimal simplicial sets with multiplications. We always assume that a simplicial $H$-set $X$ has a strict unit element~$e$. A simplicial $H$-set $X$ is called \emph{strong homotopy associative} if \begin{enumerate} \item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times X\times X\to X\quad\text{rel.\ }e\times X\times X$; \item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times X\times X\to X\quad\text{rel.\ }X\times e\times X$; and \item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times X\times X\to X\quad\text{rel.\ }X\times X\times e$, \end{enumerate} where $\mu\colon X\times X\to X$ is the multiplication. It was pointed out by J.~Stasheff that if a multiplication $\mu\colon X\times X\to X$ is homotopy associative, then there exists a multiplication $\mu'\colon X\times X\to X$ which is strong homotopy associative~\cite{Sta}. Notice that the homotopy groups $\pi_*(X)$ can be identified with the cycles in~$X$ if~$X$ is minimal, where $x\in X_n$ is a cycle if $d_jx=*$ for all~$j$. A simplicial $H$-set is said to be \emph{right (left) group-like} if~$X$ has a strict right (left) inverse map. Our main theorem is as follows. \begin{thm} Let $X$ be a connected strong homotopy associative minimal simplicial $H$-set. Then: \begin{enumerate} \item The associativity $$(ab)c=a(bc)$$ holds if one of $a,b,c$ is in $\pi_*(X)$. \item For $a\in X_n$, there exists a unique left inverse~$b$ in~$X_n$ such that $ba=e$ and there exists a unique right inverse~$c$ in~$X_n$ such that $ac=e$. \item The commutativity $$ab=ba$$ holds if $a\in\pi_*(X)$. \item $X$ is generated by $\pi_*(X)$ as a simplicial $H$-set. \item The fibration $$F_n(X)\to P_n(X)\to P_{n+1}(X)$$ is a central extension and it is also a principal $F_n(X)\cong K(\pi_n(X),n)$ bundle, where $\{P_n(X)\}_{n\ge0}$ is the Moore-Postnikov system of~$X$. \item Let $H\colon\pi_*(X)\to\bar H_*(X;Z)$ be the Hurewicz map. Then there exists a (graded) subset~$S$ of~$\pi_*(X)$ such that \begin{enumerate} \item $H(x)\ne0$ for $x\in S$; \item $H(x_1)\ne H(x_2)$ for $x_1\ne x_2$ in~$S$; and \item $X$ is generated by~$S$ as a right (or left) group-like simplicial $H$-set. \end{enumerate} \end{enumerate} \end{thm} Assertions~(1) to~(4) give a general description of the relations between the homotopy groups and the total space. Assertion~(5) shows that the Postnikov system of a connected homotopy associative minimal simplicial $H$-set is very nice. Assertion~(6) gives a relation between the total space and its homology. This supports the Moore conjecture in some sense although it is still very unclear if the exponents of the homotopy groups are related to the homology groups. We should point out that the minimal subcomplex of a loop space is non-associative in general. This paper is our starting work to understand the product structures in the ``minimal models'' for the loop spaces. The understanding of these product structures may help us to study some homotopy problems such as the Freyd conjecture, the Moore conjecture, and the Kavarre invariants problem. As an example, the complete answer when a two-stage Postnikov system~$X$ with $\pi_n(X)=\Z/2$ and $\pi_{n+1}(X)=\Z/2$ is homotopy equivalent to a minimal simplicial group is given as follows. \begin{thm} Let $n,i>0$ and let $X$ be a two-stage Postnikov system with $\pi_n(X)=\Z/2$ and $\pi_{n+i}(X)=\Z/2$. Then~$X$ is homotopy equivalent to a minimal simplicial group if and only if the Postnikov invariant of~$X$ is trivial or~$Sq^{i+1}$. \end{thm} ---------- Happy New Year! This is the one-year anniversary of this modest service. We have three new papers at Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between December 8,1995 and January 6, 1996: 1. /pub/Lesh/uass The Unstable Adams Spectral Sequence for Two-Stage Towers Kathryn Lesh University of Toledo Let KP denote a mod 2 Eilenberg-MacLane space. In this paper we study the mod 2 unstable Adams spectral sequence (UASS) of a space Y with polynomial cohomology which is obtained as the fiber of a map between simply connected spaces X \rightarrow KP to which the Massey-Peterson theorem applies. By using specially constructed Adams resolutions, we study the relationship between the UASS for X and for Y. Given conditions on X, we give information on the E_{2} term of the UASS for Y which actually gives a splitting of Ext under some circumstances. In the case that the UASS for X collapses at E_{2}, we show how to use a primary level calculation to compute almost complete information about the d_{2} differentials in the UASS for Y. 2./pub/Lesh/inf-loop Infinite Loop Spaces from Group Theory Kathryn Lesh University of Toledo Abstract: We generalize the Barratt-Priddy-Quillen theorem \Omega B (\coprod B\Sigma_{n}) is homotopy equivalent to QS^{0} by using tom Dieck's classifying spaces for a family of subgroups of a group. We show how to take a compatible choice of families F_{n} of subgroups of \Sigma_{n} and obtain an infinite loop space by group completing \coprod BF_{n}. The spaces QS^{0} and K(Z,0) are recovered as extreme cases and the infinite loop spaces obtained from other families can be thought of as interpolating between stable homotopy and integral homology. We study two special cases: the family of subgroups of the alternating groups, and the family generated by elementary abelian p-subgroups whose generators are disjoint p-cycles. We compute the infinite loop spaces which are formed in these cases and show that the latter is closely related to the cofiber of the transfer map. This paper will appear in Mathematische Zeitschrift. 3. /pub/Lesh/hybrids (The following paper is a revision of a paper already on the archive) Hybrid Spaces with Interesting Cohomology Kathryn Lesh University of Toledo Abstract: Let p be an odd prime, and let R be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to p. To such an algebra is associated a p-adic pseudoreflection group W, and we assume that W is of order prime to p and irreducible. Adjoin to R a one-dimensional element z, and give R[z] an action of the Steenrod algebra by \beta z = 0 and \beta x = (|x|/2) zx for an even dimensional element x. We show that the subalgebra of elements of R[z] consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a p-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson. This paper has appeared as TAMS 347, 3247-3262 (1995) ----------------- Five new papers this time. For some reason titles and author names are missing in most of these abstracts. I preferred them with titles and author names myself. Remember the way your abstract looks is entirely up to you. Instructions at the end. Mark Hovey Papers uploaded to Hopf between January 7,1996 and February 11, 1996: 1. /pub/Brown-Wensley/ind-nrm4 Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types Ronald Brown and Christopher D. Wensley School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, U.K. email: r.brown, c.d.wensley --- bangor.ac.uk We obtain some explicit determinations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types, and so of the second homotopy modules, of homotopy pushouts of maps of classifying spaces of discrete groups induced by certain morphisms of the groups. In some cases, the first k-invariant is calculated, using methods of free crossed resolutions. 2. /pub/Chacholski/SF12/ The outcome of this paper is two surprising facts: Theorem 1 Let F-->E-->B be any fibration over a connected space B, and h* be any homology theory. If F-->E is h* isomorphism, then B is h* acyclic Theorem 2 Let A and Y be pointed spaces and A-->X be any map. Let F be the homotopy fiber of the cofiber map X-->X/A. If A is weakly equivalent to a suspension of a connected space, then the following statements about Y are equivalent: -) map*(A,Y) is weakly contractible -) map*(F,Y) is weakly contractible 3. /pub/Johnson-Wilson/johnson-wilson-bu If $V$ is an elementary abelian $2$-group, Ossa proved that the connective $K$-theory of $BV$ splits into copies of ${\bf Z}/2$ and of the connective $K$-theory of the infinite real projective space. We give a brief proof of Ossa's theorem. 4. /pub/Moller/deterministic Deterministic p-compact group (J.M. Moller) A p-compact group is said to be deterministic if it is determined by the normalizer of a maximal torus. It is said to have N-determined automorphisms if any two of its automorphisms are determined by their restrictions to the normalizer of the maximal torus. We consider the class of p-compact groups with N-determined automorphisms and the, not unrelated, class of deterministic p-compact groups. The paper also contains some speculations that these two classes in fact comprise all p-compact groups. 5. /pub/Silverman/strip \noindent Let $\Tof{k}{f} = \sq{2^{k-1}f} \cdot \sq{2^{k-2}f} \cdot \ldots \cdot \sq{2f} \cdot \sq{f}$ in the mod-2 Steenrod algebra $\stnstar$, and let $\chi$ denote the canonical antiautomorphism of $\stnstar$. Given positive integers $k$, $\Lambda$ and $j$ with $1 \leq j \leq \Lambda$, we prove that \begin{eqnarray*} \chi \Tof{k}{2^{\Lambda}-j} & = & \Tof{\Lambda-(j-1)}{2^{j-1}(\spike{k})} \cdot \chi \Tof{k}{2^{j-1}-j}, \end{eqnarray*} generalizing formulae of Davis and the author. Given a positive integer $f$, denote by $\mu(f)$ the minimal number of summands in any representation of $f$ in the form $\sum (\spike{i_k})$. The antiautomorphism formula above implies that for $f = 2^{\Lambda}-j$,\ $1 \leq j \leq \Lambda +2$, the excess of $\chi \Tof{k}{f}$ satisfies $\ex{\chi \Tof{k}{f}} = (\spike{k})\mu(f)$ for all $k$, confirming the conjecture of the author ({\em Proc. Amer. Math. Soc.}, 119(2):657-661, 1993) for such $f$. We also prove that $\ex{\chi \Tof{k}{f}} \leq (\spike{k}) \mu(f)$ for all $f$ and $k$. ----------------- Two new papers, a revised version, and an improved abstract this time. Clarence asks me to tell everyone to please send him e-mail when you upload a file to the archive. I have given up on trying to make html versions of these messages. This has the advantage that my home page will now always have the latest messages, though in text form. Mark Hovey Papers uploaded to Hopf between February 11,1996 and April 4, 1996: 1. /pub/Brown-Golasinski-Porter-Tonks/equivcrs2.dvi Spaces of maps into classifying spaces for equivariant crossed complexes by R Brown, M Golasi\'{n}ski, T Porter, and A Tonks. ABSTRACT We give an equivariant version of the homotopy theory of crossed complexes. The applications generalize work on equivariant Eilenberg-Mac Lane spaces, including the non abelian case of dimension 1, and on local systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk and Svensson. Further, we give results not just on the homotopy classification of maps but also on the homotopy types of certain equivariant function spaces. R. Brown, T.Porter School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT United Kingdom r.brown,t.porter---bangor.ac.uk M. Golasinski Department of Mathematics Nicholas Copernicus University Torun Poland marek---mat.uni.torun.pl A. Tonks Institut Matematicas Univerdidad Autonoma Barcelona 08193 Bellaterra Barcelona Spain tonks---mat.uab.es 2. /pub/Greenlees/s1q (Note from Mark: as explained below, this paper can be downloaded whole or in 4 parts, with obvious titles) Title: Rational $S^1$-equivariant stable homotopy theory. Author: J.P.C.Greenlees Abstract: We make a systematic study of rational $S^1$-equivariant cohomology theories, or rather of their representing objects, rational $S^1$-spectra. In Part I we construct a complete algebraic model for the homotopy category of $S^1$-spectra, reminiscent of the localization theorem. The model is of homological dimension one, and simple enough to allow practical calculations; in particular we obtain a classification of rational $S^1$-equivariant cohomology theories. The model for semifree spectra is the derived category of the abelian category whose objects are $\Q [c]$-modules $N$ with a graded vector space $V$ giving an isomorphism $N[c^{-1}] = \Q [c , c^{-1}] \tensor V$; the model for arbitrary spectra is an appropriate generalization. In Part II we identify the algebraic counterparts of all the usual $S^1$-spectra and constructions on $S^1$-spectra. This enables us in Part III to give a rational analysis of a number of interesting phenomena, such as the Atiyah-Hirzebruch spectral sequence, the Segal conjecture, $K$-theory and topological cyclic cohomology. For reasons of size this is broken into four parts Part I (including Introduction and Table of Contents) Part II Part III Appendices (including Conventions and Index). Also available The whole thing (s1q123.dvi) Introduction and contents only (s1qIntro.dvi). 3./pub/Hovey-Palmieri-Strickland/axiomatic Axiomatic stable homotopy theory Mark Hovey, John Palmieri, and Neil Strickland This is the final version of a paper that was already on the archive. It is quite long. There are the usual bug fixes, plus an improved thick subcategory theorem, better understanding of morphisms of stable homotopy categories, a complete rewrite of the Noetherian section, with better results, and a couple of new sections on the Bousfield lattice. 4. /pub/Silverman/strip (This paper was announced last time, but the abstract was untexable. This version of the abstract is self-contained, and should be texable with a begin and end document thrown in. The paper itself has not changed.) Stripping and Conjugation in the Steenrod Algebra Judith H. Silverman Indiana University --- Purdue University Columbus judith---iu-math.math.indiana.edu To appear in Journal of Pure and Applied Algebra \noindent Let $S(k;f) = Sq^{2^{k-1}f} \cdot Sq^{2^{k-2}f} \cdot \ldots \cdot Sq^{2f} \cdot Sq^{f}$ in the mod-2 Steenrod algebra $A^*$, and let $\chi$ denote the canonical antiautomorphism of $A^*$. Given positive integers $k$, $\Lambda$ and $j$ with $1 \leq j \leq \Lambda$, we prove that \begin{eqnarray*} \chi S(k; 2^{\Lambda}-j) & = & S(\Lambda-(j-1); 2^{j-1}(2^k-1)) \cdot \chi S(k; 2^{j-1}-j), \end{eqnarray*} generalizing formulae of Davis and the author. Given a positive integer $f$, denote by $\mu(f)$ the minimal number of summands in any representation of $f$ in the form $\sum (2^{i_k}-1)$. The antiautomorphism formula above implies that for $f = 2^{\Lambda}-j$,\ $1 \leq j \leq \Lambda +2$, the excess of $\chi S(k;f)$ satisfies ${\mbox {ex}}(\chi S(k;f)) = (2^{k}-1)\mu(f)$ for all $k$, confirming the conjecture of the author ({\em Proc. Amer. Math. Soc.}, 119(2):657-661, 1993) for such $f$. We also prove that ${\mbox {ex}}(\chi S(k;f)) \leq (2^{k}-1) \mu(f)$ for all $f$ and $k$. ------------------- Six new papers this time. I remind you again that abstracts should be human-readable, and that you should send e-mail to Clarence when you upload a paper to hopf. Mark Hovey Papers uploaded to Hopf between April 4,1996 and Jun 3, 1996: 1. /pub/Broto-Crespo/hhhh H-spaces with noetherian mod two cohomology algebra by CARLOS BROTO and JUAN A.CRESPO Abstract: The object of this paper is to analyse the structure of connected H-spaces with noetherian mod two cohomology algebra. We will show that, up to 2-completion, they are, essentially, finite mod~2 $H$-spaces and their 3-connected covers, $\C P^\infty$, $B\Z/2^r$ and certain extensions of these. 2. /pub/Brown-Szczarba/szczarba \magnification=1200 \nologo \documentstyle{amsppt} \NoBlackBoxes \topmatter \title On the Rational Homotopy Type of Function Spaces\endtitle \author Edgar H. Brown, Jr. and Robert H. Szczarba\endauthor \abstract The main result of this paper is the construction of a minimal model for the function space $\Cal F(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi_*(\Cal F(X,Y))\otimes Q =\pi_*(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_*(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration. \endabstract \endtopmatter 3. /pub/Intermont/vkampen An Equivariant van Kampen Spectral Sequence Michele Intermont Mesa State College Grand Junction, CO 81502 mintermo---mesa5.mesa.colorado.edu This paper constructs an equivariant homotopy spectral sequence for any finite group G, finite dimensional representation V, and two suitably connected G-CW complexes X and Y. The spectral sequence converges to the collection of equivariant homotopy groups of the wedge of X and Y, while the E^2 term depends only on the equivariant homotopy groups of X and of Y, along with primary homotopy operations. The edge homomorphism of the spectral sequence is actually an isomorphism in a range, which is the equivariant van Kampen theorem of L.G. Lewis, Jr. When G is the trivial group, the spectral sequence reduces to that of C.R. Stover. 4. /pub/Lindenstrauss/ram26 \nopagenumbers \magnification=\magstep1 \centerline{\bf Abstract of the paper:} \centerline{\bf The Topological Hochschild Homology of the Gaussian Integers} \smallskip \centerline{by Ayelet Lindenstrauss} \bigskip The calculation in this paper gives the $2$-torsion in the topological Hochschild homology of rings of integers in quadratic extensions of the rationals. In general, it is not hard to deduce the $p$-torsion in the topological Hochschild homology of rings of integers which do not ramify at $p$ from the corresponding torsion for the integers. Thus, for the ring of Gaussian integers, and also for the integers with the square root of $2$ or $-2$ adjoined, this paper completes the calculation of the topological Hochschild homology spectrum (which is a priori known to be a product of Eilenberg-MacLane spectra). Precisely because of this a priori knowledge, the homotopy groups can be found by doing a homology calculation. A spectral sequence arising from the filtration of topological Hochschild homology by simplicial `skeleta' converges to the desired homology. If one reduces the rings in question modulo $2$, the spectral sequence collapses at its second term. This comparison bounds the non-triviality of higher differentials in the spectral sequence for the original ring. It is explicitly demonstrated, using simplicial calculations, that the higher differentials are as non-trivial as they could be, given this bound. \end 5. /pub/Morava/Luminy2.abstract Quantum generalized cohomology Jack Morava \begin{center}{Abstract for {\bf Quantum generalized cohomology}}\bigskip \end{center} \noindent There is a variant of Segal's category of Riemann surfaces, in which morphisms are stable complex algebraic curves [i.e. double points are allowed], with some smooth points marked; composition is defined by glueing at marked points. The spaces of morphisms in this category are built from the compactified moduli spaces $\overline M_{g,n}$ of Deligne, Mumford, and Knudesen; here $g$ is the genus and $n$ is the number of marked points. A generalized topological field theory taking values in the category of module-spectra over a ring-spectrum $\bf R$ is a family $$\tau_{g,n} : \overline M_{g,n} \rightarrow {\bf M} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf M} = {\bf M}^{\wedge n}$$ of maps, which respect composition of morphisms. More precisely, $\bf M$ is an $\bf R$-module spectrum, $\wedge_{\bf R}$ is the Robinson smash product, and $\bf M$ is endowed with a suitably nondegenerate bilinear form $${\bf M} \wedge_{\bf R} {\bf M} \rightarrow {\bf R}.$$ This data entails the existence of an $\bf R$-algebra structure on $\bf M$, such that $\tau_{g,1}$ is a morphism of monoids if the moduli space of curves is given the pair-of-pants product; it seems to define a natural context for quantum generalized cohomology.\medskip \noindent There is an interesting example of all this, associated to a smooth algebraic variety $V$. It is closely related to the Tate $\bf MU$-cohomology of the universal cover of the free loopspace of $V$, but it can be described more concretely in terms of the rational Novikov ring $\Lambda = {\Bbb Q} [H_{2}(V,{\Bbb Z})]$ of $V$ by setting ${\bf R} = {\bf MU} \otimes \Lambda$; then {\bf E} is the function spectrum $F(V,{\bf R})$ representing the cobordism of $V$ tensored with $\Lambda$, and the bilinear pairing is defined by Poincar\'e duality. In this case $\tau_{g,n}$ represents the cobordism class of the space of stable maps [in the sense of Kontsevich] from a curve of genus $g$, marked with $n$ ordered smooth points together with an indeterminate number of unordered smooth points, to $V$. A variant construction requires the unordered points to lie on a cycle $z$ in $V$; this defines a parameterized family of multiplications satisfying the analogue of the WDVV equation. When $V$ is a point, the resulting theory boils down to the version of topological gravity I advertised at the Adams Symposium; the coupling constant of the associated topological field theory is the cobordism analogue of Manin's exponential $$\sum_{n \geq 0} \overline M_{0,n+3} \frac {z^{n}}{n!} .$$ Although much of the machinery used here comes from fields adjacent to topology, this paper is concerned with the old problem of constructing complex cobordism out of Riemann surfaces by some analogue of the plus-construction. Having hacked through the physics background, I hope to produce a more topological account in the near future. \medskip \noindent This is to appear in Contemporary Math., in the Proceedings of the Hartford/Luminy Conference on the Renaissance of Operads, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov. \end{document} 6. /pub/Rudyak/LScategory (The following abstract was unreadable--slight modifications were made by Mark to make it slightly more readable, but only slightly) SOME REMARKS ON CATEGORY WEIGHT Yu.B. Rudyak April 1996 Abstract. We develop and apply the conceptof category weight which was introduced by Fadell and Husseini. We remark that elements of maximal category weight enable us to control the Lusternik-Schnirelmann category of a space. For example, we prove that if f : N --> M is a map of degree 1 of closed stable parallelizable manifolds and dim M 2cat M4 (sic) then cat N cat M (sic). Furthermore, we prove that category weight of every non-trivial Massey product is at least 2. In particular, if the Massey product hu1; :;:u:ni,ui 2 eH(X) (sic), is defined and hu1; : :;:uni 6= 0 (sic) then catX > 1, even if 0 2 hu1; : :;:uni (sic). Using a result of Singhof, we also prove that cat(M Sm ) = catM + 1 provided (sic) ------------------ Two new papers this time. Mark Hovey Papers uploaded to Hopf between Jun 3,1996 and Jun 5, 1996: 1. /pub/Davis-Lueck/assembly Authors: James F. Davis and Wolfgang Lueck Title: Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory Email: jfdavis---indiana.edu, lueck---topologie.mathematik.uni-mainz.de We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced group C*-algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. 2. /pub/Henn/sl3 \title{The cohomology of $SL(3,{\Z}[1/2])$} \author{Hans-Werner Henn} \date{ } \maketitle \begin{abstract} We compute the cohomology of $SL(3,{\Z}[1/2])$ with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod - $2$ congruence subgroups of $SL(3,{\Z})$ with coefficients in ${\FF}_p$ for $p>2$. Finally we compute the cohomology of $GL(3,{\Z}[1/2])$. \end{abstract} ----------------- Four new papers this time. Clarence will be out of town for a few weeks. Mark Hovey Papers uploaded to Hopf between Jun 5,1996 and Jun 26, 1996: 1. /pub/Dwyer/sharp Sharp homology decompositions for classifying spaces of finite groups W. Dwyer Suppose that G is a finite group. A homology decomposition for BG is a way of constructing BG up to mod p homology as a homotopy colimit of classifying spaces of subgroups K of G. The decomposition is said to be sharp if the mod p homology spectral sequence associated to the homotopy colimit collapses to give an isomorphism between the H^*BG and colim H^*BK. We develop techniques for showing that a decomposition is sharp, and apply the techniques to a number of examples. 2. /pub/Dwyer-Wilkerson/kahler-differentials/stein Kahler differentials, the T-functor, and a theorem of Steinberg William G. Dwyer and Clarence W. Wilkerson (Notre Dame University and Purdue University) It is shown that any component of the Lannes T-functor applied to a finitely generated connected graded polynomial algebra with an unstable action of the mod p Steenrod algebra yields a finitely generated polynomial algebra. This is a purely algebraic analogue of the observation that for a connected Lie group with no homology p-torsion, each centralizer of a sub-elementary abelian p-group also has no homology p-torsion. It has the algebraic consequence, previously proved by Nakajima, that if W is a subgroup of GL(V) such that Symm[V#]^W is polynomial algebra, then for any subset U of V, the stabilizer group W_U has the same property. Examples are given of reflection groups in positive characteristic for which stabilizer subgroups are not reflection groups. These include $W(F_4)$ at $p=3$, $W(E_8)$ for $p=5$, and $W( SU(pN)/Z/pZ)$ at $p=p$, for most N. 3. /pub/Kashiwabara/bpqs Abstract for "Brown-Peterson cohomology of QS^{2n}" by Takuji Kashiwabara In this paper we introduce the notion of $BP$-destabilization functor, and we show that BP cohomology of QS^{2n}'s (n>0) are naturally isomorphic to the destabilization of BP-cohomology of S^{2n}. This describes BP cohomology of QS^{2n} completely algebraically. In the passage we also determine their Morava K-theories. 4. /pub/Kashiwabara/bpqx Abstract for "On Brown-Peterson cohomology of QX" by Takuji Kashiwabara In this paper we compute BP cohomology of QX when BP cohomology of X satisfies the Landweber's exact functor theorem (under some other technical hypotheses). This includes the cases such as sphere's, Eilenberg-Maclane spaces, and many classifying spaces. ----------------- Clarence has been on vacation for a while, so we have 12 new papers this time. Please include the name and title of the paper in the abstract--otherwise Clarence has to sort through them all himself. Mark Hovey Papers uploaded to Hopf between Jun 26,1996 and Aug 9, 1996: 1. /pub/Arone-Mahowald/identity The Goodwillie tower of the identity functor and the unstable $v_n$-periodic homotopy of spheres. Greg Arone and Mark Mahowald Goodwillie's tower of the identity functor is a tower of fibrations converging to unstable homotopy, whose fibers are infinite loop spaces. The fibers in this tower were first described by B. Johnson. We reformulate this description and investigate the tower in the case of a sphere. The main result is that in the case of a sphere the tower is finite in $v_n$-periodic homotopy. The proof involves calculating the stable cohomology of the fibers in the tower, which may be of independent interest. It is possible that some changes will still be made. Comments on the manuscript are most welcome. 2. /pub/Blanc/mcw % % David Blanc % Mapping spaces and M-CW complexes % July 9, 1996 % abstract: The concept of ``homotopy groups with coefficients'', in which spheres are replaced by a Moore spaces as the representing objects, were first studied by Peterson, and in greater detail by Neisendorfer. Much of homotopy theory can be redone in this spirit, with an arbitrary but fixed space $\M$ and its suspensions replacing the spheres not only in the definition of homotopy groups, but also in that of a $CW$-complex, loop space, and so on. In particular, an M-CW complex is a space constructed inductively by successively attaching M-cells. Some of the properties of ordinary $CW$ complexes carry over to M-CW complexes - e.g., the Whitehead theorem - but others do not. In this note we address the question of recovering the space X from the mapping space X^M, for a special class of ``self-map resolvable'' spaces M, a question analogous to the classical one of recovering X from its n-fold loop space. Just as for loop spaces, one needs some additional structure on X^M in order to do so. Our procedure for recovering X is given recursively by a sequence of homotopy colimits. We may also think of this procedure as another construction of an M-CW approximation functor. Our approach can be made more explicit in the case of the mod k Moore space. 3. /pub/Blanc/rat % David Blanc % Homotopy operations and rational homotopy type % June 19, 1996 % abstract: % We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected CW complex X. The (integral) homotopy type of X is determined by its homotopy groups pi_*X, together with the action of all primary homotopy operations on it, and of certain higher homotopy operations. However, Whitehead products are the only non-trivial primary homotopy operations on the rational homotopy groups, and the relevant higher order operations are also simpler than in the integral case. Here we exhibit a collection of higher homotopy operations, which, together with the rational homotopy Lie algebra itself, determine the rational homotopy type of X. These higher operations are certain subsets of pi_* X which are indexed by elements in the homology of a certain inductively defined collection of differential graded Lie algebras (DGLs) defined below. Thus they take values in the corresponding cohomology groups, with coefficients in pi_* X. It is clear intuitively that cycles in the homology of a DGL L which are not generators, or products of other cycles, represent ``higher homotopy operations'' in L, in some sense. One of our objectives is to formalize this intuition within a more general framework. Moreover, if L represents the rational homotopy type of a topological space X, it is not always evident how to represent these rational operations as integral higher order operations. In order to address this problem, we must consider a somewhat ``flabbier'' model of rational homotopy than that provided by differential graded Lie algebras, namely a certain class of differential graded non-associative algebras. Thus we also provide a (somewhat incomplete) answer to the following question: what additional structure on the ordinary homotopy groups pi_* X of a simply-connected space X, beyond the Whitehead products, is needed to determine its homotopy type up to rational equivalence? 4. /pub/Christenson-Strickland/phantoms (There is a typo in Christensen's name, so this path may be corrected). Phantom Maps and Homology Theories J. Daniel Christensen and Neil P. Strickland (jdchrist---mit.edu and neil---pmms.cam.ac.uk) Keywords: phantom map, stable homotopy theory, spectrum, triangulated category Abstract: We study phantom maps and homology theories in a stable homotopy category $\cS$ via a certain Abelian category $\cA$. We express the group $\cP(X,Y)$ of phantom maps $X\ra Y$ as an $\Ext$ group in $\cA$, and give conditions on $X$ or $Y$ which guarantee that it vanishes. We also determine $\cP(X,HB)$. We show that any composite of two phantom maps is zero, and use this to reduce Margolis' axiomatisation conjecture to an extension problem. We show that a certain functor $\cS\ra\cA$ is the universal example of a homology theory with values in an AB 5 category, and compare this with some results of Freyd. 5. /pub/DDavis/pexp5 Elements of large order in pi_*(SU(n)) Donald M. Davis Last updated July 5, 1996. 50 pages long. Abstract It is proved that if p is an odd prime, then some homotopy group of SU(n) contains an element of order p^e, where e = n-1+[(n+2p-3)/p^2]+[(n+p^2-p-1)/p^3]. The method is to compute v1-periodic homotopy groups, using the unstable Novikov spectral sequence. This should be very close to the largest orders of v1-periodic elements, and we conjecture that the elements of largest order are v1-periodic. 6. /pub/Kuhn/loopspaces Let $T(j)$ be the dual of the $j^{th}$ Brown-Gitler spectrum (at the prime 2) with top class in dimension $j$. Then it is known that $T(j)$ is a retract of a suspension spectrum, is dual to a stable summand of $\Omega^2 S^3$, and that the homotopy colimit of a certain sequence $T(j) \rightarrow T(2j) \rightarrow \ldots$ is a wedge of stable summands of $K(V,1)$'s, where $V$ denotes an elementary abelian 2 group. In particular, when one starts with $T(1)$, one gets $K(Z/2,1) = RP^{\infty}$ as one of the summands. Refining a question posed by Doug Ravenel, I discuss a generalization of this picture. I consider certain finite spectra $T(n,j)$ for $n,j \geq 0$ (with $T(1,j) = T(j)$), dual to summands of $\Omega^{n+1}S^{N}$, conjecture generalizations of all of the above, and prove that all these conjectures are correct in cohomology. So, for example, $T(n,j)$ has unstable cohomology, and the cohomology of the colimit of a certain sequence $T(n,j) \rightarrow T(n,2j) \rightarrow \dots$ agrees with the cohomology of the wedge of stable summands of $K(V,n)$'s corresponding to the wedge occurring in the $n=1$ case above. One can also map the $T(n,j)$ to each other as $n$ varies, and the cohomological calculations suggest conjectures related to symmetric products of spheres. 7. /pub/Kuhn/symmetricpowers If $bF_q$ is the finite field of order $q$ and characteristic $p$, let $F(q)$ be the category whose objects are functors from finite dimensional $F_q$--vector spaces to $F_q$--vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in $F(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in F_q[\Sigma_n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma_n}$, $ S^n(V) = V^{\otimes n}/\Sigma_n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$. Fixing $F$, we discuss the problem of computing $Hom_{F(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $Hom_{F(q)}(S_m, G)$ for all $m$. When $q = p$, we get a complete answer for any functor $F$ chosen from the families listed above. Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when $F=S^n$, arose in recent work on the homology of iterated loopspaces. 8. /pub/Levi/comp Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) A comparison Criterion for certain loop spaces We study a comparison criterion for loop spaces on $p$-localized classifying spaces of certain finite $p$-perfect groups $G$. In particular we show that, under certain hypotheses, the homotopy type of those spaces is determined by the mod-$p$ cohomology of $G$ together with a finite Postnikov system. Appeared in Contemporary Math. 181, (1995) 9. /pub/Levi/conj Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) A counterexample to a conjecture of Cohen Let G be a finite p-superperfect group. A conjecture of F. Cohen suggests that \Omega BG^p is resolvable by finitely many fibrations over spheres and iterated loop spaces on spheres, where (-)^p denotes the p-completion functor of Bousfield and Kan. We produce a counter-example to this conjecture and discuss some related aspects of the homotopy type of \Omega BG^p. Appeared in Progress in Math. Vol 136, (1996) Birkhauser Verlag. 10. /pub/Levi/fin Ran Levi ran---math.nwu.edu Northwestern University On p-completed classifying spaces of discrete groups and finite complexes We show that for certain discrete $p$-perfect groups $G$, in particular for all $p$-perfect groups of finite cohomological dimension, the loop space on the $p$-completed classifying space $\lbgp$ is a retract of the loop spaces on a certain finite complex. For finite $vcd$ groups we provide a bound on the the dimension of such a complex. Submitted 11. /pub/Levi/gr Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) On Homological rate of Growth and the Homotopy type of \Omega BG^p Let G be a finite p-perfect group. We show that the mod-p homology of \Omega BG^p grows either polynomially or semi-exponentially. A conjecture due to F. Cohen states that \Omega BG^p for such groups G is spherically resolvable of finite weight. We show that any space X, which satisfies the conclusion of Cohen's conjecture has the property that its homology grows at most hyper-polynomially of finite degree. Thus we conclude that if a group $G$ satisfies the Cohen conjecture then the homology of \Omega BG^p grows polynomially. This enable us to produce counter examples to the conjecture. We study some further homotopy properties of our examples. We also show that the mod p homology of \Omega BG^p is a finitely generated, Lie nilpotent algebra provided it grows polynomially. Preprint 12. /pub/Levi/lsht Ran Levi Torsion in Loop Space Homology of Rationally Contractible Spaces Abstract Let $\R$ be a torsion free principal ideal domain. We study the growth of torsion in loop space homology of simply-connected $\dg\R$-coalgebras $C$, whose homology admits an exponent $r$ in $R$. Here by loop space homology we mean the homology of the loop algebra construction on $C$. We compute a bound on the growth of torsion in such objects and show that in general this bound is best possible. Our methods are applied to certain simply-connected spaces associated with classifying spaces of finite groups, where we are able to deduce the existence of global exponents in loop space homology. ----------------- Clarence has switched http daemons, and would appreciate hearing from users about any changes you notice. The new software is supposed to make downloading quicker and more reliable. We have just one new paper this time. Mark Hovey Papers uploaded to Hopf between Aug 9,1996 and Aug 18, 1996: 1. /pub/Gorbunov-Mahowald-Symons/peterme Infinite subgroups of the Morava stabilizer groups V. Gorbounov M. Mahowald P. Symonds We discuss certain infinite subgroups of the Morava stabilizer groups and outline some applications in homotopy theory. Consider a cyclic algebra ${\Bbb D}$ over ${\Bbb Q_p}$ of index $p-1$ with Hasse invariant ${1\over {p-1}}$. Let ${\Bbb S}l$ be the group of strict units of ${\Bbb D}$ of reduced norm one. The main result is the following: \begin{thm}\label{one} There is a p-th root of unity $\alpha\in {\Bbb S}l$ and a (p-1)-st root of $-p$ in ${Bbb D}$ such that \begin{enumerate} \item \{$\displaystyle \alpha^{\omega^i}$, $1\leq i\leq p-1$\} generate a subgroup $G$ of ${\Bbb S}l$, which is isomorphic to a free product of $p-1$ copies of ${\Bbb Z/p}$: ${\Bbb Z/p}*\cdots *{\Bbb Z/p}$ \item $G$ is dense in ${\Bbb S}l$. \end{enumerate} \end{thm} ------------------ A few words from Clarence. Hopf is now running new software and hardware. It is possible that the "gzip-on-the-fly" and other ftp tricks will be broken for a short while. Any problems should be reported to Clarence. It would be very helpful for Clarence if all abstracts had author and title lines that look like the following example: Great facts about Lie groups A. Borel, S. Lie, and H. Weyl Notice there are no \author or \title commands, all the authors are on the same line, and it is lower case. So please submit your abstracts in this form from now on. Note that if you want to use TeX (which I don't recommend), you can still make life easier for Clarence, I think, by using \title{ Fermat's Last Theorem } \author{ P. Fermat } There are 8 new papers this time. Mark Hovey Papers uploaded to Hopf between Aug 18,1996 and Sep 17, 1996: 1. /pub/Arkowitz-Lupton/ark_lup7 Title: Rational Obstruction Theory and Applications to Homotopy Sets. Authors: Martin Arkowitz and Gregory Lupton. Abstract: We develop an obstruction theory for homotopy of homomorphisms $f,g : {\Cal M }\to{\Cal N }$ between minimal differential graded algebras. We assume that ${\Cal M }=\Lambda V$ has an obstruction decomposition given by $V=V_0\oplus V_1$ and that $f$ and $g$ are homotopic on $\Lambda V_0$. An obstruction is then obtained as a vector space homomorphism $V_1\to H^*({\Cal N})$. We investigate the relationship between the condition that $f$ and $g$ are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study various questions about the set of homotopy classes of maps $[{\Cal M },{\Cal N }]$. We study cohomologically trivial homotopy classes of maps from ${\Cal M }$ to ${\Cal N }$. We investigate a conjecture of Copeland-Shar on the homotopy set $[{\Cal M },{\Cal N }]$. We give examples of minimal algebras that have few homotopy classes of self-maps. Because of the equivalence of the homotopy category of minimal algebras and the homotopy category of rational spaces, this study yields analogous results for rational spaces. By exploiting basic properties of rationalization, we de-localize some of the results about rational spaces to obtain information on the set of homotopy classes of maps between two finite complexes. 2. /pub/Arone-Kankaanrinta/Liehomology The homology of certain subgroups of the symmetric group with coefficients in Lie(n). Greg Arone and Marja Kankaanrinta Abstract: The authors compute the mod p homology of groups of the form \Sigma_{n_1}\times\cdots\times\Sigma_{n_k} with twisted coefficients in the module Lie(n), where n_1+\cdots+n_k=n. 3. /pub/Gorbunov-Siegel-Symonds/mor-stab-at2 The Cohomology of the Morava Stabilizer Group $\Bbb S_2$ at the Prime $3$ Vassily Gorbounov Stephen F. Siegel Peter Symonds We compute the cohomology of the Morava stabilizer group {\small $\Bbb S_2$} at the prime $3$ by resolving it by a free product ${\Bbb Z}/3*{\Bbb Z}/3$ and analyzing the ``relation module.'' 4. /pub/Gorbunov-Symonds/higherEO2 (This file has associated with it three .ps files which contain pictures. These are called fign.ps, wher n=1,2,3. If you get the .ps file, instead of the .dvi file, these will already be included. Mark) Toward the homotopy groups of the higher real $K$-theory $EO_2$ Vassily Gorbounov Peter Symonds The higher real $K$-theories $EO_n$ have been constructed by Hopkins and Miller recently. When $n=2$ this construction suggests a way of defining an "integral elliptic cohomology", which yields the usual elliptic cohomology when 6 is inverted. Our main result is the calculation of the initial term of the spectral sequence $H^*(G,E)^{\mbox{\rm \small Gal}} \Rightarrow \pi _*EO_2$, which converges to the homotopy groups of the "elliptic cohomology" spectrum localized at the prime 3. Here $G$ is the group $\Bbb Z/3\rtimes\Bbb Z/4$ and $E$ is an infinitely generated module for $G$ over $\Bbb Z_3$, which arises from the theory of formal groups. We also show how the integral modular forms of Deligne appear naturally in this initial term. This calculation was originally sketched by Hopkins and Miller, but the details were never published, so we have proceeded by our own methods. 5. /pub/Greenlees/augmentation_ideals ``Augmentation ideals of equivariant cohomology rings.'' J.P.C.Greenlees School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. j.greenlees------sheffield.ac.uk The purpose of this note is to establish a number of useful results about the augmentation ideal $J$ for the coefficient ring $F_G^*$ of a Noetherian complex orientable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions. 6. /pub/Greenlees/Rational_O2_equivariant J.P.C.Greenlees ``Rational O(2)-equivariant cohomology theories.'' 8 pp A complete algebraic model is given for rational O(2)-spectra. The main input is the model for SO(2)-spectra described in ``Rational S1-equivariant stable homotopy theory'', and the special case of ``Rational Mackey functors for compact Lie groups I'' applying to O(2). It turns out that any O(2)-spectrum is described by an SO(2)-spectrum with O(2)/SO(2)-action, together with an equivariant sheaf over the space of dihedral subgroups of O(2). ---------------------------------------------------------------------- | John Greenlees | Tel : (0114)-282 4437 direct | | Mathematics and Statistics | (0114)-276 8555 central | | University of Sheffield | | | Hicks Building | Fax : (0114)-282 4292 | | Sheffield S3 7RH | Email: j.greenlees---sheffield.ac.uk | ---------------------------------------------------------------------- | WWW Homepage | |http://www.shef.ac.uk/uni/academic/I-M/ms/staff/greenlees/index.html| ---------------------------------------------------------------------- 7. /pub/Greenlees/Rational_Mackey_functors_I (This is a new version of a paper that was already on the archive.) J.P.C.Greenlees ``Rational Mackey functors for compact Lie groups I.'' 32pp The category of rational Mackey functors is shown to be equivalent to the category of equivariant sheaves on the Weyl-toral category. The advantage is that the latter category only has morphisms in one direction, and may thus be seen to have global dimension equal to the rank of the group. This version is a simplified and corrected version of an earlier preprint. ---------------------------------------------------------------------- | John Greenlees | Tel : (0114)-282 4437 direct | | Mathematics and Statistics | (0114)-276 8555 central | | University of Sheffield | | | Hicks Building | Fax : (0114)-282 4292 | | Sheffield S3 7RH | Email: j.greenlees---sheffield.ac.uk | ---------------------------------------------------------------------- | WWW Homepage | |http://www.shef.ac.uk/uni/academic/I-M/ms/staff/greenlees/index.html| ---------------------------------------------------------------------- 8. /pub/Ishiguro/pairing Pairings of p-compact groups and H-structures on the classifying spaces of finite loop spaces Kenshi Ishiguro Fukuoka University, Fukuoka 814-80, Japan We consider the maps between classifying spaces of p-compact groups of the form $BX \times BY --->>> BZ$. The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z. An application implies that, for a finite loop space X, its classifying space BX is an H-space (Hopf space) if and only if X is the product of a torus and a finite abelian group. It is also shown, for a compact Lie group G, exactly when the p-completed space $(BG)\p$ has an H-structure. ---------------- There are 2 new papers this time. Mark Hovey Papers uploaded to Hopf between Sep 18,1996 and Oct 2, 1996: 1. /pub/Arkowitz-Gutierrez/ark_gut1 \title COMULTIPLICATIONS ON FREE GROUPS AND WEDGES OF CIRCLES \endtitle \author Martin Arkowitz and Mauricio Gutierrez \endauthor \address Mathematics Department, Dartmouth College, Hanover, NH 03755\endaddress \email Martin.Arkowitz\---Dartmouth.EDU\endemail \address Mathematics Department, Tufts University, Medford, MA 02155 \endaddress \email mgutierr\---tufts.edu\endemail \abstract By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group $F$. We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$. For a comultiplication $m$ of $F$ we define a subset $\Delta_m \subseteq F$ of quasi-diagonal elements which is basic to our investigation of associativity. The subset $\Delta _m$ can be determined algorithmically and contains the set of diagonal elements $D_m$. We show that $D_m$ is a basis for the largest subgroup $A_m$ of $F$ on which $m$ is associative and that $A_m$ is a free factor of $F$. We also give necessary and sufficient conditions for a comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\aut F$ on the set of comultiplications of $F$. We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles. \endabstract 2. /pub/Costenoble-Waner/Surgery_for_compact_G The equivariant Spivak normal bundle and equivariant surgery for compact Lie groups Steven R. Costenoble and Stefan Waner Dept. of Mathematics 103 Hofstra University Hempstead, NY 11550 We generalize the results of [Costenoble and Waner, The equivariant Spivak normal bundle and equivariant surgery, Mich. Math. J. 39 (1992), 415-424] to compact Lie groups. Using a suitable ordinary equivariant homology and cohomology, we define equivariant Poincar\'e complexes with the properties that (1) every compact $G$-manifold is an equivariant Poincar\'e complex, (2) every finite equivariant Poincar\'e complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the $\pi$-$\pi$ Theorem holds for equivariant Poincar\'e pairs under suitable gap hypotheses. ----------------- There is one new paper this time. Mark Hovey Papers uploaded to Hopf between Oct 2,1996 and Oct 10, 1996: 1. /pub/Palmieri/palmieri-f-iso Quillen stratification for the Steenrod algebra John H. Palmieri Let A be the mod 2 Steenrod algebra, and let Q denote the category of exterior sub-Hopf algebras of A, where the morphisms are given by inclusions. The restriction maps Ext_A (Z/2,Z/2) --> Ext_E (Z/2,Z/2), for E in Q, can be assembled into a map i : Ext_A (Z/2, Z/2) --> lim_Q Ext_E (Z/2,Z/2). There is an action of A on this inverse limit, and i factors through the invariants under this action, giving a map g : Ext_A (Z/2, Z/2) --> ( lim_Q Ext_E (Z/2,Z/2) )^A. We show that g is an F-isomorphism. ---------------- We have five new papers this time. Mark Hovey Papers uploaded to Hopf between Oct 10,1996 and Nov 1, 1996: 1./pub/DDavis/emb3 Embeddings of real projective spaces Donald M. Davis Abstract We tabulate known results on embeddings of real projective spaces, and prove one family of new results: P^n can be embedded in R^{2n-4} if n = 2^i + 3. 2. /pub/Feldman-Wilce/fibdegen Title: Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group: An application of homotopy theory to general topology Authors: David Feldman and Alexander Wilce Abstract: If p: E ----->B is a continuous surjection between completely regular spaces E and B , we may apply the Stone-Cech compactification functor \beta to obtain a surjection \beta p: \beta E ----->\beta B . It is well-known that if E = B x F where F is a finite set and p is projection on the first factor, then \beta E = \beta B x \beta F , and \beta p is again projection on the first factor. In this paper, we apply \beta to an n-fold covering map, that is, a local homeomorphism p: E ----->B such that p^-1 (b) has cardinality n for any b \in B . We show that the fibres of \beta p , while never exceeding n points, may degenerate to sets whose cardinality properly divides n (in contrast with the more usual, explosive sort of Stone-Cech ``pathology''). What is particularly striking about this phenomenon is that it depends on a homotopy invariant, the sectional category, of the map p . In particular, we show that if p: E ----->B is an H-bundle where H is a finite group, then \beta p has degenerate fibres iff p has infinite sectional category. In the special case where G is a p-group and p:EG ----->BG is the universal G-bundle, we can show more precisely that every possible G-orbit occurs somewhere as a fibre of \beta p . The proof of this theorem uses a weak form of the so-called generalized Sullivan conjecture, which is now a theorem of H.~Miller. It is interesting to see the structure group G manifesting itself in this way even though it is not explicitly part of the data fed to the Stone-Cech functor. Algebraic and general topology have grown far appart in recent years. Accordingly, we have tried to include enough detail to make the paper essentially self contained. Regarding the Stone-Cech compactification, we use few facts beyond the basic definitions. Readers unfamiliar with universal G -bundles should bear in mind the simplest non-trivial example, G= Z/2Z . The double cover of the infinite real projective space RP^\infty is a universal Z/2Z -bundle. No other finite group has a universal bundle which is so easily pictured; it is this case which motivated some of our terminology. 3. /pub/Hovey-Sadofsky/lnpic Invertible spectra in the $E(n)$-local stable homotopy category Mark Hovey and Hal Sadofsky Recall that the Picard group, first introduced into stable homotopy theory by Hopkins, is the group of isomorphism classes of smash-invertible spectra. For the ordinary stable homotopy category, this group is just Z on the spheres S^n. For localized categories,however the situation may be more complex. Hopkins, Mahowald, and Sadofsky, and also Strickland, have studied the Picard group of the K(n)-local category. In this paper we study the Picard group of the L_n-local, or E(n)-local, stable homotopy category. We find that if n is large relative to the prime p, the answer is just Z again. The proof involves a few general results that should be of independant interest. In particular, we give two proofs of a generalized Miller-Ravenel change of rings theorem, one of which depends on a general (unfortuately unpublished, but not tremendously difficult) change of rings theorem due to Hopkins. We also give an E(n) version of the Landweber filtration theorem, and show that the E(n) Adams spectral sequence always converges. We conclude the paper with the simplest example where n is not large relative to p, when n=1 and p=2. Here the Picard group is Z plus Z/2. 4. /pub/Mitchell/thomason Author: Stephen A. Mitchell Title: Hypercohomology spectra and Thomason's descent theorem Email: mitchell---math.washington.edu This is an expository paper on Thomason's etale descent theorem for Bott-periodic algebraic K-theory. It is mainly concerned with the proof of the theorem, and the machinery that goes into the proof. It includes an exposition of Jardine's closed model category structure on presheaves of spectra. Much background material is provided on Grothendieck sites, sheaf cohomology, etc. The paper is arranged so that readers who are not familiar with schemes or etale cohomology can still read the first and last chapters, which deal with Thomason's theorem for fields. This paper will appear in the proceedings of the 1996 Great Lakes K-theory conference. 5. /pub/Nassau/pnpnalg On the structure of $P(n)_\ast(P(n))$ for $p=2$ Christian Nassau We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$ are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$ is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed. ------------------ We have five new papers again this time. Mark Hovey Papers uploaded to Hopf between Nov 1,1996 and Nov 18, 1996: 1. /pub/Buchstaber-Ray/decfmqd ABSTRACT FOR: DOUBLE COBORDISM, FLAG MANIFOLDS, AND QUANTUM DOUBLES Victor M Buchsaber and Nigel Ray Drinfeld's construction of quantum doubles is one of several recent advances in the theory of Hopf algebras (and their actions on rings) which may be attractively presented within the framework of complex cobordism; these developments were pioneered by S P Novikov and the first author. Here we extend their programme by discussing the geometric and homotopy theoretical interpretations of the quantum double of the Landweber-Novikov algebra, as represented by a subalgebra of operations in double complex cobordism. We base our study on certain families of bounded flag manifolds with double complex structure, originally introduced into cobordism theory by the second author. We give background information on double complex cobordism, and discuss the cell structure of the flag manifolds by analogy with the classic Schubert decomposition, allowing us to describe their complex oriented cohomological properties (already implicit in the Schubert calculus of Bressler and Evens). This yields a geometrical realization of the basic algebraic structures of the dual of the Landweber-Novikov algebra, as well as its quantum double. We work in the context of Boardman's eightfold way, which clarifies the relationship between the quantum double and the standard machinery of Hopf algebroids of homology cooperations. 2. /pub/Dwyer/Exotic.Cohomology.GLnZhalf Exotic cohomology for GL(n,Z[1/2]) by W. G. Dwyer We show that for n=32 the mod 2 group cohomology of GL(n,Z[1/2]) is not detected on the subgroup of diagonal matrices. This disproves an old conjecture, and suggests that the cohomology of these general linear groups may in general be difficult to understand. 3. /pub/Green-Leary/SpectrumChern Title: The spectrum of the Chern subring Authors: David J. Green, IEM Essen, Germany. david---exp-math.uni-essen.de Ian J. Leary, Univ. of Southampton, UK. ijl---maths.soton.ac.uk Status: Submitted for publication. Abstract: For certain subrings of the mod-$p$-cohomology of a compact Lie group, we give a description of the spectrum, analogous to Quillen's description of the spectrum of the whole cohomology ring. Subrings to which our theorem applies include the Chern subring. Corollaries include a characterization of those groups for which the Chern subring is F-isomorphic to the cohomology ring. 1991 Classification: Primary 20J06; Secondary 20D15, 55R40. 4. /pub/Green-Minh/gm_transfer Title: Transfer and Chern Classes for Extraspecial $p$-Groups Authors: David J. Green, IEM Essen, Germany. david---exp-math.uni-essen.de Pham Anh Minh, University of Hue, Vietnam. Status: Submitted for publication Abstract: In the cohomology ring of an extraspecial $p$-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers. 1991 Classification: Primary 20J06; Secondary 20D15, 55R40. 5. /pub/Hung-Peterson/spherical Spherical classes and the Dickson algebra Nguyen H. V. Hu'ng and Franklin P. Peterson We attack the conjecture that the only spherical classes in the homology of $Q_0S^0$ are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the $E^2$-term of the unstable Adams spectral sequence converging to $\pi_*(Q_0S^0)$ using results about the Dickson algebra and by studying the Lannes--Zarati homomorphism. --------------- We have one new paper this time, plus a whole lot of new pictures taken and scanned by Clarence. I'll give you Clarence's message about the pictures first, then the abstract of the new paper. Mark Hovey Clarence wrote: >In the directory /pub/pictures/95-96 on hopf, there are pictures >of people from the CMS Vancouver meeting, my stay at MIT, my stay in >Zurich, and the Fields Inst in May 96. >If anyone has requests for pictures of particular people/places >or if people would like to contribute pictures, just e-mail me. >Mine were taken on color print film, finished at the local >drug store, and 4x5 paper prints scanned in and eye corrected >for density and contrast, with some trimming also. >Here's the current list of the new crop: > aadem.jpg 21-Nov-96 14:39 28k > berrick.jpg 21-Nov-96 14:42 24k > billrich.gif 21-Nov-96 09:17 79k > campbell-goerss-van.jpg21-Nov-96 14:33 61k > campbell-peterson.gif 21-Nov-96 09:22 125k > carlos.jpg 21-Nov-96 14:42 23k > cww-as-cubscout.gif 21-Nov-96 14:35 163k > cww-mit.gif 21-Nov-96 09:17 123k > drav.jpg 21-Nov-96 14:42 20k > goerss.gif 21-Nov-96 09:24 73k > gottmit.gif 21-Nov-96 09:20 136k > gray-jardine.gif 21-Nov-96 09:20 94k > harper-moore.gif 21-Nov-96 09:20 98k > jardine.gif 21-Nov-96 09:20 62k > jeanner.jpg 21-Nov-96 14:42 39k > jhubb.gif 21-Nov-96 09:21 85k > jmcclure.gif 21-Nov-96 09:21 132k > klesh.jpg 21-Nov-96 14:46 36k > lin-kane.jpg 21-Nov-96 14:42 49k > miller-peterson.gif 21-Nov-96 09:22 388k > moller.jpg 21-Nov-96 14:50 28k > notbohm.jpg 21-Nov-96 14:54 36k > pengal.jpg 21-Nov-96 14:58 43k > rlevi.jpg 21-Nov-96 15:02 14k > sadofsky-minami.gif 21-Nov-96 09:22 70k > shipley-smith.gif 21-Nov-96 09:20 111k > suter.gif 21-Nov-96 09:25 97k > toronto1.jpg 21-Nov-96 15:05 20k > vincent.jpg 21-Nov-96 15:05 37k > wgd-lec.jpg 21-Nov-96 15:09 48k > wgd-van.gif 21-Nov-96 09:27 80k Papers uploaded to Hopf between Nov. 18,1996 and Dec. 19, 1996: 1. /pub/Silverman/hit_polys_and_conjugation Hit Polynomials and Conjugation in the Steenrod Algebra and its Dual Judith H. Silverman judith---iu-math.math.indiana.edu Let $A^*$ be the mod-2 Steenrod algebra of cohomology operations and $\chi$ its canonical antiautomorphism. For all positive integers $f$ and $k$, we show that the excess of the element $\chi[Sq(2^{k-1}f) \cdot Sq(2^{k-2}f) \cdots Sq(2f) \cdot Sq(f)]$ is $(2^k-1) \mu(f)$, where $\mu(f)$ denotes the minimal number of summands in any representation of $f$ as a sum of numbers of the form $2^i-1$. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products $Sq(a_1) \ldots Sq(a_n)$ and $\chi[Sq(a_1) \ldots Sq(a_n)]$ in terms of the cardinalities of certain sets of matrices. For $s \geq 1$, let $P_s = F_2[x_1, \ldots, x_s]$ be the mod-2 cohomology of the $s$-fold product of $RP^{\infty}$ with itself, with its usual structure as an $A^*$-module. A polynomial $P \in P_s$ is {\em hit} if it is in the image of the action $\overline{A^*} \otimes P_s \longrightarrow P_s$, where $\overline{A^*}$ is the augmentation ideal of $A^*$. We prove that if the integers $e$, $f$, and $k$ satisfy $e<(2^k-1)\mu(f)$, then for any polynomials $E$ and $F$ of degrees $e$ and $f$ respectively, the product $E \cdot F^{2^k}$ is hit. This generalizes a result of Wood, conjectured by Peterson, and proves a conjecture of Singer and Silverman. ---------------- We have 6 new papers on hopf this time. Happy New Year! Mark Hovey New papers uploaded to Hopf between 12/20/96 and 12/30/96: 1. /pub/Dwyer/Exotic.Cohomology.GLnZhalf This is a new version of a paper already on the archive. I don't know how extensive the changes are. 2. /pub/Neumann-Neusel-Smith/ag1 Title: Rings of Generalized and Stable Invariants of Pseudoreflections and Pseudoreflection Groups Authors: Frank Neumann, Mara D. Neusel and Larry Smith Abstract: Let \rho: G --> GL(n, F) be a representation of a finite group G over the field F and F[V] the space of polynomial functions on V=F^n. We associate to G an ideal J_\infty(G) of F[V] called the ideal of stable invariants of \rho. If S is a set of pseudoreflections we associate to S the ideal I(S) of F[V] called the ideal of generalized invariants of S in the sense of Kac and Peterson. When G is a pseudoreflection group we investigate I(S) for various choices of S and the relation between J_\infty(G) and I(S). To a representation \rho, respectively to a set S of pseudoreflections, we also associate the rings gr_J_\infty(G) respectively gr_I(S) of stable and generalized invariants. We show that gr_I(S) is always a polynomial algebra over F and whenever \rho(G) is generated by semisimple pseudoreflections S that gr_J_\infty(G)=gr_I(S). This is the version which is published in J. of Algebra 182 (1996), 85-122. 3. /pub/Neumann-Neusel-Smith/ag4 (The fonts are weird in this file--I had trouble with my dvi viewer--Mark) Title: Rings of Generalized and Stable Invariants and Classifying Spaces of Compact Lie Groups Authors: Frank Neumann, Mara D. Neusel and Larry Smith Abstract: Let G be a compact connected Lie group with maximal torus T and Weyl group W(G). We show that the Eilenberg-Moore spectral sequence mod p (p odd prime) of the fibration G --> G/T --> BT collapses at the term E_2. This gives as corollary a different proof of the theorem of Kac, that the Serre Spectral sequence of the fibration T --> G --> G/T collapses at the term E_3. As an important step in the proof we show that the kernel of the induced map H^*(BT, F_p) --> H^*(G/T, F_p) can be identified with the ideal J_\infty (W(G)) in H^*(BT, F_p) of stable invariants of the Weyl group. The result can be applied to study torsion questions of H^*(BG, Z) in terms of the Weyl group action. This is the old version of the paper submitted to Inv. Math. 4. /pub/Shank/fmodsi Formal Modular Seminvariants R. James Shank Abstract: We construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We then observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p-3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p-2. 5. /pub/Strickland/mult Products on MU-modules by Neil Strickland We use the new categories of spectra and $MU$-modules constructed by Elmendorf, Kriz, Mandell and May to get improved results about multiplicative structures on spectra such as $P(n)$ and $E(n)$, particularly in the case $p=2$. 6. /pub/Strickland/poly Morava E-theory of symmetric groups by Neil Strickland We compute the completed $E(n)$ cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups. ---------------- We have 2 new papers on hopf this time. Mark Hovey New papers uploaded to Hopf between 12/30/96 and 1/14/97: 1. /pub/Greenlees-Strickland/chromatic Varieties and local cohomology for chromatic group cohomology rings John Greenlees and Neil Strickland Following Quillen, we use the methods of algebraic geometry to study the ring E^*(BG) where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of Abelian p-subgroups of G. This also gives information about the associated homology of BG. For example if E is the complete 2-periodic version of the Johnson-Wilson theory E(n) the irreducible components of the variety of the quotient E^*(BG)/I_k by the invariant prime ideal I_k=(p,v_1,...,v_{k-1}) correspond to conjugacy classes of Abelian p-subgroups of rank <= n-k. Furthermore, if we invert v_k the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition into connected components, corresponding to the fact that the ring splits as a product. 2. /pub/Grodal/postnikov Title: The transcendence degree of the mod p cohomology of finite Postnikov systems Author: Jesper Grodal University of Copenhagen jg---math.ku.dk Abstract: We examine the transcendence degree of the mod p cohomology of a finite Postnikov system E. We prove that, under mild assumptions on E, the transcendence degree of H^*(E;F_p) is always positive, and give a complete classification of the Postnikov systems where the transcendence degree of H^*(E;F_p) is finite. More precisely we prove that H^*(E;F_p) is of finite transcendence degree iff E is F_p-equivalent to the classifying space of a p-toral group. To obtain the results we establish a general formula for determining the transcendence degree of an unstable algebra given in terms of the growth of certain 'unstable Betti numbers'. This formula is easily applicable and has for instance Quillen's theorem about the Krull dimension of the mod p cohomology ring of a finite group as an immediate consequence. As an application of these results we derive statements about the n-connected cover X of a finite complex X. We show for instance that, under suitable connectivity assumptions on X, the LS category of X is always infinite assuming X \neq X. Finally we discuss generalizations of the obtained results to polyGEMs. ---------------- We have 12, count 'em 12, new papers on hopf this time. Mark Hovey New papers uploaded to Hopf between 1/14/97 and 2/19/97: 1. /pub/Bahri-Bendersky/ko-toric Authors: Anthony Bahri and Martin Bendersky Title: The KO-theory of Toric Manifolds Toric manifolds, the topological analogue of toric varieties, are determined by an $n$-dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. We use the Adams spectral sequence to compute the $KO$-theory of all toric manifolds and certain toric varieties. 2. /pub/Becker-Gottlieb/vfspace Spaces of Local Vector Fields by J. C. Becker and D. H. Gottlieb A local vector field is defined on an open set of a smooth manifold M . There is a natural topology which can be put on them to create a space. Each local vector field which is nonzero on the frontier of the domain has a transfer. The map assigning each vector field to its transfer is a continuous map from the space of local vector fields to Q(M+), the space of infinite loops and infinite suspensions of M plus a disjoint base point. This map is an n - 1 equivalence where the dimension of M is n . Thus stably it gives a weak homotopy equivalence. An equivariant version is discussed. 3. /pub/Goerss/complete \centerline{Comparing Completions of a Space at a Prime} \centerline{Paul G. Goerss} \bigskip\bigskip There are two practical ways to complete an abelian group $A$ at a prime $p$. One could complete $A$ with respect to the neighborhood base of zero given by the subgroups $p^nA$. This completion, called $p$-completion, is $\lim A/p^nA$. Or one could complete $A$ with respect to the neighborhood base of zero given by subgroups $B\subseteq A$ of finite $p$-power index. This completion, called $p$-pro-finite completion, is $\lim A_\alpha$ where $A_\alpha$ runs over the finite $p$-group quotients of $A$. They agree for finitely generated groups, but not in general: if $V$ is an $F_p$ vector space it is $p$-complete, but its $p$-pro-finite completion is the double dual $V^{\ast\ast}$. Also, the former, $p$-completion, is easier to define, but it is neither left nor right exact and the category of $p$-complete groups is not abelian -- it is not closed under cokernels. The latter, $p$-pro-finite completion, is initially less tractable, but it is right exact and the category of $p$-pro-finite abelian groups is an abelian category. There are two corresponding completions for topological spaces. The analog of $p$-completion is Bousfield-Kan completion [3] and the analog of $p$-pro-finite completion is related to Sullivan's pro-finite completion and has recently been given a homotopical definition by Morel [13]. The purpose of this note is to compare these two completions; in addition, we seek to give Morel's completion the same sort of computational footing that the Bousfield-Kan completion enjoys. \end 4. /pub/Goerss/limits \centerline{\bf The Homology of Homotopy Inverse Limits} \bigskip \centerline{by Paul G. Goerss} \bigskip\bigskip {\bf Abstract:} The homology of a homotopy inverse limit can be studied by a spectral sequence with has $E_2$ term the derived functors of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonn\'e modules if one has a diagram of connected abelian Hopf algebras. \end 5. /pub/Goerss-Jardine/localize Authors: P.G. Goerss and J.F. Jardine Title: Localization theories for simplicial presheaves Email: pgoerss---math.washington.edu, jardine---uwo.ca We demonstrate that most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, a theory of localization along a geometric topos morphism, and in particular a method of localizing a space or spectrum at a generalized \'etale cohomology theory. We further show that the f-localization concept has an analog for simplicial presheaves. This theory is used to answer a question of Soul\'e concerning integral homology localizations for diagrams of spaces. The manuscript was typeset in LAMSTeX, and therefore requires the lams* fonts for printing, available at http://www.math.uwo.ca/~jardine/papers/ A postscript file for this paper is available at the same address. 6. /pub/Goerss-Turner/cartan \centerline{\bf Homotopy and Homology for} \centerline{\bf Simplicial Abelian Hopf Algebras} \bigskip \centerline{by} \centerline{Paul G. Goerss and James Turner} \bigskip\bigskip \centerline{\bf Abstract} \medskip Let $A$ be a simplicial bicommutative Hopf algebra over the field $F_2$ with the property that $\pi_0A \cong F_2$. We show that $\pi_\ast A$ is a functor of the Andr\'e-Quillen homology of $A$, where $A$ is regarded as an $F_2$ algebra. Then we give a method for calculating that Andr\'e-Quillen homology independent of knowledge of $\pi_\ast A$. \end 7. /pub/Goerss-Turner/dieu \centerline{Homotopy Theory of Simplicial Abelian Hopf Algebras} \centerline{Paul Goerss and James Turner} \centerline{\bf Abstract} \smallskip We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field $F_p$, $p>0$, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its Andr\'e-Quillen homology, which, in turn, can be easily computed from the homotopy groups of the associated simplicial Dieudonn\'e module. \end 8. /pub/MWeiss/emb1 Author: Michael Weiss Title: Embeddings from the point of view of immersion theory, Part I Email: weiss.13---nd.edu Abstract: The main objects of study are spaces of smooth embeddings emb(M,N), where M and N are smooth manifolds without boundary. We study emb(M,N) by studying the contravariant functor V |--> emb(V,N) from the open subsets V of M to spaces. Some of its properties are abstracted, and a suitable calculus of such cofunctors is developed, with Taylor series and so on. The calculus serves to calculate values of such cofunctors. It works when the cofunctors in question satisfy certain higher excision conditions. In the case of the cofunctor V |--> emb(V,N), the conditions are indeed satisfied when the codimension is at least 3, according to a theorem proved recently by Goodwillie. 9. /pub/MWeiss/emb2 Authors: Thomas Goodwillie and Michael Weiss Title: Embeddings from the point of view of immersion theory, Part II Email: weiss.13---nd.edu, tomg---math.brown.edu Abstract: See also the abstract for Part I, by Weiss. Part II contains the proof of the main theorem stating that the Taylor series of the contravariant functor V |-- loop space of emb(V,N) (where V runs through the open subsets of a fixed codimension >2 submanifold M of N) converges to the functor. As an illustration, spaces of embeddings of intervals are discussed. The main theorem is deduced from an excision estimate due to Goodwillie (here Thm. 1.1, still in the making and moreover slightly extrapolated by Weiss - quote with care if at all). 10. /pub/Stasheff/ascona \\ Title: Deformation Theory and the Batalin-Vilkovisky Master Equation Author: Jim Stasheff, UNC Math, Chapel Hill NC 27599-3250 Notes: AMS-Latex, with one figure of spectral seq differentials to appear in Proceedings of a Conference on Defoprmation Theory,etc, Ascona Report-no: UNC-CH Math 97-01 \\ The Batalin-Vilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the Batalin-Vilkovisky approach is here translated into the language of deformation theory. \\ 11. /pub/Stasheff/hrcpa-final \\ Title: Homological Reduction of Constrained Poisson Algebras Author: Jim Stasheff, UNC-Math, Chjapel Hill NC 27599-3250 Notes: AMS-Latex, 16 pages Report-no: University of North Carolina preprint UNC-MATH-97-1 \\ Reduction of a Hamiltonian system with symmetry and/or constraints has a long history. There are several reduction procedures, all of which agree in ``nice'' cases. Some have a geometric emphasis - reducing a (symplectic) space of states, while others are algebraic - reducing a (Poisson) algebra of observables. The relation between symmetry and constraints is particularly tight in the case Dirac calls ``first class''. The present paper is concerned entirely with this first class case and deals with the reduction of a Poisson algebra via homological methods, due to Batalin-Fradkin-Vilkovisky in another language, but here recast in terms of the models central to rational homotopy theory. \\ 12. /pub/Strom Two Special Cases of Ganea's Conjecture Jeffrey A. Strom University of Wisconsin-Madison strom---math.wisc.edu Ganea conjectured that for any finite CW complex and any k>0, cat(X x S^k) = cat(X) + 1. In this paper we prove two special cases of this conjecture. The main result is the following. Let X be a (p-1) connected n dimensional CW complex (not necessarily finite). If cat(X) = [n/p] + 1 and n is not congruent to -1 mod p (which implies p > 1), then cat(X x S^k) = cat(X) + 1. This is proved by showing that wcat(X x S^k) = wcat(X) + 1 in a much larger range, and then showing that under the conditions imposed, cat(X) = wcat(X). The second special case is an extension of Singhof's earlier result for manifolds. ----------------- Two new papers this time. Mark Hovey New papers uploaded to Hopf between 2/19/97 and 3/2/97: 1. /pub/Turner/opsseq Operations and Spectral Sequences I James M. Turner Department of Mathematics University of Virginia Charlottesville, VA 22903 Abstract: This is the first in a series of papers which examines a general type a chain complex (over F_2) whose homology supports a well-defined action of operations. We call such complexes Dold algebras, which include the singular cochain complex of a space and the singular chain complex of an infinite loop space, and we give conditions on filtrations of such objects so that there is a compatible action of operations on the associated spectral sequences. For applications, we recover W. Singer's result of the action of Steenrod operations on the Serre spectral sequence and we extend A. Bahri's action of Dyer-Lashof operations on the Eilenberg-Moore spectral sequence. 2. /pub/Turner/relations Relations in the Homotopy of Simplicial Abelian Hopf Algebras James M. Turner Department of Mathematics University of Virginia Charlottesville, VA 22903 Abstract: Given a simplicial bicommutative Hopf algebra, over the field F_2, the homotopy is both a commutative algebra over the Cartan-Bousfield-Dwyer operations and a cocommutative coalebra over the Steenrod algebra. We show that the Hopf condition guarantees that these two structures interact via certain Nishida-type relations. This is a revision of the author's PhD thesis and will appear in the Journal of Pure and Applied Algebra. ---------------- Eleven new papers this time. This seems like a good time to remind you all: if you want people to download your papers, you have to make the abstract readable. No personal TeX macros, no word processor code. I had to do more editing than usual this time. Mark Hovey New papers uploaded to Hopf between 3/2/97 and 4/4/97: 1. /pub/Baker/ell-ext On the Adams E_2-term for elliptic cohomology Andrew Baker University of Glasgow andy---maths.gla.ac.uk http://www.maths.gla.ac.uk/~andy/ We investigate the E_2 term of Adams spectral sequence based on elliptic homology. The main results describe this E_2 term from a `chromatic' perspective. At a prime p>3, the Bousfield class of Ell is the same as that of K(0) V K(1) V K(2). Using delicate facts due to Katz (which also play a major r\^ole in work on the structure Ell_*Ell by the author, Clarke, Johnson and Laures) as well as our description of supersingular elliptic cohomology in terms of K(2) theory, we show that the E_2 term chromatically of length 2 and totally determined by the 0, 1 and 2 columns of the usual chromatic sectral sequence for BP. We apply our results to recover some of those of Clarke, Johnson and Laures as well as extending them to completely determine this Adams E_2 term. We also give a proof of a generalization of Katz's result which allows a similar analysis of the chromatic spectral sequence for the E_2 term of the Adams spectral sequence based on E(2). This approach may be of some use in connection with the more general case associated to E(n) for n>2. 2. /pub/Foskey/splitdlo Split Dual Dyer-Lashof Operations Mark Foskey Jacksonville University 2800 University Boulevard North Jacksonville, FL 32211 For each admissible monomial of Dyer-Lashof operations $Q_I$, we define a corresponding natural function $\wh Q_I\: T\bar H_*(X) \to H^*(\Omega^n \Sigma^n X)$, called a {\em Dyer-Lashof splitting}. For every homogeneous class $x$ in $H^*(X)$, a Dyer-Lashof splitting $\wh Q_I$ determines a canonical element $y$ in $H^*(\Omega^n \Sigma^n X)$ so that $y$ is connected to $x$ by the dual homomorphism to the operation $Q_I$. The union of the images of all the admissible Dyer-Lashof splittings contains a complete set of algebra generators for $H^*(\Omega^n \Sigma^n X)$. 3./pub/Henderson/Cohomology_A1 The Cohomology Algebra of a Subalgebra of the Steenrod Algebra Gregory D. Henderson Pennsylvania State University We compute the cohomology algebra of A(1), the subalgebra of the Steenrod algebra generated by P^1 and P^p. This completes a partial result given by Arunas Liulevicius in 1962 and provides explicit representatives in the cobar costruction for all but one of the algebra generators. 4. /pub/Henn/ls Title: A variant of the proof of the Landweber-Stong conjecture Author: Hans-Werner Henn We describe a variant of the recent proof by Bourguiba and Zarati of the conjecture of Landweber and Stong concerning the depth of rings of invariants. 5. /pub/Henn/profin From Mark: this paper appears to be a new version of a previous paper. The abstract has not been updated however, so I am not certain. I will include the title only right now: CENTRALIZERS OF ELEMENTARY ABELIAN p - SUBGROUPS AND MOD - p - COHOMOLOGY OF PROFINITE GROUPS by Hans-Werner Henn 6. /pub/Henn/survey1 Title: Unstable modules over the Steenrod algebra and the cohomology of groups Author: Hans-Werner Henn Let $p$ be a prime. In this survey we attempt to explain how the presence of Steenrod operations in the mod - $p$ cohomology ring $H^*(G;\FF_p)$ of a group $G$ allows to understand qualitative features of this ring, at least for a large class of groups including all finite groups but also many discrete groups like arithmetic groups, mapping class groups and automorphism groups of free groups. We will also discuss how the general theory can be applied to do actual calculations. 7. /pub/Nadiradze/CharClasses (Abstract truncated by Mark) CHARACTERISTIC CLASSES IN THE SC -THEORY AND THEIR APPLICATIONS roin nadiradze In this paper we describe some series of generators mainly in the cobordisms of self-conjugate cobordism ring [SmSt1], [G2],as well as some series of relations in SC and in Sp. The results below have been announced in [Na1] and [Na2]. In all the cases the generators are constructed as "DSCPi()jM; where M is an SC-manifold, is an SC-bundle over M, Pi() is a characteristic class in the SC -theory [Bu1], DSC is the Atiyah-Poincare duality [At1] and " is an augmentation. 8. /pub/Nadiradze/FormalGroups (Abstract truncated and edited by Mark) ONE CLASS OF FORMAL GROUPS AND COHOMOLOGY THEORIES roin nadiradze The Elliptic Cohomology and its analog arise recently, with a development of the quantum field theory. These theories are connected with specific formal groups. Similar formal groups arise in some particular cases in 1. Trigonometry (X- XV century). 2. Theory of Elliptic functions (XVIII- XIX century). 3. Algebra(middle of XX century). 4. Topology (middle of 1960-1970 years). 5. Theoretical physics (1970-1990 years). My aim is to describe these formal groups and to consider the corresponding cohomology theories for future application. In particular, we consider the following gadgets: Classification theorem. Genera and two- valued formal groups. Coefficient rings of formal groups. Cooperations and Hopf algebra structure. Some applications to the Adams- Novikov spectral sequence (ANSS). 9. /pub/Notbohm/polyinv FOR WHICH PSEUDO REFLECTION GROUPS ARE THE p-ADIC POLYNOMIAL INVARIANTS AGAIN A POLYNOMIAL ALGEBRA ? by Dietrich Notbohm Let $W$ be a finite group acting on lattice $L$ over the $p$--adic integers $\z\p$. The analysis of the ring of invariants of the associated $W$--action on the algebra $\z\p[L]$ of polynomial functions on $L$ is a classical question of invariant theory. If $p$ is coprime to the order of $W$, classical results show that $W$ is a \prg , if and only if the ring of invariants is again polynomial. We analysis the situation for those odd primes dividing the order of $W$ and, in particular, determine those \prg s for which the ring of invariants $\z\p[L]^W$ is a polynomial algebra. Mathematisches Institut, Bunsenstr. 3-5, 37073 G"ottingen, Germany e-mail: notbohm at cfgauss.uni-math.gwdg.de 10. /pub/Notbohm/type2a TOPOLOGICAL REALZATION OF A FAMILY OF PSEUDO REFLECTION GROUPS by Dietrich Notbohm We are interested in a topological realization of a family of pseudo reflection groups $G \subset \Gl (n,F_p)$; i.e. we are looking for topological spaces whose mod--$p$ cohomology is isomorphic to the ring of invariants $F_p[x_1,...,x_n]^G$. Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over $F_p$ might appear as the mod-$p$ cohomology of a space. The family under consideration is given by pseudo reflection groups which are subgroups of the wreath product $Z/q\wr \Sigma_n$ where $q$ divides $p-1$ and where $p$ is odd. Let $G$ be such a subgroup acting on the polynomial algebra $A:= \fp [x_1,...,x_n]$. We show, that there exists a space $X$ such that $H^*(X;F_p)\cong A^G$ which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod--$p$ cohomology of the classifying spaces of special orthogonal groups or of symplectic groups. The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group $G$. Using this construction we also show that the homotopy type of the $p$--adic completion of these spaces is completely determined by the mod--$p$ cohomology considered as algebra over the Steenrod algebra. Moreover we calculate the set of homotopy classes of self maps of the completed spaces. Mathematisches Institut, Bunsenstr. 3-5, 37073 G"ottingen, Germany e-mail: notbohm at cfgauss.uni-math.gwdg.de 11. /pub/Reedy/reedy (Note from Mark: I believe this is an old paper of Reedy's, maybe his thesis, that was never actually published but has served as an important reference for fans of model categories. I believe we owe this electronic transcription to Phil Hirschhorn.) \title{Homotopy Theory of Model Categories} \author{C. L. Reedy} In this paper some questions in the homotopy theory of model categories are answered. The important results of this paper are Theorems~B, C and D which state that pushouts, sequential direct limits, and realizations of simplicial objects respect weak equivalences, provided sufficient cofibrancy is present. Section~1 presents a model category structure on the simplicial objects over a model category. This is done partly to provide a justification for the term cofibrant, as applied to certain simplicial objects, and also to show that any object can be ``approximated'' by a cofibrant object. The lemmas in this section are presented without proof, since the proofs are easy, and of an entirely category theoretic nature. In section~2 it is shown that, in model categories, weak equivalences respect pushouts and sequential direct limits. That weak equivalences respect pushouts is particularly important since this shows that any closed model category is a suitable category for homology theory. In section~3 the result that the realization of a weak equivalence of cofibrant objects is a weak equivalence is proven. Certain stronger results in the case of simplicial topological spaces are also mentioned. Section~4 discusses special results about simplicial simplicial sets, including the result that realization is isomorphic to the diagonal, a useful result which is not widely known, and not original with the author. -------------- So many new papers this time that I have decided to split this message into two parts. There are 11 papers in this part, and 11 in the next. Other news: I have a one-year job at Wesleyan University in Connecticut, where my wife has worked for years. The two-body problem still looks as if it will never be resolved. My new e-mail address is mhovey---wesleyan.edu. I will also be replacing my home page at MIT with a link to my home page at Wesleyan. The latest issues of this little newsletter will not be at MIT. My new home page URL is http://www.cs.wesleyan.edu/Math/Guests/Mark . Mark Hovey New papers uploaded to Hopf between 4/4/97 and 7/17/97, part 1: 1. /pub/Arkowitz-Maruyama/Arkowitz-Muruyama1 TITLE: Self Homotopy Equivalences Which Induce the Identity on Homology, Cohomology or Homotopy Groups AUTHORS: Martin Arkowitz and Ken-ichi Maruyama ABSTRACT: For a based, 1-connected, finite CW-complex $X$, we study the following subgroups of the group of homotopy classes of self homotopy equivalences of $X$: $\Cal E_*(X)$, the subgroup of homotopy classes which induce the identity on homology groups, $\Cal E^*(X)$, the subgroup of homotopy classes which induce the identity on cohomology groups and $\Cal E_\#^{\roman{dim}+ r}(X)$, the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions $\leq \roman{dim}\,X +r$. We investigate these groups when $X$ is a Moore space and when $X$ is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on $X$ such that $\Cal E_*(X) = \Cal E^*(X)$ and obtain a class of spaces (including compact, oriented manifolds and $H$-spaces) for which this holds. Finally, we examine $\Cal E_\#^{\roman{dim}+ r}(X)$ for certain spaces $X$ and completely determine the group when $X = S^m \times S^n$ and $X = CP^n \vee S^{2n}$. 2./pub/Arlettaz-Mimura-Nakahata-Yagita/Arlettaz-Mi-Na-Ya Authors: Dominique Arlettaz, Mamoru Mimura, Koji Nakahata, Nobuaki Yagita Title: The mod 2 cohomology of the linear groups over the ring of integers Email: dominique.arlettaz---ima.unil.ch mimura---math.okayama-u.ac.jp koji.nakahata---ima.unil.ch yagita---mito.ipc.ibaraki.ac.jp This paper completely determines the ring structure, with an explicit description of the generators, of the mod 2 cohomology of the linear groups GL(Z), SL(Z) and St(Z) and its module structure over the Steenrod algebra. 3. /pub/Baker/ell-calc Hecke operations and the Adams E_2-term Andrew Baker University of Glasgow andy---maths.gla.ac.uk http://www.maths.gla.ac.uk/~andy/ We use Hecke operators to investigate part of the E_2 term of Adams spectral sequence based on elliptic homology. The main result is a rederivation of Ext^1 which combines use of `classical' Hecke operators and p-adic Hecke operators due to Serre. 4./pub/Berrick-Casacuberta/pluscon Title: A UNIVERSAL SPACE FOR PLUS-CONSTRUCTIONS Authors: A. J. BERRICK, Department of Mathematics, National University of Singapore, Kent Ridge 119260, Singapore; berrick---math.nus.sg Carles CASACUBERTA, Departament de Matem\`atiques, Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Spain; casac---mat.uab.es Last revision: July 8, 1996 Abstract: We exhibit a two-dimensional, acyclic, Eilenberg--Mac Lane space $W$ such that, for every space $X$, the plus-construction $X^+$ with respect to the largest perfect subgroup of $\pi_1(X)$ coincides, up to homotopy, with the $W$-nullification of $X$; that is, the natural map $X\to X^+$ is homotopy initial among maps $X\to Y$ where the based mapping space ${\rm map}_*(W,Y)$ is weakly contractible. Furthermore, we describe the effect of $W$-nullification for any acyclic $W$, and show that some of its properties imply, in their turn, the acyclicity of $W$. 5./pub/Christensen/ideals Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta J. Daniel Christensen jdc---math.jhu.edu Abstract: We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik-Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A-infinity modules over an A-infinity ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim^1 group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially. This article will appear in Advances in Mathematics. A final version will be sent off shortly, so comments and corrections are particularly welcome. 6./pub/Christensen-Strickland/phantoms (This is a revised version of a paper already on the archive--Mark) Phantom Maps and Homology Theories J. Daniel Christensen and Neil P. Strickland (jdc---math.jhu.edu and neil---pmms.cam.ac.uk) Keywords: phantom map, stable homotopy theory, spectrum, triangulated category Abstract: We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X --> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis' axiomatisation conjecture to an extension problem. We show that a certain functor S --> A is the universal example of a homology theory with values in an AB 5 category, and compare this with some results of Freyd. This paper will appear in Topology. 7./pub/Cohen-Levi/surv-bg+ ON THE HOMOTOPY THEORY OF p-COMPLETED CLASSIFYING SPACES By Frederick R. Cohen and Ran Levi Abstract This paper is mostly a survey of work done on the subject of the title. In particular we concentrate on the study of p-completed classifying spaces of finite p-perfect groups and the associated loop spaces. We consider homological properties as well as various homotopy theoretic properties of these spaces and relate those properties to other questions in and outside of homotopy theory. A substantial number of examples, open questions and conjectures is included. 8./pub/Dadarlat-McClure/dadarlat_mcclure When are two commutative C*-algebras stably homotopy equivalent? Marius Dadarlat and James McClure Department of Mathematics Purdue University, West Lafayette IN 47907--1395 There is a close relationship between compact topological spaces and commutative C* algebras: a topological space X determines the C* algebra, denoted C(X), of continuous complex-valued functions on X, and all commutative C* algebras arise in this way for some X (specifically, one can let X be the maximal-ideal spectrum of the C* algebra). The basic definitions of homotopy theory have analogs for C* algebras, and it is natural to ask how the homotopy-theoretic properties of X relate to those of C(X). A standard procedure in the theory of C* algebras is to tensor a given algebra with the algebra of compact operators on a separable Hilbert space: this process is called ``stabilization'' (and is unrelated to homotopy-theoretic stablization). Dadarlat and Nemethi have shown that when C(X) is stabilized then its homotopy-theoretic properties are essentially determined by the spectrum bu \smash X (but not simply by the homotopy groups of this spectrum). In this paper we use the computational methods introduced by Robinson and refined by Elmendorf, Kriz, Mandell and May to give explicit answers to several questions concerning the homotopy theory of the stabilization of C(X). One result that may be of independent interest is a long exact sequence for calculating Ext over the polynomial ring Z[x]. 9./pub/Dwyer-Dror-Farjoun-Ravenel Bousfield localizations of classifying spaces of nilpotent groups by W. G. Dwyer, E. Dror-Farjoun, and D. Ravenel Let $G$ be a finitely generated nilpotent group. We show that the localization of $BG$ with respect to a multiplicative complex oriented homology theory $h_*$ is again a space of type $K(\pi,1)$; in fact, it is the same as the localization of $BG$ with respect to the ordinary homology theory determined by the ring $h_0$. 10./pub/Dwyer-Wilkerson/kahler-differentials/stein Kahler differentials, the T-functor, and a theorem of Steinberg W. G. Dwyer and C. W. Wilkerson Let $T$ be the functor on the category of unstable algebras over the Steenrod algebra constructed by Lannes. We use an argument involving Kahler differentials to show that $T$ preserves polynomial algebras. This leads to new and relatively simple proofs of some topological and algebraic theorems. ( This is a somewhat shorter ( examples deleted ) of the original version posted ). 11./pub/Dwyer-Wilkerson/lie/liegroups The elementary geometric structure of compact Lie groups by W. G. Dwyer and C. W. Wilkerson We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of $p$-compact groups. ------------- 11 new papers in this part. Mark Hovey New papers uploaded to Hopf between 4/4/97 and 7/17/97, part 2: 12./pub/Elmendorf/CWstable Stabilization as a CW approximation by A. D. Elmendorf, Purdue University Calumet. Abstract: This paper describes a peculiar property of the category of $S$-modules constructed by the author, Kriz, Mandell, and May: the full subcategory of suspension spectra (which are all $S$-modules) forms a precise copy of the category of topological spaces. Consequently, the ``classical'' homotopy category of $S$-modules with morphisms the actual homotopy classes of maps contains a copy of unstable homotopy theory. Stabilization and stable homotopy are then induced by CW approximation as $S$-modules. One consequence is that CW complexes whose suspension spectra are CW $S$-modules must be contractible. 13./pub/Henderson/Cohomology_P1 (Your moderator is confused about this paper: it has exactly the same abstract as Cohomology_A1, except A(1) is replaced by P(1). It is exactly 60 bytes longer than Cohomology_A1, which means it must be the same paper. Perhaps Greg can tell us what is going on?) The Cohomology Algebra of a Subalgebra of the Steenrod Algebra Gregory D. Henderson Pennsylvania State University We compute the cohomology algebra of P(1), the subalgebra of the Steenrod algebra generated by P^1 and P^p. This completes a partial result given by Arunas Liulevicius in 1962 and provides explicit representatives in the cobar costruction for all but one of the algebra generators. 14./pub/Hovey-Sadofsky/lnpic This is a revised version of Invertible spectra in the $E(n)$-local stable homotopy category by Mark Hovey and Hal Sadofsky I (Mark) think this version is considerably better: it has better proofs of the splitting of L_K(n)E(m) for m>=n and of L_K(n)BP, and it has a proof of Hopkins' general Hopf-algebroid change-of-rings theorem. 15. /pub/Hovey-Strickland/kn (From Mark: I am very happy this paper is finally ready. It is about 100 pages long though, so will take some time to download.) Morava K-theories and localisation by Mark Hovey and Neil P. Strickland We study the structure of the categories of $K(n)$-local and $E(n)$-local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and $K(n)$-nilpotent spectra. We give a number of useful extensions to the theory of $v_n$ self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible $K(n)$-local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the Brown-Comenetz dual of $M_nS$ lies in the Picard group. We give a detailed analysis of some examples when $n=1$ or $2$, and a list of open problems. 16./pub/Jeanneret-Osse/ktheory TITLE: The K-Theory of $p$-compact groups AUTHORS: A. Jeanneret Mathematisches Institut Sidlerstrasse 5 CH-3012 BERN (Switzerland)/ A. Osse Institut de Mathematiques Rue Emile Argand 11 CH-2007 NEUCHATEL (Switzerland) ABSTRACT: In this paper, we show that the $p$-adic K-theory of a connected $p$-compact group is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is $\lambda$-isomorphic to the K-theory of some $BG$, where $G$ is a compact connected Lie group. 17./pub/Kuhn/genstability Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA Title: Rational cohomology and cohomological stability in generic representation theory Email: njk4x---virginia.edu With Fq a finite field of characteristic p, let F(q) be the category whose objects are functors from finite dimensional Fq vector spaces to vector spaces over the algebraic closure of Fp. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space V_m induces an exact functor from F(q) to the category of modules over GL(m,q). E.Friedlander and A.Suslin have introduced a category P of ``strict polynomial functors'' which has the same relationship to the category of rational GL_m modules that F(q) has to the category of GL(m,q) modules. Our main theorem says that, for all finite objects F,G in P, and all s, the natural restriction Ext^s_{P}(F[k], G[k]) ---> \Ext^s_{F(q)}(F,G) is an isomophism for all large enough k and q. Here F[k] denotes F twisted by the Frobenious k times. This theorem is an analogue of an old stability theorem of E.Cline, B.Parshall, L.Scott, and W.van der Kallen relating rational GL_m modules to GL(m,q) modules. These two theorems then combine with an observation of Friedlander and Suslin to show that, for all finite F,G in P, and all s, the natural map Ext^s_{F(q)}(F,G) ---> Ext^s_{GL(m,q)}(F(V_m),G(V_m)) is an isomorphism for all large enough m and q. Thus group cohomology of the finite general linear groups in the stable range (a.k.a. stable K-theory of Fq with twisted coefficients) has often been identified with (the more computable) MacLane cohomology. 18./pub/Levi/sl3z A NOTE ON THE HOMOTOPY TYPE OF BSL_3(Z)^2 By Ran Levi To appear in the Mathematical Proceedings of the Cambridge Philosophical Society Abstract It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BG^p has the property that \Omega BG^p is a retract of the loop space on a simply-connected, Fp- finite, p-complete space. In this note we consider a particular example where this theorem applies, namely we study the homotopy type of BSL_3(Z)^2. It particular we analyze \Omega BSt_3(Z)^2 a double cover of \Omega BSL_3(Z)^2, and obtain a splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2^r maps on spheres. We also give a formula for the Poincar\'e series of H_*(\Omega B\Gamma^p; Fp) for a general group \Gamma, as above, in terms of possibly simpler components. This formula is used to calculate the mod-2 homology of \Omega B\Gamma^2 for \Gamma= SL_3(Z) or St_3(Z)^2 as modules over a certain tensor subalgebra. 19./pub/Lueck-Mueller/lueck061197 Author: Wolfgang Lueck and Andreas Mueller Title: Existence of finitely dominated $CW$-complexes with G_1(X) = \pi_1(X) and non-vanishing finiteness obstruction Email: lueck---math.uni-muenster.de We show for a finite abelian group G and any element in the image of the Swan homomorphism Z/|G|* \to \widetilde{K}_0(ZG) that it can be realized as the finiteness obstruction of a finitely dominated connected CW-complex X with fundamental group \pi_1(X) = G such that \pi_1(X) is equal to the subgroup G_1(X) defined by Gottlieb. This is motivated by the observation that any H-space X satisfies \pi_1(X) = G_1(X) and still the problem is open whether any finitely dominated H-space is up to homotopy finite. 20./pub/Notbohm/polyspaces SPACES WITH POLYNOMIAL MOD-P COHOMOLOGY Dietrich Notbohm In the early seventieth, Steenrod posed the question which polynomial algebras over the Steenrod algebra appear as the cohomology ring of a topological space. For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson showed that all such algebras are given as the mod-p reduction of the invariants of a pseudo-reflection group acting on a polynomial algebra over the $p$--adic integers. We show that this necessary condition is also sufficient for finding a realization of such a polynomial algebra and, for odd primes, answer Steenrod's question completely. e-mail: notbohm at cfgauss.uni-math.gwdg.de 21./pub/Rudyak/AnalyticApps On analytical applications of stable homotopy (the Arnold conjecture, critical points) Yuli B. Rudyak April 1997 address: Mathematisches Institut, Universit\"at Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg 1, Germany \abstract We prove the Arnold conjecture for closed symplectic manifolds with $\pi_2(M)=0$ and $cat M=dim M$. Furthermore, we prove an analog of the Lusternik--Schnirelmann theorem for functions with ``generalized hyperbolicity'' property. \endabstract 22./pub/Strickland/strickland_subgp (This is a revised version of a paper already on the archive). Finite Subgroups of Formal Groups Neil Strickland In this paper we discuss various moduli problems involving the classification of finite subgroups or related structures on formal groups of finite height. Analogous problems for elliptic curves have of course been widely studied. The moduli spaces which we consider turn out to be surprisingly well-behaved. They are all Cohen-Macaulay, and most of them are smooth. The original motivation for this work came from algebraic topology, in particular the study of power operations in certain homology theories constructed by Morava. I learnt most of what I know about these questions from Mike Hopkins, and a great deal of the theory presented here was developed in discussions with him. See section~\ref{se:at} for a brief discussion of how moduli problems arise in algebraic topology. --------------- There are 2 new papers in the last couple of days. By the way, I was right about the Henderson paper from last time: it was just a change of title. Mark Hovey New papers uploaded to Hopf between 7/17/97 and 7/21/97: 1. /pub/EBrown/ed \title Short Complete Proofs of the Serre\\ Spectral Sequence Theorems\endtitle A new improved "Simple complete proofs of the Serre spectral sequence theorems". by Edgar H. Brown, Jr. April 1,1997 In (\cite{B}) I set forth a proof of the Serre calculation of $E^{p,q}$ and claimed among other things that unlike my previous attempts to prove this in graduate course lectures, this proof was routine. On presenting this material in class, I discovered it was not as routine as I had imagined. With the help of Pallavi Jayawart and Saso Strle I have drastically improved and simplified the presentation, reducing proofs of the Serre Spectral Sequence (SSS) theorems (\cite{S}) to a collection of lemmas provable by straightforward mechanical checking which is left to the reader. In addition it offers some motivation for the definitions. A knowledge of the standard material on singular homology and cohomology, including the Eilenberg-Zilber theorem, is sufficient to prove the lemmas. We do homology first and then add variations, including cohomology, as exercises. 2./pub/Rudyak/LScategory (This is a new version of a paper already on the archive. The abstract is completely unreadable by humans, so I will content myself with giving you the title) SOME REMARKS ON CATEGORY WEIGHT Yu.B. Rudyak ------------- Suddenly people are uploading papers to Hopf every day, so here is another installment. 4 papers this time. Mark Hovey New papers uploaded to Hopf between 7/21/97 and 7/24/97: 1./pub/Babenko-Katz-Suciu/freebks DVI File: freebks.dvi Title: Volumes, middle-dimensional systoles, and Whitehead products Authors: Ivan K. Babenko, Mikhail G. Katz and Alexander I. Suciu Subj-class: 53C23 (Primary) 55Q15 (Secondary) Comments: LaTeX2e, 10 pages. Also available at dg-ga/9707016 Let (X,g) be a closed, orientable Riemannian manifold of dimension 2m. The k-systole of (X,g), sys_k(g), is the infimum of areas of non-bounding cycles represented by maps of k-dimensional manifolds into X. We are interested in the following question: Does there exist a constant, C, such that every metric g on X satisfies (*) sys_{m}^{2}(g) <= C vol_{2m}(g) ? If there is no such C, we say that X is ``systolically free." In the case of surfaces of positive genus the answer to (*) is affirmative. In the case m >= 2, this question has been referred to by M. Gromov as the ``basic systolic problem." Progress on the problem became possible once Gromov described a special family of metrics on S^1 x S^3 and surgical procedures suitable for generalizations. In this note, we prove that closed manifolds of dimension 2m >= 6 with torsion-free middle-dimensional homology are systolically free. An underlying theme of this paper is the influence of homotopy theory on the geometric inequality (*). Our basic topological tools are the Hilton-Milnor theorem and theorems of Eckmann and G. Whitehead on composition maps in homotopy groups of spheres. Our geometric tools are the coarea inequality and pullback arguments for simplicial metrics. 2./pub/Fresse/CartanOp Benoit Fresse Institut de Recherche Math\'ematique Avanc\'ee, 7 rue Ren\'e-Descartes, 67084 Strasbourg Cedex, France fresse---math.u-strasbg.fr By a result due to H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra. 3./pub/Oprea-Rudyak/LScatOfSymplectic On the Listernik-Schnirlemann categroy of symplectic manifolds and the Arnold conjecture. Yuli B. Rudyak and John Oprea We prove that, for every closed connected symplectic manifold $(M,\omega)$, we have $\cat M=\dim M$ provided that $\omega$ vanishes on the second homotopy group. 4./pub/Sadofsky-Wilson/moravahopfalgebras Commutative Morava homology Hopf algebras Hal Sadofsky University of Oregon Eugene, OR, 97403 sadofsky---math.uoregon.edu W. Stephen Wilson Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu We give the Dieudonn\'e module theory for Z/2(p^n-1)-graded bicommutative Hopf algebras over F_p. These objects arise as the Morava K-theory of homotopy associative, homotopy commutative H-spaces. ---------------- 2 papers this time. Mark Hovey New papers uploaded to Hopf between 7/24/97 and 7/28/97: 1./pub/Casacuberta-Rodriguez/maps TITLE: On weak homotopy equivalences between mapping spaces AUTHORS: Carles Casacuberta (casac---mat.uab.es) Jose L. Rodriguez (jlrodri---mat.uab.es) ABSTRACT: Let $S^n_+$ denote the $n$-sphere with a disjoint basepoint. We give conditions ensuring that a map $h: X\to Y$ that induces bijections of homotopy classes of maps $[S^n_+,X]\cong [S^n_+,Y]$ for all $n\ge 0$ is a weak homotopy equivalence. For this to hold, it is sufficient that the fundamental groups of all path-connected components of $X$ and $Y$ be inverse limits of nilpotent groups. This condition is fulfilled by any map between based mapping spaces $h: map_*(B,W)\to map_*(A,V)$ if $A$ and $B$ are connected CW-complexes. The assumption that $A$ and $B$ be connected can be dropped if $W=V$ and the map $h$ is induced by a map $A\to B$. From the latter fact we infer that, for each map $f$, the class of $f$-local spaces is precisely the class of spaces orthogonal to $f$ and $f\wedge S^n_+$ for $n\ge 1$ in the based homotopy category. This has useful implications in the theory of homotopical localization. (This article will appear in Topology.) 2. /pub/Turner/vanishing On Simplicial Commutative Rings with Vanishing Andr\'e-Quillen Homology James M. Turner Department of Mathematics College of the Holy Cross One College Street Worcester, MA 01610-2395 We propose a generalization of a conjecture of D. Quillen on the vanishing of Andr\'e-Quillen homology to the simplicial commutative ring setting. This conjecture characterizes a notion of local complete intersection, extended to the simplicial setting, under a suitable hypothesis on the local characteristic. Under a suitable finiteness condition, we then prove the conjecture in the case of a simplicial commutative algebra augmented over a field. ---------------- The AMS has a cool new email forwarding system. You can now send me email at the address hovey---member.ams.org and it will automatically forward it to my real email address, whatever that happens to be at the time (so you won't have to remember mhovey---wesleyan.edu or hovey---unemployed.mathematician.com or whatever my email address is next year). You can sign up for this service at the URL http://www.ams.org/committee/profession/mail-forward.html . Last I checked you had to go this specific site--there were no links to it. 3 new papers this time. Mark Hovey New papers uploaded to Hopf between 7/28/97 and 8/4/97: 1. /pub/Morgenroth/BOn Homotopy Uniqueness of Products of Orthogonal Groups Helmut Morgenroth (Goettingen) In this paper we prove the homotopy uniqueness of the classifying space of products of orthogonal groups at the prime 2. We use methods of Notbohm from his proof of the homotopy uniqueness of classifying spaces of certain compact connected Lie groups at primes dividing the order of the Weyl groups. By replacing the torus SO(2)^n by O(2)^n, we make these methods work in our case, too. 2./pub/Monks/Grass TITLE: Groebner Bases and the Cohomology of Grassmann Manifolds with Application to Immersion AUTHOR: Ken Monks Department of Mathematics University of Scranton Scranton, PA 18510 email: monks---uofs.edu FILENAME: GRASS.DVI ABSTRACT: Let G_{k,n} be the Grassmann manifold of k-planes in R^{n+k}. Borel showed that H^*( G_{k,n}; Z_2) = Z_2[w_1,...,w_k] /I_{k,n} where I_{k,n} is the ideal generated by the dual Stiefel-Whitney classes \overline{w}_{n+1},...,\overline{w}_{n+k}. We compute Groebner bases for the ideals I_{2,2^i-3} and I_{2,2^i-4} and use these results along with the theory of modified Postnikov towers to prove new immersion results, namely that G_{2,2^i-3} immerses in R^{2^{i+2}-15}. As a benefit of the Groebner basis theory we also obtain a simple description of H^*(G_{2,2^i-3};Z_2) and H^*(G_{2,2^i-4};Z_2) and use these results to give a simple proof of some non-immersion results of Oproui. 3. /pub/Oprea-Rudyak/LScatOfSymplectic (This is a new abstract of a paper I announced last week) On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture Yuli B. Rudyak and John Oprea We prove that the Lusternik-Schnirelmann category $\text{cat}(M)$ of a closed symplectic manifold $(M, \omega)$ equals the dimension $\text{dim}(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $\pi_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. A modified form of the conjecture, replacing the latter quantity (via Lusternik-Schnirelmann theory) by $\text{cup}(M) + 1$, has been proved recently by various authors using techniques of Floer. The first author has also recently shown that the {\it original\/} form of the conjecture holds when $\text{cat}(M) =\text{dim}(M)$. Thus, this paper completes the proof of the original Arnold conjecture for closed symplectic manifolds with, for example, $\pi_2(M)=0$. ----------------- I am going on vacation for about a week, so even if you post a closed formula for the stable homotopy groups of spheres, it won't appear on this list right away. I also won't answer my mail for a while. Clarence has a new directory of pictures, /pub/Pictures-97, containing several subdirectories. Also, /pub/Goerss-Jardine/goerss-jardine.html is a link to Paul Goerss and J. F. Jardine's book, Simplicial Homotopy Theory. I do not include the table of contents as it was already posted to this list. There are also new versions of the following 2 papers. I include the abstracts in case you have forgotten them. Mark Hovey New papers uploaded to Hopf between 8/4/97 and 8/11/97: 1. /pub/Turner/opsseq1 Operations and Spectral Sequences I James M. Turner Department of Mathematics College of the Holy Cross One College Street Worcester, MA 01610-2395 e-mail: jmturner---math.holycross.edu Abstract: This is the first in a series of papers which examines a general type a chain complex (over F_2) whose homology supports a well-defined action of operations. We call such complexes Dold algebras, which include the singular cochain complex of a space and the singular chain complex of an infinite loop space, and we give conditions on filtrations of such objects so that there is a compatible action of operations on the associated spectral sequences. For applications, we recover W. Singer's result of the action of Steenrod operations on the Serre spectral sequence and we extend A. Bahri's action of Dyer-Lashof operations on the Eilenberg-Moore spectral sequence. 2. /pub/Turner/vanishing On Simplicial Commutative Rings with Vanishing Andr\'e-Quillen Homology by James M. Turner Department of Mathematics College of the Holy Cross One College Street Worcester, MA 01610-2395 e-mail: jmturner---math.holycross.edu Comments: 14 pages, LaTeX2e Keywords and phrases: simplicial commutative algebras, Andr\'e-Quillen homology, complete intersections, Serre spectral sequences, simplicial dimension, Postnikov envelopes, Poincar\'e series Abstract: We propose a generalization of a conjecture of D. Quillen, on the vanishing of Andr\'e-Quillen homology, to simplicial commutative rings. This conjecture characterizes a notion of local complete intersection, extended to the simplicial setting, under a suitable hypothesis on the local characteristic. Further, under the condition of finite-type homology, we then prove the conjecture in the case of a simplicial commutative algebra augmented over a field of non-zero characteristic. As a consequence, we obtain a proof of Quillen's conjecture for a Noetherian commutative algebra - again augmented over a field of non-zero characteristic. ---------------- Clarence has been remodelling Hopf recently, as you have seen in his messages to the list. In particular, directory names have changed so as to be easier to find--e.g Ed Brown is now BrownE instead of EBrown. He also has detailed instructions for how he (and I) would like an abstract for a paper submitted to Hopf to look. 7 new papers this time. Mark Hovey New papers uploaded to Hopf between 8/11/97 and 8/24/97: 1. /pub/Hodgkin/topglz Topological K-theory of GL(Z) at the prime 2 Luke Hodgkin Kings College, Strand, London, WC2R 2LS lukec.hodkin---kcl.ac.uk Recent results of Voevodsky and others have effectively led to the proof of the Lichtenbaum-Quillen conjectures at the prime 2, and consequently made it possible to determine the 2-local homotopy type of the K-theory spectra of various number rings. The basic case is that of BGL(Z); in this note we use these results to determine the 2-local (topological) K-theory of the space BGL(Z), which can be described as a completed tensor product of two quite simple components; one corresponds to a real 'image of J' space, the other to BBSO. 2. /pub/Hovey/model Model categories by Mark Hovey Warning: this is a 1 Mb dvi file of a 200 page book. Each chapter is available separately on my home page (see end of message for URL). This book began with the question: when is the homotopy category of a model category a stable homotopy category in the sense of Hovey-Palmieri-Strickland? It grew into a monograph developing the theory of model categories from the ground up to an answer to this question. Chapters: I. Model categories: standard stuff, except I consider the 2-category of model categories and show the homotopy category is part of a "pseudo-2-functor". II. Examples: cofibrantly generated model categories, chain complexes of R-modules, topological spaces, chain complexes of B-comodules, where B is a commutative Hopf algebra. III. Simplicial sets: Based on excellent book of Goerss-Jardine. IV. Quillen rings: model categories with an internal tensor product and Hom functor, compatible with model structure. V. Framings: the homotopy category of any model category looks like the homotopy category of a simplicial model category. VI. Pointed model categories: cofiber and fiber sequences as in Quillen, taking into account the simplicial structure of Chapter V. VII. Stable model categories and triangulated categories: New and stronger definition of triangulated category that applies in all known cases. The homotopy category of a pointed model category is triangulated if and only if the suspension is an equivalence. Generators in the homotopy category. VIII. Vistas: miscellaneous wild-eyed speculations. +Bibliography and Index. 3. /pub/HunterD-Kuhn/hunterkuhn1 Abstract for "Mahowaldean families of elements in stable homotopy groups revisited" by David J. Hunter and Nicholas J. Kuhn Beginning with Mahowald's work in his 1977 Topology "eta_j" paper, various infinite families of elements in the stable homotopy groups of spheres have been constructed in which all elements have a fixed Adams filtration. We revisited these constructions, figuring that the hindsight of 20 years and a "modern" understanding of both the Segal conjecture and unstable A-module technology might lead to clarification of this work. What resulted is the following. We isolate the two crucial results from the older literature, and present these stripped of extraneous detours. We then reorganize how these are used, together with the idea the BZ/p should be an intermdiate space in the constructions. This leads to streamlined and unified constructions of Mahowald's 2-primary filtration 2 family, Bruner's 2-primary fitration 3 family, and R.Cohen's odd primary filtration 3 families. In the odd prime case improvements are most dramatic: generously counting, we recover the main results of Cohen's Memoir with a saving of about 45 pages. All of our techniques were available before 1980. Our unified presentation reveals that Cohen's odd prime families are the exact analogue of Bruner's 2-primary family, and hint that the odd prime families h_0h_j in Ext_A^2(Z/p,Z/p) may behave more like the 2-primary NONpermanent cycles h_2h_j, than the permanent cycles h_1h_j. A new example shown is that, at odd primes, k_0h_0h_j in Ext_A^4(Z/p,Z/p) is a permanent cycle. 4. /pub/HunterD-Kuhn/hunterkuhn2 Abstract for "Characterizations of spectra with U-injective cohomology which satisfy the Brown-Gitler property" by David J. Hunter and Nicholas J. Kuhn We work in the stable homotopy category of p--complete connective spectra having mod p homology of finite type. H^*(X) means cohomology with Z/p coefficients, and is a left module over the Steenrod algebra A. A spectrum Z is called "spacelike" if it is a wedge summand of a suspension spectrum, and a spectrum X "satisfies the Brown--Gitler property" if the natural map [X,Z] ---> Hom_A(H^*(Z),H^*(X)) is onto, for all spacelike Z It is known that there exist spectra T(n) satisfying the Brown--Gitler property, and with H^*(T(n)) isomorphic to the injective envelope of H^*(S^n) in the category U of unstable A--modules. Call a spectrum X "standard" if it is a wedge of spectra of the form (L smash T(n)), where L is a stable wedge summand of the classifying space of some elementary abelian p--group. Such spectra have U--injective cohomology, and all U--injectives appear in this way. Working directly with the two properties of T(n) stated above, we clarify and extend earlier work by many people on Brown--Gitler spectra. Our main theorem is that, if X is a spectrum with U--injective cohomology, the following conditions are equivalent: (A) there exists a spectrum Y whose cohomology is a reduced U--injective, and a map X --> Y that is epic in cohomology, (B) there exists a spacelike spectrum Z, and a map X --> Z that is epic in cohomology, (C) $\epsilon:\Sigma^{\infty}\Omega^{\infty}X --->>> X$ is monic in cohomology, (D) X satisfies the Brown--Gitler property, (E) X is spacelike, (F) X is standard. (An unstable module is "reduced" if it has no nontrivial submodule which is a suspension.) As an application, we prove that the Snaith summands of $\Omega^2S^3$ are Brown--Gitler spectra -- a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture. Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an n--connected space X admits a map to an n--fold suspension that is monic in mod p homology, then $\epsilon: \Sigma^n\Omega^n X --->>> X$ is onto in mod $p$ homology. 5. /pub/Ravenel-Wilson-Yagita/rav-wil-yag Brown-Peterson cohomology from Morava $K$-theory This is the "final" version, Aug 1997, of a paper already on the archive from Nov 1995. Douglas C. Ravenel University of Rochester Rochester, New York 14627 drav---troi.cc.rochester.edu} W. Stephen Wilson Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu Nobuaki Yagita Ibaraki University Mito, Ibaraki, Japan yagita---mito.ipc.ibaraki.ac.jp We give some structure to the Brown-Peterson cohomology (or its $p$-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even dimensional. We can say that the Brown-Peterson cohomology is even dimensional (concentrated in even degrees) and is flat as a $BP^*$-module for the category of finitely presented $BP^*(BP)$-modules. At first glance this would seem to be a very restricted class of spaces but the world abounds with naturally occurring examples: Eilenberg-MacLane spaces, loops of finite Postnikov systems, classifying spaces of most finite groups whose Morava K-theory is known (including the symmetric groups), $QS^{2n}$, $BO(n)$, $MO(n)$, $BO$, $\ImJ$, etc. We finish with an explicit algebraic construction of the Brown-Peterson cohomology of a product of Eilenberg-MacLane spaces and a general K\"unneth isomorphism applicable to our situation. 6. /pub/Voronov/moduli Title: Stability of the Rational Homotopy Type of Moduli Spaces Authors: Alexander A. Voronov (RIMS and M.I.T.) Comments: 7 pages, 1 figure Report-no: RIMS-1157 Subj-class: 14H10 (Primary) 32G15, 55P62 (Secondary) We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational homotopy type of M_{g,n} as g goes to infinity and the formality of it. (Please get EPS file graph.eps also from /pub/Voronov/graph.eps ) 7. /pub/WilsonWS/wilrwy2 Brown-Peterson cohomology from Morava $K$-theory, II W. Stephen Wilson Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu We improve on some results with Ravenel and Yagita in a paper by the same name. In particular, we generalize when injectivity, surjectivity, and exactness of Morava $K$-theory implies the same for Brown-Peterson cohomology. A type of flatness is no longer assumed, but instead it is a consequence of weaker assumptions. The main application is an easier proof that $QS^{2k+1}$ has this flatness property. ----------------- Just one new paper this time. Mark Hovey New papers uploaded to Hopf between 8/24/97 and 8/28/97: 1./pub/Duflot/quantumgrp2 Lots of Hopf Algebras Jeanne Duflot Department of Mathematics, Colorado State University Fort Collins, Colorado, 80523 duflot---math.colostate.edu Abstract: The purpose of this paper is to give a "universal" construction of Hopf algebras over any commutative ring. Some special examples of these Hopf algebras, related to the mod-p Steenrod algebras, are stud- ied in more detail. -------------- A couple of people have had trouble getting the papers I announce here. To solve this, I will now put down the complete URL necessary to download the dvi file in question. You can cut and paste this URL into your web browser's "Open Location..." dialog box and it will automatically download the file. 6 new papers this time. Mark Hovey New papers uploaded to Hopf between 8/28/97 and 9/18/97: 1. ftp://hopf.math.purdue.edu/pub/Ando-Morava-Sadofsky/AndoMoravaSadofsky.dvi Title: "Completions of Z/p-Tate cohomology of periodic spectra" Authors: Matthew Ando, Jack Morava, Hal Sadofsky. AMS Primary classification: 55N22, 55P60, secondary 14L05 Respective addresses: Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA; Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA; Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. Email: ma2m---faraday.clas.Virginia.EDU, jack---math.jhu.edu, sadofsky---math.uoregon.edu ABSTRACT: We construct splittings of some completions of the Z/p-Tate cohomology spectra of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/p-Tate cohomology spectrum (i.e. the suspension of the Mahowald inverse limit, lim (P_{-k} \smash E(n))). We also give a multiplicative splitting of tE(n) after a suitable base extension. 2. ftp://hopf.math.purdue.edu/pub/BrownR/gpdscrs.dvi (There is also a file ftp://hopf.math.purdue.edu/pub/BrownR/gpdscrs2.dvi which is 30K larger. You may want to get that one, as it has a more recent date). Title: Groupoids and crossed objects in algebraic topology Author: Ronald Brown r.brown---bangor.ac.uk http://www.bangor.ac.uk/~mas010/ Abstract: These are the 63 pages of lecture notes for my six lectures at the Summer School on the Foundations of Algebraic Topology, Grenoble, June 16 - July 5, 1997. The notes concentrate on the background, intuition, proof and applications of the 2-dimensional Van Kampen Theorem (for the fundamental crossed module of a pair), with sketches of extensions to higher dimensions. One of the points stressed is how the extension from groups to groupoids leads to an extension from the abelian homotopy groups to non abelian higher dimensional generalisations of the fundamental group, as was sought by the topologists of the early part of this century. This links with J.H.C. Whitehead's efforts to extend combinatorial group theory to higher dimensions in terms of combinatorial homotopy theory, and which analogously motivated his simple homotopy theory. 3. ftp://hopf.math.purdue.edu/pub/Cortinas/holimhocolimequalshocolimholim.dvi (The title says it all, but heres the abstract anyway :-) ) Title: On interchanging homotopy limits and colimits in CAT. Author:Guillermo Corti\~nas Email:willie---mate.unlp.edu.ar We consider functors of the form $C:I--->>>CAT$ going from a small category $I$ to the category of small categories. We show that if $I$ has a final object and $C:I\times J--->>>CAT$ is a functor, then holim_Ihocolim_JC and hocolim_Jholim_IC are homotopy equivalent. Here hocolim_IC is the Grothendieck construction \int_IC, and holim_IC is the category of sections of the natural projection hocolim_IC--->>>I. 4. ftp://hopf.math.purdue.edu/pub/Hovey-Palmieri/bousfield.dvi Some questions about Bousfield classes by Mark Hovey and John Palmieri This paper attempts to present a global view of the Bousfield lattice. We re-examine a number of conjectures on the subject, proving many of them to be equivalent. The point of view expressed in the paper is that the Bousfield lattice would like to be a Boolean algebra on the Bousfield classes of K(n) for 0 <= n <= infinity. The only thing stopping it from being so is "strange" spectra, exemplified by I, the Brown-Comenetz dual of the sphere. 5. ftp://hopf.math.purdue.edu/pub/Rudyak/arnold.dvi On an alternative proof of the Arnold conjecture for symplectic manifolds with $\pi_2(M)=0$. Yuli B. Rudyak Let $(M,\omega)$ be a closed symplectic manifold. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. Recently author and Oprea proved this conjecture for symplectic manifolds with $\omega|\pi_2(M)=0=c_1|\pi_2(M)$. Here we indicate another proof (probably, more simple) of this result. 6. ftp://hopf.math.purdue.edu/pub/Tamanoi/bp-image.dvi Title: The image of the BP Thom map for Eilenberg--Mac Lane spaces Author: Hirotaka Tamanoi Address: Department of Mathematics University of California Santa Cruz, CA 95064 tamanoi---math.ucsc.edu ABSTRACT. We define fundamental classes in BP cohomology of Eilenberg--Mac Lane spaces. We then determine the image of the BP Thom map from BP cohomology to mod-p cohomology for any Eilenberg--Mac Lane space. The image turns out to be a polynomial subalgebra of the mod-p cohomology ring with infinitely many generators obtained by applying maximum number of Milnor primitives to the fundamental class in mod-p cohomology. This subalgebra is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We use a description of mod-p cohomology rings of Eilenberg--Mac Lane spaces in terms of the Milnor basis rather than in terms of admissible monomials. We also show that BP cohomology determines Morava K cohomology for Eilenberg--Mac Lane spaces. ----------------- Two final versions this time. Mark Hovey New papers uploaded to Hopf between 9/18/97 and 9/22/97: 1. ftp://hopf.math.purdue.edu/pub/Dwyer-Wilkerson/lie/liegroups.dvi The elementary geometric structure of compact Lie groups by W. G. Dwyer and C. W. Wilkerson We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of $p$-compact groups. (7/15/97 Final version, submitted to LMS Bulletin ) 2. ftp://hopf.math.purdue.edu/pub/Dwyer-Wilkerson/kahler-differentials/stein.dvi Kahler differentials, the T-functor, and a theorem of Steinberg W. G. Dwyer and C. W. Wilkerson Let $T$ be the functor on the category of unstable algebras over the Steenrod algebra constructed by Lannes. We use an argument involving Kahler differentials to show that $T$ preserves polynomial algebras. This leads to new and relatively simple proofs of some topological and algebraic theorems. ( This is a somewhat shorter ( examples deleted ) of the original version posted 7/16/97, CWW). (This is the final version of the paper. 9/15/97. CWW) -------------- One new paper. But also the file ftp://hopf.math.purdue.edu/pub/Ando-Morava-Sadofsky/AndoMoravaSadofsky.dvi is no longer completely in italics. So if that was driving you crazy, you can download the latest version. Mark Hovey New papers uploaded to Hopf between 9/22/97 and 9/29/97: 1. ftp://hopf.math.purdue.edu/pub/Clarke-Hunton-Ray/euc2.dvi Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems Primary 05A40; % Umbral calculus Secondary 55N22. % Bordism and cobordism theories, formal group laws \author{Francis Clarke} \address{Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales} \email{F.Clarke---Swansea.ac.uk} \author{John Hunton} \address{Department of Mathematics and Computer Science, University Road, Leicester LE1 7RH, England} \email{jrh7---mcs.le.ac.uk} \author{Nigel Ray} \address{Department of Mathematics, Manchester University, Manchester M13 9PL, England} \email{nige---mathematics.manchester.ac.uk} We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring $E_{*}$ with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where $E_{*}$ is free of additive torsion, in which context the central issues are number-theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions which are motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications; we also explain how the theory of Sheffer sequences fits naturally into this framework. Part One appeared as [N. Ray, Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. Math. 61 (1986), 49-100]. ---------------- Four new papers this time, one by Morava, one by Rudyak, and two by Tamanoi. I can't resist adding a commercial: Bill Dwyer and John Palmieri have just written this great program called bibweb that runs bibtex on your file, takes any entries not found and downloads them from the MathSciNet database and puts them in a .bib file for you! Isn't that great? It is a new program, so must be considered beta, but I tried it and it works, though it is kind of slow. It requires Unix. You can download it from John's web page: http://www.nd.edu/~jpalmier Mark Hovey New papers uploaded to Hopf between 9/29/97 and 10/21/97: 1. ftp://hopf.math.purdue.edu/pub/Morava/Schur.dvi Schur cohomology and a Kontsevich-Witten genus Jack Morava AMS Classification: Primary 14H10, Secondary 55N35, 81R10 Johns Hopkins University: jack---math.jhu.edu ABSTRACT: Two-dimensional topological gravity is a kind of physicist's interpretation of the rational cohomology of the group completion of the monoid of Riemann surfaces under glueing. It has a natural algebra of operations, which look vaguely like the operations in complex cobordism, and Witten has raised the question of their possible homotopy-theoretic interpretation. Over the integers this theory turns out to have an interesting model, which looks a lot like (a double of) the cohomology of Sp/U. There is an associated formal-group-like object, which looks unfamiliar because its coordinate seems to be centered at infinity, corresponding to asymptotic expansions of interest in physics. [This paper is a kind of sequel to 'Generalized quantum cohomology' posted previously on {\bf Hopf}, which has since appeared [in Contemporary Math. 202, Proceedings of the operads renaissance conference, ed. Loday, Stasheff, & Voronov] 2. ftp://hopf.math.purdue.edu/pub/Rudyak/AnalyticApps.dvi Title: On Analytical Applications of Stable Homotopy (The Arnold Conjecture, Critical points) Author: Yuli B. Rudyak Address: Mathematisches Institut, Universit"at Heidelberg, Im Neuenheimer Feld 288, D- 69120 Heidelberg 1, Germany E-mail address: july---mathi.uni-heidelberg.de Abstract. We prove the Arnold conjecture for closed symplectic manifolds with \pi_2(M) = 0 and catM = dimM. Furthermore, we prove an analog of the Lusternik- Schnirelmann theorem for functions with "generalized hyperbolicity" property. 3. ftp://hopf.math.purdue.edu/pub/Tamanoi/Eilenberg-MacLane.dvi Title : Q-subalgebras, Milnor Basis, and Cohomology of Eilenberg--Mac Lane Spaces Author : Hirotaka Tamanoi Address: Department of Mathematics University of California Santa Cruz Santa Cruz, CA 95064, USA Email: tamanoi---math.ucsc.edu ABSTRACT. We describe mod $p$ cohomology rings of Eilenberg-Mac Lane spaces in terms of the Milnor basis rather than in terms of admissible monomials of the Steenrod algebra. We give a formula of excess for the Milnor basis elements, correcting Kraines' formula in odd prime case. Using the Milnor basis description, we study and characterize certain polynomial subalgebras generated by elements obtained by applying maximum number of Milnor primitives on mod p fundamental classes of Eilenberg-Mac Lane spaces. A simple and interesting unstable pattern emerges. These subalgebras are exact images of the BP-Thom map into mod p cohomology rings. 4. ftp://hopf.math.purdue.edu/pub/Tamanoi/msp.dvi Title: Multiplicative Indecomposable Splittings of MSp_{[2]} Author: Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz, CA 95064 Email: tamanoi---math.ucsc.edu Abstract: When 2 is inverted, the symplectic cobordism cohomology theory becomes complex oriented with respect to the Buhstaber orientation. We study multiplicative idempotents in this theory in detail. Such multiplicative idempotents can only annihilate polynomial generators in degrees -4n, where 2n+1 is not a prime power. We split off a multiplicatively indecomposable smallest possible nontrivial cohomology theory from the above theory. The coefficient ring of this theory has polynomial generators in degrees of the form -2(p^k-1) for any positive k and for any odd prime p. --------------- Seven new papers this time. Mark Hovey New papers uploaded to Hopf between 9/29/97 and 10/21/97: 1. ftp://hopf.math.purdue.edu/pub/Goerss-Jardine/localize6.dvi Localization theories for simplicial presheaves P.G Goerss and J.F. Jardine This work was motivated in part by the following question of Soule: given a simplicial presheaf X on a site C, how does one produce a map of simplicial presheaves X to LX in such a way that each of the maps in sections X(U) to LX(U) for U in C, is an integral homology localization map in the sense of Bousfield? Secondly, if Y is a simplicial presheaf which is integrally homology local in a suitable sense, is it the case that the map X to LX induces an isomorphism [LX; Y ] ~=[X; Y ] relating sets of morphisms in the homotopy category of simplicial presheaves on C? These questions are related to the definition of the K-theory of simplicial sheaves that appears in [8]. The first of these questions is easily answered by observing that associated fibrations in the closed model category describing Bousfield's homology localizations are created with small object constructions and are therefore natural; in particular there is a functorial method of picking out a fibrant model Y to LY for arbitrary simplicial sets Y , which restricts in particular to a natural simplicial presheaf map X(U) to LX(U), U in C. The second question involves homotopy coherence, and is therefore much more subtle: the analogous space-level problem can be solved by Bousfield's original techniques, but this does not imply the functorial global solution that Soule requires. The problem is solved by using methods introduced in this paper, and in particular by applying Theorem 3.9 below. In the case corresponding to the identity functor on the site C, the chaotic topology on C and the constant presheaf of spectra associated to the Eilenberg-Mac Lane spectrum HZ, Theorem 3.9 implies that there is a closed simplicial model structure in the sense of Quillen on the category SPre(C) of simplicial presheaves on C such that the cofibrations are the pointwise monomorphisms and the weak equivalences are the pointwise integral homology isomorphisms. The map i : X to LX is then just a choice of fibrant model (ie. trivial cofibration, taking values in fibrant object) for this closed model structure, and the induced maps in sections i : X(U) to LX(U) are fibrant models for the corresponding theory on simplicial sets (ie. integral homology localizations in Bousfield's original sense), because the U-sections functor has a left adjoint which preserves cofibrations and takes integral homology isomorphisms to pointwise in- tegral homology isomorphisms. Furthermore, if we say that a simplicial presheaf Y is integral homology local if it's fibrant with respect to this new closed model structure on SPre(C), then Y is globally fibrant in the traditional sense, and the closed simplicial model structure gives an isomorphism ss(LX; Y ) ~=ss(X; Y ) in naive homotopy classes of maps which is induced by the HZ-trivial cofibration i (here ss(X; Y ) = ss0hom (X; Y ), for example). 2. ftp://hopf.math.purdue.edu/pub/Jeanneret-Osse/pHomogeneous.dvi TITLE: The Eilenberg-Moore spectral sequence for K-theory; applications to $p$-compact homogeneous spaces. AUTHORS: A. Jeanneret \& A. Osse AMS SUBJECT CLASSIFICATION (1991): 55T20, 55N15, 55P35, 57T35. ADDRESSES OF AUTHORS: A. Jeanneret Mathematisches Institut Sidlerstrasse CH-3012 BERN (Switzerland)/ A. Osse Institut de Mathematiques Rue Emile Argand 11 CH-2007 NEUCHATEL (Switzerland) Email ADDRESS OF AUTHORS: jeanner---math-stat.unibe.ch / AKImou.osse---maths.unine.ch ABSTRACT: We construct the Eilenberg-Moore spectral sequence for some generalized cohomology theories, along the lines of Smith and Hodgkin. We prove its multiplicativity and give some sufficient conditions for its convergence to the desired target. As applications, we compute the K-theory of various spaces associated to $p$-compact groups. 3. ftp://hopf.math.purdue.edu/pub/Lesh/ident-inf-loop-spaces.dvi Identification of infinite loop spaces arising from group theory Kathryn Lesh Let F_{n} be a family of subgroups of \Sigma_{n} which is closed under taking subgroups and conjugates. Such a family has a classifying space, BF_{n}, and we showed in an earlier work that a compatible choice of F_{n} for each n gives a simplicial monoid \coprod_{n} BF_{n} which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian p-groups, and the second gives a new expression for the desuspension of quotients of symmetric powers of spheres as suspension spectra. Address: Department of Mathematics, University of Toledo, Toledo OH 43606 klesh---uoft02.utoledo.edu 20 Aug 97 - 20 Aug 98: Department of Mathematics, Room 2-235, MIT, Cambridge, MA 02139 klesh---math.mit.edu 4. ftp://hopf.math.purdue.edu/pub/Moller/pup.dvi Title of Paper: PU(p) as a p-compact group Author: J.M. M{\o}ller AMS Classification numbers 55R35, 55P15, 55P10 Addresses of Author: Department of Mathematics Universitetsparken 5 DK-2100 Copenhagen Email address of Author: moller---math.ku.dk Text of Abstract: It has been conjectured that p-compact groups are determined by the normalizer of the maximal torus. Here, I consider the special case where the p-compact group is the adjoint form PU(p) of the unitary group. It is shown that PU(p) considered as a p-compact group indeed is determined by the normalizer of the maximal torus. General results then imply that the same is true for related p-compact groups such as U(p) and SU(p) and products of these. It also leads to cohomological uniqueness results and to an alternative construction and classification of self maps of BPU(p) completed at the prime p. 5. ftp://hopf.math.purdue.edu/pub/Tamanoi/Vn-series.dvi Title: Spectra of BP-linear relations, $v_n$-series, and BP cohomology of Eilenberg--Mac Lane spaces Author: Hirotaka Tamanoi Address: Department of Mathematics University of California Santa Cruz, CA 95064 Email: tamanoi---math.ucsc.edu Abstract: On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod $p$ cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod $p$ cohomology. We prove main relations in the BP cohomology of Eilenberg--Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define $v_n$-series which are $v_n$-analogues of the $p$-series. Finally we show that the above main relations come from the $v_n$-series. 6. ftp://hopf.math.purdue.edu/pub/Vershinin/hobr.dvi Homology of Braid Groups and Their Generalizations Vladimir V. Vershinin Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia In the paper we give a survey of (co)homologies of braid groups and groups connected with them. Among these groups are pure braid groups and generalized braid groups. We present explicit formulations of some theorems of V. I. Arnold, E. Brieskorn, D. B. Fuks, F. Cohen, V. V. Goryunov and others. The ideas of some proofs are outlined. As an application of (co)homologies of braid groups we study the Thom spectra of these groups. 7. ftp://hopf.math.purdue.edu/pub/Vershinin/thogebr.dvi Thom Spectra of Generalized Braid Groups Vladimir V. Vershinin Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg-MacLane spectrum for Z/2. Precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This paring generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over Z/2 on one generator of dimension one: Z/2[s]. ----------------- Nine new papers this time. Mark Hovey New papers uploaded to Hopf between 11/19/97 and 12/11/97: 1. ftp://hopf.math.purdue.edu/pub/Anton/quillen.dvi Author:Marian F. Anton Title: On a Conjecture of Quillen at the Prime 3 Email:manton1.nd.edu In this paper we prove a conjecture of Quillen about GL(n,A), where A is a ring of S-integers containing 1/3 and the third root of 1, for n=2 and disprove the same conjecture for n>26. 2. ftp://hopf.math.purdue.edu/pub/Iwase/ls-cat4h.dvi Title of Paper: Ganea's conjecture on Lusternik-Schnirelmann category Author: Norio Iwase AMS Classification numbers: Primary 55M30, Secondary 55P35, 55Q25, 55R35, 55S36. Addresses of Author: Graduate School of Mathematics, Kyushu University, Japan. Email addresses of Authors: n.iwase---maths.abdn.ac.uk Text of Abstract: A series of complexes $Q_p$ indexed by all primes $p$ is constructed with $\cat{Q_p} = 2$ and $\cat{Q_p{\times}S^n} = 2$ for either $n \geq 2$ or $n = 1$ and $p = 2$. This disproves Ganea's conjecture on Lusternik-Schnirelmann category. 3. ftp://hopf.math.purdue.edu/pub/Iwase-Saito-Sumi/hom_co-h.dvi (You might want to change the .dvi to .ps.gz or to .pdf, as this one has postscript diagrams that will not appear in the .dvi file) Title of Paper: Homology of the universal covering of a co-H-space Authors: Norio Iwase, Shiroshi Saito and Toshio Sumi AMS Classification numbers: Primary 55P45, Secondary 19A13 Addresses of Authors: (N.Iwase) Graduate School of Mathematics, Kyushu University, Fukuoka, Japan. (S.Saito) Department of Mathematics, Shinshu University, Matsumoto, Japan. (T.Sumi) Department of Art and Information Design, Kyushu Institute of Design, Fukuoka, Japan. Email addresses of Authors: (N.Iwase) n.iwase---maths.abdn.ac.uk (T.Sumi) sumi---kyushu-id.ac.jp Included EPS or PS files: coh-fig1.eps, coh-fig2.eps and coh-fig3.eps Text of Abstract: The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, dual notion to a Hopf space. We show a homological property of co-H-spaces in a slightly general situation. As its corollary, the conjecture is verified for co-H-spaces of finite type with homology groups concentrated in dimensions 1 and $*$, $n+1 \leq * \leq n+2$, for some $n \geq 1$. Also, such a co-H-space has the homotopy type of a wedge sum of circles, Moore spaces of type $(A,n+1)$ and $(B,n+2)$, for some finitely generated abelian groups $A$ and $B$. 4. ftp://hopf.math.purdue.edu/pub/Moreno/cdivisor.dvi Title:The zero divisors of the Cayley-Dickson algebras over the real numbers Author: Guillermo Moreno R. Institution: Department of Mathematics CINVESTAV-IPN Mexico., Mexico. comments: plain tex, latex 31 pages. tex, 31 pages In this paper we describe algebraically the zero divisors of the Cayley- Dickson algebras $\a_{n}=\erre^{2^n}$ for $n \ge 4$ over the real numbers. 5. ftp://hopf.math.purdue.edu/pub/Palmieri/palmieri-steenrod.dvi Stable homotopy over the Steenrod algebra John H. Palmieri In this paper I apply the results of "Axiomatic stable homotopy theory" (Hovey-Palmieri-Strickland) to the study of the Steenrod algebra A and its cohomology Ext_A. To do this, I work in the category Stable(A), in which the objects are cochain complexes of injective comodules over the dual of A. In this category, one can set up basic homotopy theoretic tools (like Postnikov towers); using these, I generalize some well-known results about A-modules to the category Stable(A) (like the vanishing line theorems of Anderson-Davis and Miller-Wilkerson.) I then use the methods and philosophy of modern stable homotopy theory to study Stable(A). This gives generalizations of deeper results about A-modules (such as the nilpotence theorems of the author), new proofs of old results (such as "chromatic convergence" in Stable(A)), and new results (such as a disproof of a version of the telescope conjecture in the category Stable(A)). 6. ftp://hopf.math.purdue.edu/pub/Pengelley-Williams/fwilliams.dvi Sheared algebra maps and operation Hopf algebras for mod 2 homology and cohomology David J. Pengelley email:davidp---nmsu.edu Frank Williams email:frank---nmsu.edu New Mexico State University Las Cruces, NM 88003 The mod 2 Steenrod algebra A and Dyer-Lashof algebra R have both striking similarities and differences, arising from their common origins in ``lower-indexed'' algebraic operations. These algebraic operations and their relations generate a Hopf algebra K, whose module actions are equivalent to, but quite different from, those of A and R. The exact relationships emerge as ``sheared algebra bijections'', which also illuminate the role of the cohomology of K. As a Hopf algebra, K^* has a particularly attractive and potentially useful structure, providing a bridge between those of A^* and R^*, and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras. 7. ftp://hopf.math.purdue.edu/pub/Strom/ecwgt.dvi Essential Category Weight Jeffrey A. Strom 55M30, 55P50 1175 F/AB 656 W. Kirby Wayne State University Detroit, MI 48202 strom---math.wayne.edu Abstract: The purpose of this paper is to introduce a new concept, the essential category weight of a map f : X---> Y, and to develop a basic theory, which includes nearly all of the classical lower bounds on cat(X) as trivial corollaries of significantly stronger theorems. Furthermore, we prove some results which seem to be unique in the literature. The theory leads to a proof of a special case of Ganea's X x S^k conjecture. 8. ftp://hopf.math.purdue.edu/pub/Strom/ecwgtclass.dvi (These two papers are definitely different, despite the identical titles) Essential Category Weight Jeffrey A. Strom 55M30, 55P50 1175 F/AB 656 W. Kirby Wayne State University Detroit, MI 48202 strom---math.wayne.edu Abstract: In this paper we prove a useful characterization of those maps f : BG ---> Y with essential category weight at least N. This leads to simple geometric proofs of results of Whitehead, Ginsburg, Toomer, Krasnoselkii, Eilenberg and Ganea. We discover that the infinity term of the Eilenberg-Moore spectral sequence is the graded algebra on H^*(X) associated to the filtration by essential category weight. 9. ftp://hopf.math.purdue.edu/pub/Strounine/hmlgdcmp.dvi \title{Homology decompositions for classifying spaces of compact Lie groups} \author{Alexei Strounine} \address{Mathematics Department, University of Notre Dame, Notre Dame, IN 46556} \email{alexei.strounine.1---nd.edu} \subjclass{Primary 55R35; Secondary 55R40} Let $p$ be a prime number and $G$ be a compact Lie group. A homology decomposition for the classifying space $BG$ is a way of building $BG$ up to mod $p$ homology as a homotopy colimit of classifying spaces of subgroups of $G$. In this paper we develop techniques for constructing such homology decompositions. In \cite{JMO} Jackowski, McClure and Oliver construct a homology decomposition of $BG$ by classifying spaces of $p$-stubborn subgroups of $G$. Their decomposition is based on the existence of a finite-dimensional mod $p$ acyclic $G-CW$-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the $p$-stubborn decomposition of $BG$ which does not use this geometric construction. ----------------- -------------------------------- The latest from Hopf: We have seven new papers this time, including, at last, the Hovey-Shipley-Smith paper on symmetric spectra. Clarence invites anyone unhappy with their abstracts on Hopf to submit new ones to him. He has also been working on the system: the latest version of the hopf interface is always at http://hopf.math.purdue.edu/pub/working.html The URL http://hopf.math.purdue.edu will give you the old version. The new interface has recent papers in reverse chronological order, and older papers in alphabetical order. If you click on a paper name in the new interface, you get various options for downloading it, including a new option of downloading the first two pages of the dvi file, thus allowing you to read the intro without reading the whole paper. You can also search the archive from the new interface. Besides the 7 new papers listed below, there is also a new version of John Greenlees bibliography: ftp://hopf.math.purdue.edu.pub/Greenlees/greenleesbibliography.dvi Mark Hovey New papers uploaded to Hopf between 12/11/97 and 1/17/98: 1. ftp://hopf.math.purdue.edu/pub/Arlettaz-Pointet/Arlettaz-Pointet.dvi Authors: Dominique Arlettaz and Nicole Pointet-Tischler Authors' addresses: Dominique Arlettaz, Institut de mathematiques, Universite de Lausanne, CH-1015 L ausanne, Switzerland Nicole Pointet-Tischler, Max-Planck-Institute for Mathematics,Gottfried-Clarenst rasse 26, D-53225 Bonn, Germany Title: Postnikov invariants of H-spaces Email: dominique.arlettaz---ima.unil.ch, pointet---mpim-bonn.mpg.de It is known that the order of all Postnikov k-invariants of an H-space of finite type is finite. This paper establishes the finiteness of the order of the k-invariants of (n-1)-connected H-spaces which are not necessarily of finite type, up to dimension 2n. Similar results hold more generally for higher k-invariants of iterated loop spaces. Moreover, we provide in all cases explicit universal upper bounds for the order of these k-invariants. 2. ftp://hopf.math.purdue.edu/pub/Cole-Greenlees-Kriz/Afgl.dvi Title: EQUIVARIANT FORMAL GROUP LAWS. Authors: MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-0001 E-mail address: mmcole---math.lsa.umich.edu School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: j.greenlees---sheffield.ac.uk Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 E-mail address: ikriz---math.lsa.umich.edu Abstract: Motivated by complex oriented theories we define A-equivariant formal group laws for any abelian compact Lie group A, show there is a representing ring for them, and begin the investigation of it. We examine a number of topological cases, including K-theory in some detail. 3. ftp://hopf.math.purdue.edu/pub/Greenlees/kuG.dvi \title{Equivariant forms of connective K theory} \author{J.P.C.Greenlees} \address{School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} Let $G$ be a group of order $p$, and consider the following properties of a $G$-spectrum $E$. 1. $E$ is a split ring spectrum, and nonequivariantly $ku$. 2. $E[v^{-1}]\simeq K$ (equivariantly) where $v$ is the degree 2 Bott element arising from the split structure. 3. $E$ is complex orientable (equivariantly). 4. $E^G_*$ is concentrated in even degrees. 5. $E_G^*$ is a Noetherian ring. Property 1 simply states that $E$ is an equivariant form of connective K-theory, and is therefore not negotiable. \begin{thm} There is a $G$-spectrum $ku$ with the five properties above. Its coefficient ring is the Rees ring $$ku_G^*=R(G)[v,y]/(vy = \chi (\alpha ), y\rho )$$ where $\alpha$ is the natural representation of $G$, $\chi (\alpha )= 1-\alpha$ is its K-theory Euler class and $\rho =1 + \alpha + \cdots + \alpha^{p-1}$ is the regular representation. The Bott element $v$ is in degree 2, and the element $y$ is in degree $-2$. \end{thm} 4. ftp://hopf.math.purdue.edu/pub/Greenlees/s1q.dvi \title{Rational $S^1$-equivariant stable homotopy theory.} \author{J.P.C.Greenlees} \address{School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} \begin{abstract} We make a systematic study of rational $S^1$-equivariant cohomology theories, or rather of their representing objects, rational $S^1$-spectra. In Part I we construct a complete algebraic model for the homotopy category of $S^1$-spectra, reminiscent of the localization theorem. The model is of homological dimension one, and simple enough to allow practical calculations; in particular we obtain a classification of rational $S^1$-equivariant cohomology theories. In Part II we identify the algebraic counterparts of all the usual change of groups functors associated to $S^1$-spectra. This enables us in Part III to give a rational analysis of a number of interesting phenomena, such as ordinary cohomology, the Atiyah-Hirzebruch spectral sequence, the Segal conjecture, $K$-theory and topological cyclic homology. Finally in Part IV we make a more thorough study of the algebraic models. This culminates in an identification of the algebraic models of the smash product and function spectrum. \end{abstract} \begin{comments} This is a revised version of the paper submitted to the archive in 1996. It is substantially reorganized and improved, and supercedes the 1996 version. The largest change is that the two chapters on smash products and function spectra have now expanded to form Part IV. If you want to download it in smaller pieces, go to my preprint page from http://www.shef.ac.uk/~ms/staff/greenlees/ \end{comments} 5. ftp://hopf.math.purdue.edu/pub/Hovey-Shipley-Smith/symm.dvi Symmetric spectra Mark Hovey, Brooke Shipley, and Jeff Smith The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum. 6. ftp://hopf.math.purdue.edu/pub/Schwede-Shipley/last.dvi Algebras and modules in monoidal model categories Stefan Schwede and Brooke E. Shipley We construct model category structures for monoids and modules in symmetric monoidal model categories, with applications to symmetric spectra and $\Gamma$-spaces. 7. ftp://hopf.math.purdue.edu/pub/Shipley/whole.dvi Title: SYMMETRIC RING SPECTRA AND TOPOLOGICAL HOCHSCHILD HOMOLOGY Author: BROOKE SHIPLEY The category of symmetric spectra introduced by Jeff Smith is a closed symmetric monoidal category whose associated homotopy category is equivalent to the traditional stable homotopy category, see [HSS]. In this paper, we study symmetric ring spectra, i.e., the monoids in the category of symmetric spectra. Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address: bshipley---math.uchicago.edu ---------------- We have three new papers on Hopf this time. In addition, Clarence has gone through and checked all the URL's on hopf--there have been 3 phantom papers in the archive for quite a while now, which are now back among the living. If you have been unable to download the Hunter-Kuhn papers or the latest Duflot paper, all announced here long ago, try again now. Mark Hovey New papers uploaded to Hopf between 1/17/98 and 1/22/98: 1. ftp://hopf.math.purdue.edu/pub/Gerstenhaber-Wilkerson/dga-deform.dvi Authors: Murray Gerstenhaber and Clarence Wilkerson Title: On the deformation of rings and algebras, V: Deformation of differential graded algebras From the introduction: In this paper we consider the deformation theory of differential graded modules (DGM's) and differential graded algebras (DGA's), where only the differential varies, the underlying module or algebra structure remaining fixed. At the outset we consider only individual modules or algebras and afterwards we examine deformations of sheaves. Note: this will appear in the Stasheff Fest volume. 2. ftp://hopf.math.purdue.edu/pub/Hodgkin/KBGamma3.dvi Title: K-theory of mapping class groups III: Odd torsion Author: Luke Hodgkin Address: King's College, Strand, London WC2R 2LS, UK. Email: luke.hodgkin---kcl.ac.uk This is the long-awaited calculation of the odd-torsion in K^\ast(B\Gamma^n) (mapping-class groups for punctured spheres). The size and location of the torsion (it's all in K^1) is completely calculated, together with where it comes from and why; and there is information about the module structure over K^\ast(BSO(3)). Methods are: (a) the description of B\Gamma in terms of function spaces due to Bodigheimer,Cohen and Peim and (b) author's earlier calculation of the structure of K^\ast mod torsion (Math.Z. 218, 611-634). This is my last communication on this subject; if anyone wants to find the 2-torsion, good luck to them. (Now, that's what I call an abstract! Mark :) ) 3. ftp://hopf.math.purdue.edu/pub/Hovey-Palmieri/bousfield.dvi (This is a greatly modified version of a paper already on the archive. This version is more like John's talk at Baltimore.) Title: The structure of the Bousfield lattice Authors: by Mark Hovey and John Palmieri Using Ohkawa's theorem that the collection of Bousfield classes is a set, we perform a number of constructions with Bousfield classes. In particular, we describe a greatest lower bound operator; we also note that a certain subset DL of the Bousfield lattice is a frame, and we examine some consequences of this observation. We make several conjectures about the structure of the Bousfield lattice and DL. In particular, we conjecture that DL is obtained by killing "strange" spectra, such as the Brown-Comenetz dual of the sphere. We introduce a new "Boolean algebra of spectra" cBA, which contains Bousfield's BA and is complete. Our conjectures allow us to identify cBA as being isomorphic to the complete atomic Boolean algebra on {K(n) : n>= 0}, {A(n) : n>= 2}, and HF_p. Our conjectures imply that BA is the subBoolean algebra consisting of finite wedges of the K(n) and A(n), and their complements. ---------------- There are two new papers on the archive this time. Notice that I have changed the URLs: if you use these URLs you will get to a page that will give you choices for what you want to download. If you still like ftp, replace "http" with "ftp", delete "cgi-bin/generate?/", and add a suffix like ".dvi" at the end. I also want to tell readers of my "Symmetric spectra" paper, with Brooke Shipley and Jeff Smith, that the last section of that paper, on topological symmetric spectra, has some errors. The category of topological symmetric spectra is just not as nice as simplicial symmetric spectra. We will be submitting a revised version. Mark Hovey New papers uploaded to Hopf between 1/22/98 and 2/5/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/cmceinf The Homotopy Theory of \einf Algebras Michael A. Mandell Let $k$ be a commutative ring and let $\oC$ be the operad of differential graded $k$-modules obtained as the singular $k$-chains of the linear isometries operad \cite[\S V.9]{km}. We show that the category of $\oC$-algebras is a proper closed model category. We use the amenable description of the coproduct in this category \cite[V.3.4]{km} to analyze the coproduct of and develop a homotopy theory for algebras over an arbitrary \einf operad. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/einf $E_{\infty}$-Algebras and $p$-Adic Homotopy Theory Michael A. Mandell Let $\FP$ denote the field with $p$ elements and $\FPbar$ its algebraic closure. We show that the singular cochain functor with coefficients in $\FPbar$ induces a contravariant equivalence between the homotopy category of connected $p$-complete nilpotent spaces of finite $p$-type and a full subcategory of the homotopy category of \einf $\FPbar$-algebras. --------------- Just one new paper this time. Mark Hovey New papers uploaded to Hopf between 2/5/98 and 2/15/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-Thompson/k1bkss Abstract for ``The Bousfield-Kan spectral sequence for periodic homology theories'' by Martin Bendersky and Rob Thompson. AMS classification numbers: Primary 55Q40; Secondary 55P60, 55T15, 55H20, 19L99. Author addresses: Hunter College, CUNY Department of Mathematics and Statistics 695 Park Ave. New York, NY 10021 mbenders---shiva.hunter.cuny.edu thompson---math.hunter.cuny.edu http://math.hunter.cuny.edu/~benders http://math.hunter.cuny.edu/~thompson ------------------------------------------------------------------------- In this paper we construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain non-connective periodic homology theories E such as complex periodic K-theory, and define an E-completion of a space X. For X = S^{2n+1} and E = K we calculate the E_2-term and show that the spectral sequence converges to the homotopy groups of the K-completion of the sphere. This also determines all of the homotopy groups of the (unstable) K-theory localization of S^{2n+1} including three divisible groups in negative stems. --------------- We have five new papers this time. I also want to announce that I will post annoucements of new papers on the algebraic topology part of the xxx archive as well, at least when they have not previously appeared on hopf. Mark Hovey New papers uploaded to Hopf between 2/15/98 and 2/26/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbounov-Mahowald/eon2 Formal completion of the Jacobian of plane curves and higher real K-theories by V. Gorbounov and M. Mahowald. In early sixties Manin proved that every formal group of finite height defined over a field of finite characteristic is a summand in the formal completion of the Jacobian of a certain curve. It turns out that a universal lift of a formal group of height $p-1$ over an algebraically closed field of characteristic $p$ comes as a summand in the formal completion of the Jacobian of a certain curve with $p$ marked points, defined over the Lubin-Tate deformation space. These curves generalize the Legendre family of elliptic curves. As an immediate application, we will describe the representation which is crucial for calculating the initial term of the spectral sequence, converging to the homotopy groups of the higher real $K$-theories, introduced recently by Hopkins and Miller. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/mon-mod Monoidal model categories by Mark Hovey A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider monoids and modules over a given monoid. We would like to be able to study the homotopy theory of these monoids and modules. This question was first addressed by Stefan Schwede and Brooke Shipley in "Algebras and modules in monoidal model categories", who showed that under certain conditions, there are model categories of monoids and of modules over a given monoid. This paper is a follow-up to that one. We study what happens when the conditions of Schwede-Shipley do not hold. This will happen in any topological situation, and in particular, in topological symmetric spectra. We find that, with no conditions on our monoidal model category except that it be cofibrantly generated and that the unit be cofibrant, we still obtain a homotopy category of monoids, and that this homotopy category is homotopy invariant in an appropriate sense. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moller/di2 Title: The 3-compact group DI(2) Author: Jesper M. M{\o}ller AMS Class: 55R35 Address: Matematisk Institut Universitetsparken 5 DK-2100 Copenhagen Denmark e-mail: moller---math.ku.dk The 3-complete space BDI(2) realizes the mod 3 rank 2 Dickson invariants. We investigate DI(2) considered as a 3-compact group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WilsonWS/bpfromkn2 title: Brown-Peterson cohomology from Morava K-theory, II author: W. Stephen Wilson address: Johns Hopkins University, Baltimore, Maryland 21218 email: wsw---math.jhu.edu abstract: We improve on some results with Ravenel and Yagita in a paper by the same name. In particular, we generalize when injectivity, surjectivity, and exactness of Morava K-theory implies the same for Brown-Peterson cohomology. A type of flatness is no longer assumed, but instead it is a consequence of weaker assumptions. The main application is an easier proof that QS^{2k+1} has this flatness property. In addition, we show that if elements in the Brown-Peterson cohomology generate all of the Morava K-theories, then they also generate the Brown-Peterson cohomology and it is Landweber flat. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WilsonWS/knequiv Title: K(n+1) equivalence implies K(n) equivalence Author: W. Stephen Wilson Address: Johns Hopkins University, Baltimore, Maryland 21218 email: wsw---math.jhu.edu abstract: We give an entirely different proof of a recent result of Bousfield's which states that if there is a map of spaces inducing an isomorphism on the (n+1)st Morava K-theory then it also induces an isomorphism on the n-th Morava K-theory. The result relies heavily on the fundamentals introduced to prove the results in Ravenel-Wilson-Yagita which in turn relies on the forthcoming paper by Boardman-Wilson containing a generalization of Quillen's theorem that MU^*(X) is generated by non-negative degree elements when X is a finite complex. ---------------- There are six new papers this time. Mark Hovey New papers uploaded to Hopf between 2/26/98 and 3/5/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adin-Blanc/Adin_Blanc Resolutions of Associative and Lie algebras Ron Adin Bar Ilan University David Blanc University of Haifa We here describe certain explicit canonical resolutions for free associative and free (graded) Lie algebras, in the category of non-associative algebras. Both resolutions are based on the combinatorics of suitable collections of leaf-labeled trees. The Lie case was needed for the second author's description of higher homotopy operations in rational homotopy theory: it turns out that in order to describe all such higher operations, one must resolve the rational differential graded Lie algebra L_* (representing the rational homotopy type of a given space X) simplicially, by suitable free (differential) graded Lie algebras. The higher homotopy operations correspond to relations and syzygies for these free graded Lie algebras, thought of as non-associative algebras. Since we must replace all the Lie algebras by the corresponding free differential algebras in a functorial manner (to preserve the simplicial structure of the original resolution of L_* we need canonical resolutions of free Lie algebras in the category of non-associative algebras, as described in this paper. The construction is closely related to ``strongly homotopy Lie algebras'' Our main interest is indeed in the Lie case. The associative case, which is based on work of Stasheff, is included mainly as a preliminary illustration of the ideas involved, and to fix notation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc_Cwres CW simplicial resolutions of spaces, with an application to loop spaces David Blanc University of Haifa (blanc---mathcs2.haifa.ac.il) A simplicial resolution of a space X by wedges of spheres is a simplicial space W_. such that (a) each space W_n is homotopy equivalent to a wedge of spheres, and (b) for each k>0, the augmented simplicial group \pi_k W_.->\pi_k X is acyclic. Such resolutions were first constructed by Stover, and have a number of applications. However, the Stover construction yields very large resolutions, which do not lend themselves readily to computation. We show here that in fact any space X has such resolutions, which may be constructed from purely algebraic data, consisting of an (arbitrary) simplicial resolution of \pi_* X as a \Pi-algebra ( \ -- \ that is, as a graded group with an action on the primary homotopy operations on it), and in fact every such algebraic resolution of \pi_* X is realizable topologically. Moreover, such resolutions can be given a convenient ``CW structure''. There is an analogous result for maps. As an application of such CW resolutions, we describe an obstruction theory for deciding whether a given space X is a loop space, in terms of higher homotopy operations. The present approach does not require a given $H$-space structure on $\X$, and may be adapted also to the existence of A_{n}-structures. Note: the second part of this paper contains the second part of the preprint previously posted under the title: "Loop spaces and homotopy operations". 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc_Loop Loop spaces and homotopy operations David Blanc University of Haifa (blanc---mathcs2.haifa.ac.il) The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{*-1} X (which are isomorphic to \pi_* Y, if X=\Omega\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing \Pi-algebras, which is simplified here by showing that any (suitable) \Delta-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts). Note: this paper is an expanded version of the first half of the preprint previously posted under the same name. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/mon-mod This is a slightly revised version of the paper announced here last time (Monoidal model categories). The main new feature is a proof that the monoid axiom holds in any category where everything is fibrant and there is a well-behaved "unit interval" I. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey-Shipley-Smith/symm This is a revised version of the paper "Symmetric spectra" that was previously announced. The prior version had several errors in the section on topological symmetric spectra that are now cleared up. The rest of the paper has changed only minimally. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lydakis/smash_gamma Abstract for "Smash products and $\Gamma$-spaces" by Manos Lydakis Fakultaet fuer Mathematik Universitaet Bielefeld Postfach 100131 33501 Bielefeld Germany manos---math206.mathematik.uni-bielefeld.de We study a symmetric monoidal smash product of $\Gamma$-spaces, which corresponds to the smash product of spectra under the Bousfield-Friedlander equivalence between the homotopy categories of connective spectra and $\Gamma$-spaces. --------------- There are two new papers this time: the first one is not on Hopf yet but is available from xxx. I expect it to be available soon from Hopf as well, and will give you the expected Hopf URL. I will also give you the URL to download the .dvi file from xxx. In figuring out this URL I learned, to my chagrin, that xxx not only requires you to submit your .tex file, but also allows anyone else to download it! I had not realized this before. Mark Hovey New papers uploaded to Hopf (and xxx) between 3/5/98 and 3/17/98: 1. http://xxx.lanl.gov/dvi/math.AT/9803068 (If you want postscript, change "dvi" to "ps". If you want pdf, change "dvi" to "pdf". If you want the tex source, you will have to figure out what to do for yourself.) Should also be available on Hopf soon at http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hopkins-Palmieri-Smith/vanish Paper: math.AT/9803068 From: John H. Palmieri Date: Mon, 16 Mar 1998 14:58:38 GMT (6kb) Title: Vanishing lines in Adams spectral sequences are generic Authors: M. J. Hopkins, J. H. Palmieri, J. H. Smith Comments: 6 pages Subj-class: Algebraic Topology MSC-class: 55T15; 55P42 \\ We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property. \\ ( http://xxx.lanl.gov/abs/math/9803068 , 6kb) 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/Gamma_algebra `Stable homotopical algebra and $\Gamma$-spaces' by Stefan Schwede Department of Mathematics Massachusetts Insitute of Technology Cambridge, MA 02141 schwede---math.mit.edu In this paper we advertise the category of $\Gamma$-spaces as a convenient framework for doing `algebra' over `rings' in stable homotopy theory. $\Gamma$-spaces were introduced by Segal who showed that they give rise to a homotopy category equivalent to the homotopy category of connective spectra. Bousfield and Friedlander later provided model category structures for $\Gamma$-spaces, and Lydakis recently introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences, and compare simplicial modules and algebras to their Eilenberg-MacLane spectra counterparts. We believe that one advantage of the $\Gamma$-space approach is its simplicity. The definitions of the stable equivalences, the smash product, and the `rings' (which we call `Gamma-rings') only take a few pages. Furthermore $\Gamma$-spaces are nicely compatible with classical rings and modules. There is an Eilenberg-MacLane functor which embeds the category of simplicial abelian groups as a full subcategory of the category of $\Gamma$-spaces. The embedding has a strong symmetric monoidal left adjoint which models spectrum homology. We can give a quick proof that modules and algebras over a simplicial ring have the same homotopy theory as their counterparts over the associated Eilenberg-MacLane Gamma-ring. One intrinsic limitation of this approach comes from the fact that $\Gamma$-spaces only model connective spectra. This rules out applications in certain areas of stable homotopy theory, but it is no essential restriction for the purpose of algebraic K-theory, topological Hochschild homology and topological cyclic homology. This paper will appear in the Mathematical Proceedings of the Cambridge Philosophical Society. ---------------- Greg Kuperberg gave me some info on why xxx allows users to download tex source. 1. Almost every other archive except us does so--we and the K-theory archive are almost the only ones who routinely use dvi. 2. No one has reported any problems with plaigarism, and xxx is itself some defense against it, since it keeps the TeX files unchanged and dated. 3. Modern technology will soon, if it does not do so already, allow you to cut and paste from a dvi file to a tex file anyway. So I apologize for overreacting--the issue is more complicated than I originally thought. There is one new paper this time, plus the Hopkins-Palmieri-Smith paper has landed on Hopf in the expected place, as I reported it would last time. Mark Hovey New papers uploaded to Hopf (and xxx) between 3/17/98 and 3/19/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ravenel-Wilson-Yagita/rav-wil-yag Brown-Peterson cohomology from Morava $K$-theory This is the "final" version of this paper. It is significantly changed from the version which has been on the archive. Douglas C. Ravenel University of Rochester Rochester, New York 14627 drav---troi.cc.rochester.edu} W. Stephen Wilson Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu Nobuaki Yagita Ibaraki University Mito, Ibaraki, Japan yagita---mito.ipc.ibaraki.ac.jp We give some structure to the Brown-Peterson cohomology (or its $p$-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even dimensional. We can say that the Brown-Peterson cohomology is even dimensional (concentrated in even degrees) and is flat as a $BP^*$-module for the category of finitely presented $BP^*(BP)$-modules. At first glance this would seem to be a very restricted class of spaces but the world abounds with naturally occurring examples: Eilenberg-MacLane spaces, loops of finite Postnikov systems, classifying spaces of most finite groups whose Morava K-theory is known (including the symmetric groups), $QS^{2n}$, $BO(n)$, $MO(n)$, $BO$, $\ImJ$, etc. We finish with an explicit algebraic construction of the Brown-Peterson cohomology of a product of Eilenberg-MacLane spaces and a general K\"unneth isomorphism applicable to our situation. ---------------- Two new papers this time. Mark Hovey New papers uploaded to Hopf between 3/19/98 and 3/25/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/DavisD-Zelov/vitaly Some embeddings and nonimmersions of real projective spaces Donald M. Davis and Vitaly Zelov submitted to Boardman conference proceedings Abstract We prove the following new results. If alpha(n)=2, then RP^{16n+8} cannot be immersed in R^{32n+3}, and RP^{16n+10} cannot be immersed in R^{32n+11}. If alpha(n)>2, then RP^{8n+4} can be immersed in R^{16n+1}. The method is obstruction theory. The main novelty is careful consideration of secondary indeterminacy for the nonimmersions, and a combining of two methods for the embeddings. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WuJ/newsimplicialgroup_1 On Combinatorial Descriptions of Homotopy Groups of Certain Spaces This is the revised version of the paper title "On Combinatorial Descriptions of Homotopy Groups of K(ss; 1)" on the archive. It is significantly changed from the version which has been on the archive, including adding some new results. Jie Wu Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 USA jiewu---math.upenn.edu Included file: newsimplicialgroup_1.dvi Abstract: 1) We give combinatorial groups which occur naturally for which the homotopy groups of the suspension of $K(\pi,1)$ for general $\pi$ are the centers. [Theorem 1.5]. 2) We give explicit groups which occur naturally for which all of the homotopy groups of the 3-sphere are the centers [Theorem 1.2]. 3) Furthermore we give an explicit finitely presented nilpotent group for which the (general) higher homotopy group of the 3-sphere is the torsion of the center [Proposition 4.9]. In other words, there are explicit finite 2-complexes of which the (general) higher homotopy groups of the $3$-sphere are the torsion of the centers of the fundamental groups. 4) Our descriptions are NOT yet calculations of higher homotopy groups. But it is expected to have uses of computer for studying the centers of these combinatorial groups. 5) We have a group theoretical description of the torsion of homotopy groups of any simply connected space [Theorem 2.22]. We do not yet have an explicit combinatorial description of homotopy groups of higher spheres. 6) Our description allows us to compute the homotopy groups of Cohen's construction of the 1-sphere (the space that is considerable useful in mathematical physics and deformation theory) and to reprove Milnor's unpublished example on Moore's problem. ---------------- Four new papers this time--one at Hopf and three at xxx. Mark Hovey New papers uploaded to Hopf and xxx between 3/25/98 and 4/4/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Crossley-Whitehouse/conjinvs Title: On Conjugation Invariants in the dual Steenrod algebra Authors: M. D. Crossley and Sarah Whitehouse AMS Classification Numbers: 55S10 Address (of both authors): Departement de Mathematiques, Institut Galilee, Universite Paris 13, 93430 Villetaneuse, France. Email addresses: crossley---math.univ-paris13.fr sarah---math.univ-paris13.fr Abstract Text: We investigate the canonical conjugation, chi, of the mod 2 dual Steenrod algebra A_*, with a view to determining the subspace, A_*^chi, of elements invariant under chi. We give bounds on the dimension of this subspace for each degree and show that, after inverting xi_1, it becomes polynomial on a natural set of generators. 2. http://xxx.lanl.gov/dvi/math.DG/9803136 (This paper is cross-listed in the algebraic topology and differential geometry sections of xxx, as is the next one.) From: Michael Farber Date: Fri, 27 Mar 1998 16:34:26 GMT (23kb) Title: Witten deformation and polynomial differential forms Authors: Michael Farber and Eugenii Shustin Comments: 27 pages, 4 figures, AmsTex Subj-class: Differential Geometry; Algebraic Topology \\ As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a cylindrical structure. We prove that the cohomology of the Witten deformation acting on the complex of the polynomially growing forms can be computed as the relative cohomology of the manifold with respect to the negative remote fiber of the function. We show that the assumptions of our main theorem are satisfied in a number of interesting special cases, including generic real polynomials. 3. http://xxx.lanl.gov/dvi/math.DG/9803137 From: Michael Farber Date: Fri, 27 Mar 1998 16:48:58 GMT (27kb) Title: Poincar\'e - Reidemeister metric, Euler structures, and torsion Authors: Michael Farber and Vladimir Turaev Comments: 3 figures, AmsTex Subj-class: Differential Geometry; Algebraic Topology \\ In this paper we define a Poincar\'e-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler structures, introduced earlier by the second author. We show that the sign of our PR-scalar product is determined by the Stiefel-Whitney classes and the semi-characteristic of the manifold. As an application, we compute the Ray-Singer analytic torsion via the torsions of Euler structures. Another application: a computation of the twisted semi-characteristic in terms of the Stiefel-Whitney classes. 4. http://xxx.lanl.gov/dvi/math.AT/9803156 From: Jim Stasheff Date: Tue, 31 Mar 1998 20:12:53 GMT (11kb) Title: Grafting Boardman's Cherry Trees to Quantum Field Theory Authors: Jim Stasheff Comments: 8 pages, 9 figures Subj-class: Algebraic Topology; Quantum Algebra \\ Michael Boardman has been a major contributor to the theory of infinite loop spaces and higher homotopy algebra. Indeed Boardman was the first to refer to `homotopy everything'. One particular contribution which has had major progeny is his use of `geometric' trees, combinatorial tress with lengths attached to edges. Here is a modified version of the talk given to honor Mike on the occassion of his 60th birthday. It is an idiosyncratic survey of parts of homotopy algebra from Poincar\'e to the present day, with emphasis on Boardman's original ideas, starting with his cubical subdivision of the associahedra through recent applications in mathematical physics via compactifications of moduli spaces. ---------------- One new paper this time. Mark Hovey New papers uploaded to Hopf between 4/4/98 and 4/10/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lueck/luecktype0498 Title: ``The type of the classifying space for a family of subgroups'' Author: Wolfgang Lueck AMS-classification number: 55R35 Address: Wolfgang Lueck Fachbereich Mathematik und Informatik Westf\"alische Wilhelms-Universitaet Muenster Einsteinstr. 62 48149 M\"unster Germany email: lueck---math.uni-muenster.de Abstract: The classifying space E(G,F) for a family F of subgroups of a group G is defined up to G-homotopy as a G-CW-complex E(G,F) such that E(G,F)^H is contractible if H belongs to F and is empty otherwise. The space E(G,F) occurs in the Baum-Connes-Conjecture, The Isomorphism-Conjecture in algebraic K-theory and L-theory of Farrell and Jones and in the version of the Atiyah-Segal-Completion-Theorem for proper cocompact actions of discrete groups. We investigate the question whether there is a finite-dimensional G-CW-model, a finite G-CW-model or a G-CW-model of finite type for E(G,F) focusing on the case where F is the family FIN of finite subgroups. -------------- One new paper this time. Mark Hovey New papers uploaded to Hopf between 4/10/98 and 4/16/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/stanley1 Title:``Determining Closed Model Category Structures'' Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley---math.fu-berlin.de Abstract: In this paper we give conditions under which a closed model category structure is determined on a category. The conditions are more general than cofibrantly generated. The paper contains two advances in tequniques of giving categories closed model structures. The first is the proof of the lifting property of acyclic cofibrations with respect to the maps in the corresponding factorization. This allows us to have a generating class, as oposed to a set, of acyclic cofibrations. The second is the definition of fibrations as retracts of compositions of maps that are the second maps in the two factorizations. This makes it easier to construct the needed liftings. At the end of the paper we apply our construction to two examples. ---------------- Here are two papers crosslisted to the algebraic topology section of xxx, submitted in the last week. Mark Hovey New papers uploaded to xxx between 4/16/98 and 4/21/98: 1. http://xxx.lanl.gov/dvi/math.AG/9804078 From: Eduard Looijenga Date: Thu, 16 Apr 1998 12:16:55 GMT (11kb) Title: M_g is a union of g-1 affine open subsets Authors: Eduard Looijenga Comments: LaTeX2e, 9 pages Subj-class: Algebraic Geometry; Algebraic Topology MSC-class: 14H15; 14F17 \\ We prove the assertion of the title. This implies that the cohomological dimension of the moduli space M_g (of smooth curves of genus g) is at most g-2 for quasi-coherent sheaves (this amplifies Diaz's theorem which says that M_g has no complete subvariety of dimension g-1) and at most 4g-5 for constructible sheaves. We also recover Harer's theorem which says that M_g has the homotopy type of finite cell complex of dimension 4g-5. We find similar results for the moduli space of n-pointed curves of genus g. \\ ( http://xxx.lanl.gov/abs/math/9804078 , 11kb) 2. http://xxx.lanl.gov/dvi/math.GN/9804085 From: Krystyna Kuperberg Date: Sun, 19 Apr 1998 20:11:23 GMT (9kb) Title: Bihomogeneity and Menger manifolds Authors: Krystyna Kuperberg (Auburn University) Comments: 9 pages Subj-class: General Topology; Algebraic Topology MSC-class: 54F35 (primary), 55M99 (secondary) Journal-ref: Top. Appl. 84 (1998), 175-184 \\ For every triple of integers a, b, and c, such that a>O, b>0, and c>1, there is a homogeneous, non-bihomogeneous continuum whose every point has a neighborhood homeomorphic the Cartesian product of three Menger compacta m^a, m^b, and m^c. In particular, there is a homogeneous, non-bihomogeneous, Peano continuum of covering dimension four. \\ ( http://xxx.lanl.gov/abs/math/9804085 , 9kb) -------------- One new paper this time. Mark Hovey New papers uploaded to hopf between 4/16/98 and 4/29/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/hkalg Algebraization of E-infinity Ring Spectra Michael A. Mandell For a commutative ring k, the homotopy category of commutative Hk-algebras (strictly unital E-infinity ring spectra under the Eilenberg-MacLane spectrum Hk) is equivalent to the homotopy category of E-infinity differential graded k-algebras. The functor from topology to algebra is a CW approximation and cellular chain functor; the inverse equivalence is constructed by Brown's representability theorem. ------------------ For those of you who are interested, I will be at Wesleyan again next year, in another one-year job. Efforts, on my part at least, to make this permanent continue. Also, I will no longer list papers cross-referenced to algebraic topology on the xxx archive. I will continue to list papers on xxx whose main classification is algebraic topology. If you disagree with this decision, please let me know. I will tell you that the paper by Looijenga on M_g as a union of g-1 affines has been withdrawn from xxx. Two new papers this time. Mark Hovey New papers uploaded to hopf between 4/29/98 and 5/8/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/DavisD/ee7 3-primary v1-periodic homotopy groups of E7 Donald M. Davis Abstract In this paper we compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. This represents the next stage in the author's goal of calculating the v1-periodic homotopy groups of all compact simple Lie groups (at least when localized at an odd prime). Most of the work goes into calculating the unstable Novikov spectral sequence of \Omega E7/Sp(2). Showing that this spectral sequence converges to the v1-periodic homotopy groups in this case utilizes recent results of Bousfield and Bendersky-Thompson. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/comparison `S-modules and symmetric spectra' by Stefan Schwede Department of Mathematics Massachusetts Insitute of Technology Cambridge, MA 02139 schwede---math.mit.edu Abstract: We study a symmetric monoidal adjoint functor pair between the category of S-modules of Elmendorff, Kriz, Mandell and May and Jeff Smith's category of symmetric spectra. The functors induce equivalences between the respective homotopy categories of spectra, module spectra and ring spectra. ----------------- Two new papers this time. Mark Hovey New papers uploaded to hopf between 5/8/98 and 5/12/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/model This is a revised version of my book, "Model categories". It is a 1-Meg download. You can download individual chapters, or you will be able to in about 2 weeks, from my home page (URL in instructions below). This is a major revision from the previous version, and is much better in my opinion. Mostly the exposition is better, but also some results are improved. The best new result is a better suffcient condition for smallness in the homotopy category. Recall the book is a thorough introduction to model categories, with detailed discussion of the major examples. The highlights include a proof that the homotopy category of an arbitrary model category is naturally a closed module over the homotopy category of simplicial sets, and sufficient conditions for the homotopy category of a model category to be a stable homotopy category in the sense of Hovey-Palmieri-Strickland. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lupton/Halpconj Title: Variations on a Conjecture of Halperin Author: Gregory Lupton AMS Classn.: 55P62 Address: Department of Mathematics Cleveland State University 2400 Euclid Avenue Cleveland OH 44115 U.S.A. E-mail address: Lupton---math.csuohio.edu Abstract: Halperin has conjectured that the Serre spectral sequence of any fibration that has fibre space a certain kind of elliptic space should collapse at the $E_2$-term. In this paper we obtain an equivalent phrasing of this conjecture, in terms of formality relations between base and total spaces in such a fibration (Theorem 3.4). Also, we obtain results on relations between various numerical invariants of the base, total and fibre spaces in these fibrations. Some of our results give weak versions of Halperin's conjecture (Remark 4.4 and Corollary 4.5). We go on to establish some of these weakened forms of the conjecture (Theorem 4.7). In the last section, we discuss extensions of our results and suggest some possibilities for future work. ---------------- Three new papers this time, two from hopf and one from xxx. Mark Hovey New papers uploaded to hopf and xxx between 5/12/98 and 5/15/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lydakis/s_functors Simplicial functors and stable homotopy theory by Manos Lydakis Fakultaet fuer Mathematik Universitaet Bielefeld 33615 Bielefeld Germany manos---math206.mathematik.uni-bielefeld.de We study a nice model for the smash product of spectra, the smash product of simplicial functors. We give a self-contained account of the required parts of stable homotopy and model categories. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scavenels/epiref TITLE: "Universal epimorphic equivalences for group localizations" AUTHORS: Jose L. Rodriguez Universitat Autonoma de Barcelona 08193 Bellaterra, Spain jlrodri---mat.uab.es http://mat.uab.es/jlrodri Dirk Scevenels Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B--3001 Heverlee, Belgium dirk.scevenels---wis.kuleuven.ac.be ABSTRACT: Recent work by Bousfield shows the existence, for any map $\phi$, of a universal space that is killed by homotopical $\phi$-localization. Nullification with respect to this so-called universal $\phi$-acyclic space is related to $\phi$-localization in the same way as Quillen's plus construction is related to homological localization. Here we construct a universal $f$-acyclic group for any group homomorphism $f$. Moreover, we prove that there is a universal epimorphism $E(f)$ that is inverted by $f$-localization. Although the kernel of the $E(f)$-localization homomorphism coincides with that of the $f$-localization homomorphism, we show that localization with respect to $E(f)$ has in general nicer properties than $f$-localization itself. 3. http://xxx.lanl.gov/dvi/math.AT/9805061 From: "Grigori L. Rybnikov" Date: Wed, 13 May 1998 11:34:38 GMT (12kb) Title: On the fundamental group and triple Massey's product Authors: Grigori Rybnikov Comments: 11 pages, Latex2e with AMSLaTeX 1.2, uses XY-pic package Subj-class: Algebraic Topology; Algebraic Geometry; Combinatorics \\ Let us say that a map of arcwise connected topological spaces (having the homotopy type of CW-complexes) is a pseudo-homeomorphism if it induces an isomorphism of the first integer homology groups and an epimorphism of the second integer homology groups. We prove that any invariant of a topological space w.r.t. pseudo-homeomorphisms is an invariant of the fundamental group of this space. We also describe a necessary condition for the fundamental groups to be distinguished by such invariants. As an example we show that the invariant used in math.AG/9805056 to distinguish the fundamental groups of combinatorially equivalent arrangements is, in fact, a form of triple Massey's product on the first integer homology group. \\ ( http://xxx.lanl.gov/abs/math/9805061 , 12kb) ----------------- Four new papers this time, including three revisions. Mark Hovey New papers uploaded to hopf between 5/15/98 and 5/21/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Morava-Sadofsky/AndoMorSad-revised (This is a revised version of a paper previously on the archive). Completions of Z/p-Tate cohomology of periodic spectra Matthew Ando, Jack Morava and Hal Sadofsky Classification numbers primary: 55N22, 55P60, secondary: 14L05 University of Virginia, Johns Hopkins University, University of Oregon ma2m---faraday.clas.Virginia.edu, jack---math.jhu.edu, sadofsky---math.uoregon.edu We construct splittings of some completions of the $\mathbf{Z}/ (p)$-Tate cohomology of $E (n)$ and some related spectra. In particular, we split (a completion of) $tE (n)$ as a (completion of) a wedge of $E (n-1)$'s as a spectrum, where $t$ is shorthand for the fixed points of the $\mathbf{Z}/ (p)$-Tate cohomology spectrum (i.e. the Mahowald inverse limit $\invlim{k}{(P_{-k} \wedge\Sigma E (n))}$). We also give a multiplicative splitting of $tE (n)$ after a suitable base extension. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lesh/uass-twostage (This is a renamed, and thus probably extensively revised, version of a paper previously on the archive) Title: The Unstable Adams Spectral Sequence for Two-Stage Towers Author: Kathryn Lesh AMS Classification: 55T15, 55Q52, 55R45 Address: Department of Mathematics University of Toledo Toledo, OH 43606 E-mail: klesh---uoft02.utoledo.edu Let KP denote a generalized mod 2 Eilenberg-MacLane space and let Y be the fiber of a map X -> KP to which the Massey-Peterson theorem applies. We study the relationship of the mod 2 unstable Adams spectral sequence (UASS) for X and for Y. Given conditions on X, we split the E_{2}-term for Y, and we use a primary level calculation to compute d_{2} for Y up to an error term. If the UASS for X collapses at E_{2} (for example, if X is an Eilenberg-MacLane space), the UASS for Y collapses at E_{3}, and we have the entire UASS for Y. We also give examples and address a conjecture of Bousfield on the UASS for the Lie group SO. To appear in: Topology and Its Applications 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mahowald-Rezk/rezk-mahowald-duality Brown-Comenetz duality and the Adams spectral sequence by Mark Mahowald and Charles Rezk (mark---math.nwu.edu, rezk---math.nwu.edu) May 15, 1998 We show that the class of $p$-complete connective spectra with finitely presented cohomology over the Steenrod algebra admits a duality theory related to Brown-Comenetz duality. This construction also produces a full-plane version of the classical Adams spectral sequence for such spectra, which converges to the homotopy groups of a ``finite'' localization. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sadofsky-Wilson/knhopf3-24-98 (This is a revised version of a paper previously on the archive) Title: Commutative Morava homology Hopf algebras Authors: Hal Sadofsky, W. Stephen Wilson Primary 16W30, 55N20 Addresses: University of Oregon, Johns Hopkins University Email: sadofsky---math.uoregon.edu, wsw---math.jhu.edu We give the Dieudonne module theory for Z/(2p^n-2)-graded bicommutative Hopf algebras over $\Fp $. These objects arise as the Morava $K$-theory of homotopy associative, homotopy commutative $H$-spaces. --------------- Its getting so I can't go out of town anymore! Twelve new papers on hopf, and one on xxx. Mark Hovey New papers uploaded to hopf and xxx between 5/21/98 and 6/17/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Abrams/abrams-cotensor Modules, comodules and cotensor products over Frobenius algebras Lowell Abrams 16D90; 16E30 Department of Mathematics, Hill Center, Rutgers University, New Brunswick, NJ 08903 labrams---math.rutgers.edu We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right comodules over A, relative to this coproduct, is isomorphic to the category of right modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular right A^e-module D to M \otimes N, viewed as a right A^e-module. Some of the properties of D are investigated, and some sample calculations are given. Finally, we show that when A is commutative or semisimple, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of M \otimes N. This paper has been submitted to the Journal of Algebra, and copyright may be transferred. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arone/SuspIterates Iterates of the suspension map and Mitchell's finite spectra with $A_k$-free cohomology Greg Arone arone---math.uchicago.edu AMS classification: 55P40, 55P42, 55P65 We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss in the context of ``Orthogonal calculus''. We show that Mithchell's finite spectra with $A_k$-free cohomology arise naturally as stabilizations of Weiss' cross-effects. Furthermore, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell's spectra. This last result is a partial generalization of a previous result of Mahowald and the author. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arone-Mahowald/ArMahowald The Goodwillie Tower of the identity functor and the unstable periodic homotopy of spheres AMS Classifiaction: 55P47, 55Q40, 55S12 Greg Arone arone---math.uchicago.edu Mark Mahowald mark---math.nwu.edu We investigate Goodwillie's ``Taylor tower'' of the identity functor from spaces to spaces. More specifically, we reformulate Johnson's description of the Goodwillie derivatives of the identity, and prove that when evaluated at an odd-dimensional sphere, the only layers in the tower that are not contractible are those indexed by a prime power. Furthermore, in the case of a sphere the tower is finite in $v_k$-pe- riodic homotopy. It has $k+1$ stages if the sphere is odd dimensional, and $2(k+1)$ stages if the sphere is even-dimensional. This is a revised version of a previously uploaded preprint. The paper has been accepted for publication, and is now in its final form. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/htbg ON THE HOMOTOPY TYPE OF BG FOR CERTAIN FINITE 2-GROUPS G Carles Broto and Ran Levi We consider the homotopy type of classifying spaces $BG$, where $G$ is a finite $p$-group and study the question, whether or not the mod $p$ cohomology of $BG$, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence determine the homotopy type of $BG$. This article is devoted to producing some families of finite 2-groups, where cohomological information determines the homotopy type of $BG$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/snk LOOP STRUCTURES ON HOMOTOPY FIBRES OF SELF MAPS OF A SPHERE By Carles Broto and Ran Levi Let $S^{2n-1}\{k\}$ denote the fibre of the degree $k$ map on the sphere $S^{2n-1}$. If $k=p^r$, where $p$ is an odd prime and $n$ divides $p-1$ then $S^{2n-1}\{k\}$ is known to be a loop space. It is also known that $S^3\{2^r\}$ is a loop space for $r\geq 3$. In this paper we study the possible loop structures on this family of spaces for all primes $p$. In particular we show that $S^3\{4\}$ is not a loop space. Our main result is that whenever $S^{2n-1}\{p^r\}$ i a loop space, the loop structure is unique up to homotopy. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ishiguro/toral Toral groups and classifying spaces of $p$--compact groups Kenshi Ishiguro (kenshi---ssat.fukuoka-u.ac.jp) Fukuoka University, Fukuoka 814-0180, Japan We show converses to some known results for the classifying spaces of $p$--toral groups or $p$--compact toral group. Suppose $G$ is a compact Lie group. The following results are included. (A) If there is a positive integer $k$ such that the $n$--th homotopy groups of $(BG)\p$ are zero for all $n \ge k$, then $(BG)\p$ is the classifying space of a $p$--compact toral group. (B) If the canonical map $Rep(G, K) --->>> [BG, BK]$ is bijective for any compact connected Lie group $K$, then $G$ is a $p$--toral group. We will also discuss the conditions of a compact Lie group that its loop space of the $p$--completed classifying space be a $p$--compact group. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/athom The Inverse Invariant Theory Problem and Steenrod Operations Mara D Neusel AMS Classification: 55S10 Steenrod Algebra, 13A50 Invariant Theory, 55XX Algebraic Topology AG Invariantentheorie, Germany University of Minnesota, School of Mathematics, 127 Vincent Hall, 206 Church Street S.E., Minneapolis MN 55455, USA mdn---sunrise.uni-math.gwdg.de maramara---steenrod.mast.queensu.ca neusel---math.umn.edu This is a pure postscript file. This paper is devoted to the study of inverse invariant theory and its relationship with the $\steenrod$--invariant prime spectrum of an unstable algebra over the Steenrod algebra. We will show that this spectrum is a chain saturated poset. Moreover we will prove the existence of Thom classes, detect a fractal of the Dickson algebra in any unstable algebra and give a counterexample to the Reverse Landweber--Stong Conjecture. Along the way to these results we will generalize the famous Adams--Wilkerson theorems to arbitrary Galois fields, have a closer look at fields and their extensions over the Steenrod algebra, and generalize some results about the unstable part of a module over the Steenrod algebra. [ARCHIVE NOTE: available in .ps.gz and .pdf formats only due to use of custom postscript fonts. I don't advise this format because it limits the onscreen viewing and search possibilities. ] 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/AnalyticApps (No abstract on archive. This paper is an updated version of a paper previously on the archive). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/ArnoldConj (This is also an update of a previously announced paper). ON STRICT CATEGORY WEIGHT AND THE ARNOLD CONJECTURE Yuli B. Rudyak Rudyak and Oprea proved the Arnold conjecture for symplectic manifolds $(M,\omega)$ with $\pi_2(M)=0$. The proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/CategoryWeight (And so is this one) CATEGORY WEIGHT: NEW IDEAS CONCERNING LUSTERNIK--SCHNIRELMANN CATEGORY Yuli B. Rudyak The concept of category weight was introduced by Fadell--Husseini and developed by Rudyak and Strom. Here we give a survey, some further development and applications of category weight. 11. http://xxx.lanl.gov/dvi/math.AT/9806021 From: Peter Saveliev Date: Thu, 4 Jun 1998 20:37:08 GMT (20kb) Title: A Lefschetz type coincidence theorem Authors: Peter Saveliev Comments: 20 pages Subj-class: Algebraic Topology MSC-class: 55M20, 55H25 \\ A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x). In particular, the theorem contains some well-known coincidence results for (i) X,Y manifolds and (ii) f with acyclic fibers. \\ ( http://xxx.lanl.gov/abs/math/9806021 , 20kb) 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Turner/looking-glass Simplicial Commutative F_p-Algebras Through the Looking-Glass of F_p-Local Spaces 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secondary: 55P60, 55P99, 55S05 James M. Turner 1395 Mathematical Sciences Building Purdue University West Lafayette, IN 47907-1395 jmt---ziplink.net Submitted to the Proceedings in Honor of Michael J. Boardman We propose a dictionary approach to studying the homotopy theory of simplicial augmented commutative F_p-algebras using the homotopy theory of connected F_p-local spaces as our guide. We indicate how standard topological tools translate to the setting of simplicial algebras. We further indicate how theorems translate as well. For example, we recall a theorem of P. Goerss giving an algebraic version of the Hilton-Milnor theorem which fits in our framework. We next propose how a theorem of J.-P. Serre on F_p-local spaces with bounded homotopy groups translates into our algebraic setting and relate it to a conjecture of D. Quillen on the vanishing of Andr\'e-Quillen homology. We also describe what a simplicial algebra version of a theorem of D. Kan and W. Thurston should look like. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Turner/vanishingnew On Simplicial Commutative Algebras with Vanishing Andr\'e-Quillen Homology 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secondary: 13D40 James M. Turner 1395 Mathematical Sciences Building Purdue University West Lafayette, IN 47907-1395 jmt---ziplink.net October 1997; Revised: June 12, 1998 In this paper, we study the Andr\'e-Quillen homology of simplicial commutative F-algebras, where F is a field of positive characteristic, with certain vanishing properties. We will show, under certain conditions on $\pi_0$ and the vanishing of the homotopy groups, that the vanishing of Andr\'e-Quillen homology implies that the simplicial commutative F-algebra in question is a homology complete intersection. As a consequence, we resolve a conjecture of D. Quillen in the case of commutative Noetherian F-algebras. --------------- We have two new papers this time and one final version. There is also a new picture, taken at the Adams Symposium in 1990, at http://hopf.math.purdue.edu/pub/new-html/contribpics.html There is a link to this page from the main hopf page. Mark Hovey New papers uploaded to hopf between 6/17/98 and 6/29/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arone-Mahowald/ArMahowald The Goodwillie Tower of the identity functor and the unstable periodic homotopy of spheres AMS Classification: 55P47, 55Q40, 55S12 Greg Arone arone---math.uchicago.edu Mark Mahowald mark---math.nwu.edu We investigate Goodwillie's ``Taylor tower'' of the identity functor from spaces to spaces. More specifically, we reformulate Johnson's description of the Goodwillie derivatives of the identity, and prove that when evaluated at an odd-dimensional sphere, the only layers in the tower that are not contractible are those indexed by a prime power. Furthermore, in the case of a sphere the tower is finite in $v_k$-pe- riodic homotopy. It has $k+1$ stages if the sphere is odd dimensional, and $2(k+1)$ stages if the sphere is even-dimensional. This is a revised version of a previously uploaded preprint. The paper has been accepted for publication, and is now in its final form. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moller/toricrep Title: Toric representations of p-compact groups. Author: J.M. M{\o}ller Department of Mathematics University of Copenhagen DK-2100 Copenhagen Denmark AMS Classification numbers: 55R35, 55S37 Address of author: Department of Mathematics University of Copenhagen DK-2100 Copenhagen Denmark e-mail: moller---math.ku.dk Abstract: We compute the mapping spaces map(X,Y) where (X,Y)=(BSU(3),BF_4), (BG_2,BF_4), and (BSU(3),BG_2). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/mugcomps Computations in Complex Equivariant Bordism Theory by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps---math.brown.edu In this paper we present computations of the ring structure of the coefficients of equivariant bordism, answering questions which have been open since these theories were first defined by Conner and Floyd and tom Dieck. We have a result which establishes an algebraic framework in which to understand equivariant bordism for any group such that any proper subgroup is contained in a proper normal subgroup. This class of groups includes abelain groups and $p$-groups. Our general result is computationally satisfying when one can find a suitable representation of $MU^G_*$. For abelian groups the map to completion at the augmentation ideal seems to be such a representation, so we make explicit computations of that map. We give applications to the geometry of lens spaces and $S^1$ actions on stably complex four-manifolds. -------------- Both Clarence and I have been out of town, consecutively, with the result that there are 13 papers to announce this time. The two papers by Jack Morava below are "postfinal" versions--each has an appendix not present in the published version! So Jack is taking Tibor Beke's suggestion, for which I confess a sentimental attraction myself, to heart. Jack also wants me to let you know that he did not intend to imply that one should not replace preliminary postings on Hopf with final versions, merely that xxx may not be the place for preliminary versions. Mark Hovey New papers uploaded to hopf between 6/29/98 and 7/17/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Hopkins-Strickland/eswgtc Elliptic spectra, the Witten genus, and the theorem of the cube. M. J. Hopkins, M. Ando, and N. P. Strickland MIT mjh---math.mit.edu University of Virginia and Johns Hopkins University ando---math.jhu.edu Trinity College, Cambridge n.strickland---dpmms.cam.ac.uk We show that every elliptic spectrum receives a natural MU<6>-orientation. For the elliptic spectrum defined by the Tate curve, this orientation specializes to the Witten genus. The naturality of the orientation implies that the modularity of the Witten genus for MU<6>-manifolds. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Strickland/pairings Weil pairings and Morava K-theory M. Ando and N. P. Strickland University of Virginia and Johns Hopkins University ando---math.jhu.edu Trinity College, Cambridge n.strickland---dpmms.cam.ac.uk An important component of joint work with M. Hopkins (Elliptic spectra, the Witten genus, and the theorem of the cube) is that the complex-orientable cohomology of BU<6> represents the group of "cubical structures on the trivial torsor over the formal group". We give a proof of this result for Morava K-theories which demonstrates the close relationship of the topological situation to the algebro-geometric situation in which the notion of cubical structure originally arose. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Casacuberta-Rodriguez-Tai/rigid TITLE: "Localizations of abelian Eilenberg--Mac Lane spaces of finite type" AUTHORS: Carles Casacuberta Universitat Autonoma de Barcelona 08193 Bellaterra, Spain casac---mat.uab.es http://mat.uab.es/casac Jose L. Rodriguez Universitat Autonoma de Barcelona 08193 Bellaterra, Spain jlrodri---mat.uab.es http://mat.uab.es/jlrodri Jin-Yen Tai Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, Jin-Yen.Tai---Dartmouth.edu ABSTRACT: Using recent techniques of unstable localization, we extend earlier results on homological localizations of Eilenberg--Mac Lane spaces, and show that several deep properties of such localizations can be explained by the preservation of certain algebraic structures under the effect of idempotent functors. We study localizations $L_f K(G,n)$ of Eilenberg--Mac Lane spaces with respect to any map $f$, where $n\ge 1$ and $G$ is abelian. We find that, if $G$ is finitely generated, then the result is a $K(A,n)$, where $A$ can be computed using cohomological data derived from $f$. If $G=\Z$, then $A$ is a commutative ring which is isomorphic to the ring $\End(A)$ of its own additive endomorphisms; such rings, which we call rigid, form a proper class which contains the set of solid rings. From this fact it follows that there is a proper class of distinct homotopical localizations of the circle $S^1$. Among other applications of our results, we show that, if $X$ is a product of abelian Eilenberg--Mac Lane spaces and $f$ is any map, then the homotopy groups $\pi_m(L_f X)$ become modules over the ring $\pi_1(L_f S^1)$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Devadoss/mosaic Tessellations of Moduli Spaces and the Mosaic Operad Satyan L. Devadoss Primary: 14H10 Secondary: 05B45, 52B11 Department of Mathematics Johns Hopkins University Baltimore, MD 21218 devadoss---math.jhu.edu The following are all EPS files: assoc bcollide blow6 braid6 btp cubes hyperbolic k2tok4 k5codim1 k6codim1 kapdc m04 m05c m05d m05pieces m06c onepoly pairpants pcollide polycomp simpose twist twistpf Abstract: We construct a new (cyclic) operad of \emph{mosaics} defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the sets \overline{\mathcal {M}}^n_0({\mathbb R}) of real points of the moduli space of punctured stable curves of genus zero, which are naturally tiled by Stasheff associahedra. We (combinatorially) describe them as iterated blow-ups and show that their fundamental groups form an operad with similarities to the operad of braid groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gilkey-Leahy-Sadofsky/GLSeigen Title: Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues Authors: Peter B. Gilkey, John V. Leahy, Hal Sadofsky AMS classification: 53B20 Address: Department of Mathematics, University of Oregon, Eugene, OR 97403. Email: gilkey---math.uoregon.edu, leahy---math.uoregon.edu, sadofsky---math.uoregon.edu Abstract: A Riemannian metric on a manifold is said to be IP if the eigenvalues of the skew-symmetric curvature operator are pointwise constant, i.e. they depend upon the point of the manifold but not upon the particular $2$ plane in the tangent bundle at that point. We classify the IP metrics for manifolds of dimensions $m=5$, $m=6$, and $m>8$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hirschowitz-Simpson/descente Title: Descente pour les $n$-champs (Descent for $n$-stacks) Authors: Andr\'e Hirschowitz, Carlos Simpson Authors' addresses: Universit\'e de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 2, France; Laboratoire Emile Picard, Universit\'e Toulouse 3, 31062 Toulouse cedex, France Authors' email addresses: ah---math.unice.fr; carlos---picard.ups-tlse.fr Subj-class: Algebraic Geometry; Algebraic Topology; Category Theory Abstract: We develop the theory of $n$-stacks (or more generally Segal $n$-stacks which are $\infty$-stacks such that the morphisms are invertible above degree $n$). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of $n$-stacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of $n$-stacks in terms of ``effectivity of descent data''; construction of the stack associated to an $n$-prestack; a strictification result saying that any ``weak'' $n$-stack is equivalent to a (strict) $n$-stack; and a descent result saying that the $(n+1)$-prestack of $n$-stacks (on a site) is an $(n+1)$-stack. As for other examples, we start from a ``left Quillen presheaf'' of cmc's and introduce the associated Segal $1$-prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of $\Oo$-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Stabilization of Model Categories by Mark Hovey Wesleyan University hovey---member.ams.org Suppose C is a (nice enough) model category, and G: C --> C is a left Quillen endofunctor of C. Think of C as the category of pointed topological spaces, and G as the suspension. Then we construct a new model category Sp(C,G), an embedding C --> Sp(C,G), and an extension of G to a Quillen EQUIVALENCE of Sp(C,G). Essentially, we have inverted the functor G, up to homotopy. When C is the category of pointed topological spaces and G is the suspension, we recover the Bousfield-Friedlander model category of spectra. The trouble with the Bousfield-Friedlander model category is that it is not symmetric monoidal, and we have the same problem with Sp(C,G). But there is also the same fix. Suppose C is a (nice enough) symmetric monoidal model category, K is a cofibrant object of C, and D is a (nice enough) C-model category. Think of C as pointed simplicial sets, D as a pointed simplicial model category, and K as the simplicial circle. Then we construct a model category Sp^Sigma(D,K), so that Sp^Sigma(C,K) is a symmetric monoidal model category, Sp^Sigma(D,K) is a Sp^Sigma(C,K)-model category, and smashing with K is a Quillen equivalence on Sp^Sigma(D,K). When C is pointed simplicial sets, and K is S^1, we get the symmetric spectra of Hovey-Shipley-Smith. The method used is the Bousfield localization technology of Hirschhorn, so the words "nice enough" mean "left proper cellular", though occasionally we also need to assume to domains of the generating cofibrations are cofibrant. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jardine/sym Title of paper: Presheaves of symmetric spectra Author: J.F. Jardine AMS Classification numbers: 55P42 18F20 55U35 Address of Author: Mathematics Department University of Western Ontario London, Ontario N6A 5B7 Canada Email: jardine---uwo.ca This paper shows that there is a proper closed simplicial model category on the category of presheaves of symmetric spectra on an arbitrary Grothendieck site, and that the resulting homotopy category is equivalent to the stable category of presheaves of spectra. The argument follows the outline established by Hovey, Shipley and Smith, while many of the techniques of proof originate in the Goerss-Jardine paper "Localization theories for simplicial presheaves". This paper was written in lamstex and requires the lamstex fonts to view or print. A postscript version is available at http://www.math.uwo.ca/~jardine/papers/ 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kuhn/kuhnloops "New relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces" Nicholas J. Kuhn AMS classification number: 55P42 Mathematics Department University of Virginia Charlottesville, VA 22903 njk4x---virginia.edu This is a revised version of the 1996 preprint "New cohomological relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces". The paper studies a bigraded family of finite spectra T(n,j), at p=2, which specialize to the dual Brown-Gitler spectra when n=1. One can take hocolimits of these as either j goes to infinity or n goes to infinity. When one lets j go to infinity, one gets in cohomology A-modules, which are shown to be related to the cohomology of K(V,n)'s in the same way that the Carlsson modules are related to the cohomology of K(V,1)'s. When one lets n go to infinity, one gets a filtration of HZ/2 that cohomologically looks like the mod 2 Whitehead conjecture filtration (a modified symmetric products of spheres filtration). A result new in the revision is that this IS the modified symmetric products of spheres filtration. Also new in the revision is an appendix which relates my constructions to work of Arone-Mahowald, and Arone-Dwyer on the Goodwillie tower of spheres. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Luminy6-final \begin{center}{Abstract for {\bf Quantum generalized cohomology}}\bigskip \end{center} \noindent There is a variant of Segal's category of Riemann surfaces, in which morphisms are stable complex algebraic curves [i.e. double points are allowed], with some smooth points marked; composition is defined by glueing at marked points. The spaces of morphisms in this category are built from the compactified moduli spaces $\overline M_{g,n}$ of Deligne, Mumford, and Knudesen; here $g$ is the genus and $n$ is the number of marked points. A generalized topological field theory taking values in the category of module-spectra over a ring-spectrum $\bf R$ is a family $$\tau_{g,n} : \overline M_{g,n} \rightarrow {\bf M} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf M} = {\bf M}^{\wedge n}$$ of maps, which respect composition of morphisms. More precisely, $\bf M$ is an $\bf R$-module spectrum, $\wedge_{\bf R}$ is the Robinson smash product, and $\bf M$ is endowed with a suitably nondegenerate bilinear form $${\bf M} \wedge_{\bf R} {\bf M} \rightarrow {\bf R}.$$ This data entails the existence of an $\bf R$-algebra structure on $\bf M$, such that $\tau_{g,1}$ is a morphism of monoids if the moduli space of curves is given the pair-of-pants product; it seems to define a natural context for quantum generalized cohomology.\medskip \noindent There is an interesting example of all this, associated to a smooth algebraic variety $V$. It is closely related to the Tate $\bf MU$-cohomology of the universal cover of the free loopspace of $V$, but it can be described more concretely in terms of the rational Novikov ring $\Lambda = {\Bbb Q} [H_{2}(V,{\Bbb Z})]$ of $V$ by setting ${\bf R} = {\bf MU} \otimes \Lambda$; then {\bf E} is the function spectrum $F(V,{\bf R})$ representing the cobordism of $V$ tensored with $\Lambda$, and the bilinear pairing is defined by Poincar\'e duality. In this case $\tau_{g,n}$ represents the cobordism class of the space of stable maps [in the sense of Kontsevich] from a curve of genus $g$, marked with $n$ ordered smooth points together with an indeterminate number of unordered smooth points, to $V$. A variant construction requires the unordered points to lie on a cycle $z$ in $V$; this defines a parameterized family of multiplications satisfying the analogue of the WDVV equation. When $V$ is a point, the resulting theory boils down to the version of topological gravity I advertised at the Adams Symposium; the coupling constant of the associated topological field theory is the cobordism analogue of Manin's exponential $$\sum_{n \geq 0} \overline M_{0,n+3} \frac {z^{n}}{n!} .$$ Although much of the machinery used here comes from fields adjacent to topology, this paper is concerned with the old problem of constructing complex cobordism out of Riemann surfaces by some analogue of the plus-construction. Having hacked through the physics background, I hope to produce a more topological account in the near future. \medskip \noindent This is to appear in Contemporary Math., in the Proceedings of the Hartford/Luminy Conference on the Renaissance of Operads, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov. \end{document} [ Jack tells me that this is the final version. I've labeled the DVI file as Luniny6-final.dvi and reclused the original version, CWW 7/13/98] 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Schur2-final Schur cohomology and a Kontsevich-Witten genus Jack Morava AMS Classification: Primary 14H10, Secondary 55N35, 81R10 Johns Hopkins University: jack---math.jhu.edu ABSTRACT: Two-dimensional topological gravity is a kind of physicist's interpretation of the rational cohomology of the group completion of the monoid of Riemann surfaces under glueing. It has a natural algebra of operations, which look vaguely like the operations in complex cobordism, and Witten has raised the question of their possible homotopy-theoretic interpretation. Over the integers this theory turns out to have an interesting model, which looks a lot like (a double of) the cohomology of Sp/U. There is an associated formal-group-like object, which looks unfamiliar because its coordinate seems to be centered at infinity, corresponding to asymptotic expansions of interest in physics. [This paper is a kind of sequel to 'Generalized quantum cohomology' posted previously on {\bf Hopf}, which has since appeared [in Contemporary Math. 202, Proceedings of the operads renaissance conference, ed. Loday, Stasheff, & Voronov] [ Jack tells me that this is the final version. I've labeled the DVI file as Schur2-final.dvi and reclused the original version, CWW 7/13/98] 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/ls1 Title:``Spaces with Lusternik-Schnirelmann category n and cone length n+1'' Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley---math.fu-berlin.de Abstract: We construct a series of spaces, $X(n)$, for each $n>0$, such that $cat(X(n))=n$ and $cl(X(n))=n+1$. We show that the Hopf invariants determine whether the category of a space goes up when attaching a cell of top dimension. We give a new proof of counterexamples to Ganea's conjecture. Also we introduce some techniques for manipulating cone decompositions. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Voronov/swiss/voronov-swiss The Swiss-Cheese Operad Alexander A. Voronov AMS Classification: Primary 55P99, 18C99; Secondary 14H10, 17A30, 17A42, 81T40 Department of Mathematics M.I.T., 2-246 77 Massachusetts Ave. Cambridge, MA 02139-4307 Email address: voronov---math.mit.edu Included EPS or PS files: disks.eps and semidisks.eps Abstract. We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the open-closed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad. ---------------- Oops! I forgot an algebraic topology paper that appeared on xxx. (Some other algebraic topology papers also appeared, but these have been annouced on Hopf). Mark Hovey New algebraic topology papers uploaded to xxx between 6/29/98 and 7/17/98: 1. http://xxx.lanl.gov/dvi/math.AT/9807053 Paper: math.AT/9807053 From: Martin Guest Date: Sat, 11 Jul 1998 08:34:54 GMT (21kb) Title: Spaces of polynomials with roots of bounded multiplicity Authors: M. A. Guest, A. Kozlowski, K. Yamaguchi Comments: 29 pages, AMS-TeX Subj-class: Algebraic Topology MSC-class: 55P35; 58D15; 57R45 \\ We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right. \\ ( http://xxx.lanl.gov/abs/math/9807053 , 21kb) --------------- There are three new papers on Hopf this time. I have decided that I do not want to compete with xxx's excellent announcement service. I will therefore remind you how to subscribe to xxx now and, permanently, in the instructions at the end, but will no longer announce new preprints on xxx. To subscribe to xxx, send e-mail to math---xxx.lanl.gov, with subject "subscribe" (not in quotes). The body of the message should consist of the line "add AT" (without quotes). You will then be subscribed to the Algebraic Topology announcement service. A similar message works for other categories on xxx--e.g. "add GT" will give you geometric topology. Mark Hovey New papers uploaded to hopf between 7/17/98 and 8/19/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Goerss/hopfring Title: Hopf rings, Dieudonn\'e modules, and $E_\ast \Omega^2S^3$. Author: Paul Goerss AMS Classification Nos: 55205, 55N20, 57T05, 16W30 Department of Mathematics Northwestern University Evanston IL 60208 pgoerss---math.washington.edu (until 8/24/98) pogerss---math.nwu.edu Abstract: Hopf algebras over the prime field with $p$ elements is an abelian category which is equivalent, by work of Schoeller, to a category of graded modules, known as Dieudonn\'e modules. Graded ring objects in Hopf algebras are called Hopf rings, and they arise in the study of unstable cohomology operations for extraordinary cohomology theories. The central point of this paper is that Hopf rings can be studied by looking at the associated ring object in Dieudonn\'e modules. They can also be computed there, and because of the relationship between Brown-Gitler spectra and Dieudonn\'e modules, calculating the Hopf ring for a homology theory $E_\ast$ comes down to computing $E_\ast\Omega^2S^3$ -- which Ravenel has done for $E = BP$. The are two major algebraic difficulties encountered in this approach. The first is to decide what a ring object is in the category of Dieudonn\'e modules, as there is no obvious symmetric monoidal pairing associated to a tensor product of modules. The second is to show that Hopf rings pass to rings in Dieudonn\'e modules. This involves studying universal examples, and here we pick up an idea suggested by Bousfield: torsion-free Hopf algebras over the $p$-adic integers with some additional structure, such as a self-Hopf-algebra map that reduces to the Frobenius, can be easily classified. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hopkins-Mahowald/eo2homotopy title: From elliptic curves to homotopy theory authors: Mike Hopkins Mark Mahowald Department of Mathematics Department of Mathematics MIT Northwestern University Cambridge MA 02139 Evanston IL 60208 AMS class: 55P42,55P60,55N20,55N22,55Q45 Addresses: Department of Mathematics Department of Mathematics MIT Northwestern University Cambridge MA 02139 Evanston IL 60208 Email: mjh------math.mit.edu} mahowald---math.nwu.edu Include: eo2homotopy.eps Abstract: A surprising connection between elliptic curves over finite fields and homotopy theory has been discovered by Hopkins. In this note we will follow this development for the prime 2 and discuss the homotopy which developed from this. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/mugcomps Computations in Complex Equivariant Bordism Theory by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps---math.brown.edu In this paper we present computations of the ring structure of the coefficients of equivariant bordism, answering questions which have been open since these theories were first defined by Conner and Floyd and tom Dieck. We have a result which establishes an algebraic framework in which to understand equivariant bordism for any group such that any proper subgroup is contained in a proper normal subgroup. This class of groups includes abelian groups and $p$-groups. Our general result is computationally satisfying when one can find a suitable representation of $MU^G_*$. For abelian groups the map to completion at the augmentation ideal seems to be such a representation, so we make explicit computations of that map. We give applications to the geometry of lens spaces and $S^1$ actions on stably complex four-manifolds. **This paper is a revised version of a previous submission. The biggest **change is the explicit naming of ring generators of $MU^G_*$. ----------------- We have 9 new papers this time. Mark Hovey New papers uploaded to hopf between 8/19/98 and 9/26/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem/amsbetti Buildings, Group extensions and the Cohomology of Congruence Subgroups A.Adem (U.Wisconsin) We use methods from group cohomology and buildings to estimate betti numbers for congruence subgroups in SL_3(Z) and Sp_4(Z). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Karagueuzian/cmhess Essential Cohomology of Finite Groups A.Adem & D.Karagueuzian (U.Wisconsin) Abstract: We prove that a finite group G has Cohen--Macaulay, undetectable mod p cohomology if and only if G is a p-group such that all its elements of order p are central. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Karagueuzian-Milgram-Umla nd/newly8 The Cohomology of the Lyons Group and Double Covers of the Alternating Groups A.Adem (U.Wisconsin), D.Karagueuzian (U.Wisconsin), R.J. Milgram (Stanford U.), K. Umland (U. New Mexico). --We compute the mod 2 cohomology of Lyons' sporadic simple group as well as that of the double covers of A_8, S_8 and A_10. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Karagueuzian-Minac/fire2 On the Cohomology of Galois Groups Determined by Witt Rings A.Adem (U.Wisconsin), D. Karagueuzian (U.Wisconsin), J.Minac (U.Western Ontario) Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group G_F (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that its mod 2 cohomology contains the mod 2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space X_F endowed with an action of an elementary abelian group E such that the computation of the cohomology of G_F reduces to calculating the E-equivariant cohomology of the space. For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Yalcin/gpext4 On Some Examples of Group Actions and Group Extensions A.Adem (U.Wisconsin) and E.Yalcin (Indiana U.) --We consider the problem of which 2-groups can acts freely on a product of equidimensional spheres and show that it relates to questions about group extensions. We apply some rather unexpected examples from group theory to do this. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/athom The Inverse Invariant Theory Problem and Steenrod Operations Mara D Neusel AMS Classification: 55S10 Steenrod Algebra, 13A50 Invariant Theory, 55XX Algebra ic Topology AG Invariantentheorie mdn---sunrise.uni-math.gwdg.de This is a pure postscript file. This is a (heavily) revised version of the paper with the same titel put on Hopf in June. I have corrected some blunders and at least 1001 typos, added more examples and rewritten big parts of the story. I hope it is now more readable. Here the original abstract once more: This paper is devoted to the study of inverse invariant theory and its relationship with the $\steenrod$--invariant prime spectrum of an unstable algebra over the Steenrod algebra. We will show that this spectrum is a chain saturated poset. Moreover we will prove the existence of Thom classes, detect a fractal of the Dickson algebra in any unstable algebra and give a counterexample to the Reverse Landweber--Stong Conjecture. Along the way to these results we will generalize the famous Adams--Wilkerson theorems to arbitrary Galois fields, have a closer look at fields and their extensions over the Steenrod algebra, and generalize some results about the unstable part of a module over the Steenrod algebra. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pakianathan/exponent Title: Exponents and the Cohomology of Finite Groups. Author: Jonathan Pakianathan AMS Classification: Primary 20J06, 17B50, 17B56 Address of Author: Department of Mathematics, University of Wisconsin, Madison, WI 53706. Email: pakianat---math.wisc.edu Status: Reprint. To appear in "The Proceedings of the A.M.S.". This paper provides an example of a p-group G which has elements of order p^3 in some of its integral cohomology groups but which also has the property that p^2 annihilates H^i(G;Z) for all sufficiently high i. This provides a counterexample to a conjecture of A. Adem which stated that if a finite group K has an element of order p^n in one of its integral cohomology groups then it has such an element in infinitely many of its cohomology groups. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Scheerer-DStanley/don Title:``On the rational LS-category of a cartesian product of maps'' Authors: Hans Scheerer and Don Stanley AMS-classification number: 55P50 Address: Hans Scheerer Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: scheerer---math.fu-berlin.de stanley---math.fu-berlin.de Abstract: We give an example of a rational map, $f$, such that $cat f=cat f\times id_{S^3}=2. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/DStanley/ls4 Title:``On the Lusternik-Schnirelmann category of maps'' Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley---math.fu-berlin.de Abstract: We give conditions when $cat(f \times g) 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pengelley-Wiliams/newlowerops SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS FOR MOD 2 HOMOLOGY AND COHOMOLOGY DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof al- gebra R have both striking similarities and differences, arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite dif- ferent from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K* has a particularly attractive and potentially useful structure, providing a bridge between those of A* and R*, and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras. New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp---nmsu.edu E-mail address: frank---nmsu.edu 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/smith.invar.biblio This is an invariant theory bibliography compiled by Larry Smith. Clarence could not get this to run through TeX, so there is no .dvi version. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WuJ/p_local_minimal Note on the Minimal Simplicial Set Models for $p$-local $H$-spaces Jie Wu Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 USA jiewu---math.upenn.edu Included file: p_local_minimal.dvi In this note, we show that there exists a commutative (but non-associative in general) multiplication on the minimal simplicial set model of any path-connected $p$-local $H$-space with $p>2$. This result is the homotopy theory analogy of the Jordan algebra in the sense that one can make a new strictly commutative multiplication from an old non-commutative multiplication when the power map $2\colon X\rTo X$ is a homotopy equivalence. We need to point out that by using the results in our previous paper "On products on minimal simplicial sets", any connected (non-associative) minimal simplicial $H$-set $X$ is nilpotent with respect to associativity in the sense that for each $X_n$ the higher associators with length sufficiently large are trivial. More precisely, the nilpotency degree (with respect to the associator length) of the $(n+1)$-st Postnikov section of $X$ is at most one bigger that the nilpotency degree of the $n$-th Postnikov section of $X$. ---------------- Seven new papers this time. I made a typo last time: it was Pengelley-Williams, not Pengelley-Wiliams. Mark Hovey New papers uploaded to hopf between 10/26/98 and 11/7/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Scheerer/Beralgebra (No title or author list in this abstract) Let the space X be a 1-connected cogroup. If R is a ring, then a cohomology (flat) product H^{p+1}(X;R) X H^{q+1}(X;R) --->H^{p+q+1}(X;R) was defined by Arkowitz. If we set A^p(X;R)=H^{p+1}(X;R) for p>0 and A^0(X;R)=R,then A*(X;R) is a graded algebra. Berstein has defined a coalgebra B_*(X;K) and dual algebra B*(X;K) when X is a cogroup and K is a field. Our main result is that A*(X;K) and B*(X;K) are isomorphic algebras if X has finite type over K. It follows that the conilpotency class of X is bounded below by the length of the longest product in the algebras B*(X;K). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/all-or-nothing Phantom maps and chromatic phantom maps J. Daniel Christensen and Mark Hovey jdc---math.jhu.edu and hovey---member.ams.org Keywords: Phantom map, chromatic phantom map, n-phantom map, cohomotopy, stable homotopy, spectrum, n-finite type. Abstract: In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V smash X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor W_{n-1} defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions under which the natural map Y --> W_{n-1}^2 Y is an isomorphism. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/wyoming TITEL: Localizations over the Steenrod Algebra. The lost Chapter AUTHOR: Mara D. Neusel EMAIL: mdn---sunrise.uni-math.gwdg.de mneusel---cfgauss.uni-math.gwdg.de neusel---math.umn.edu maramara---steenrod.mast.queensu.ca AMS CODE: 55S10 Steenrod Algebra, 13BXX Ring Extensions and Related Topics, 55XX Algebraic Topology, 13XX Commutative Rings and Algebras KEY WORDS: Steenrod Algebra, Unstable Algebras over the Steenrod Algebras, Unstable Part, Localizations, Noetherianess, Integral Closure, Dickson Algebra ABSTRACT: Let H be an unstable algebra over the Steenrod algebra, and let S\subset \H be a multiplicatively closed subset. The localization at S, i.e. S^{-1}H, inherits an action of the Steenrod algebra from H, which is, however, in general no longer unstable. In this note we consider the following three statements. (1) H is Noetherian, (2) the integral closure, \overline{H_{S^{-1}H}}, of H in the localization with respect to S is Noetherian, (3) \overline{H_{S^{-1}H}}= Un(S^{-1}H). where Un(-) denotes the unstable part. If the set S contains only (nonzero) non zero divisors and the algebras are reduced then (1) is equivalent to (2). If S contains zero divisors, then only (1) \Rightarrow (2) remains true, to show the converse is false we construct a counter example. The implication (2) \Rightarrow (3) is always true, while its converse (3) \Rightarrow (2) needs a weird bunch of technical assumptions to remain true. However, none of them can be removed: we illustrate this also with examples. Finally, as a technical tool, we characterize Delta-finite algebras. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-ho-models "A model for the homotopy theory of homotopy theory" Charles Rezk (Primary 55U35; Secondary 18G30) Department of Mathematics Northwestern University Evanston, IL 60208 rezk---math.nwu.edu November 3, 1998 We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-sharp-maps "Fibrations and homotopy colimits of simplicial sheaves" Charles Rezk (Primary 18G30; Secondary 18B25, 55R99) Department of Mathematics Northwestern University Evanston, IL 60208 rezk---math.nwu.edu November 3, 1998 We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scevenels/homology TITLE: "Homology equivalences inducing an epimorphism on the fundamental group" AUTHORS: Jose L. Rodriguez Universitat Autonoma de Barcelona E--08193 Bellaterra, Spain jlrodri---mat.uab.es http://mat.uab.es/jlrodri Dirk Scevenels Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B--3001 Heverlee, Belgium dirk.scevenels---wis.kuleuven.ac.be ABSTRACT: Quillen's plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strom/EPhant Essential Category Weight and Phantom Maps Jeffrey A. Strom Wayne State University strom---math.wayne.edu This purpose of this paper is to study the relationship between maps with infinite essential category weight and phantom maps. The essential category weight of a map f: X --> Y is the least N such that fg is nullhomotopic whenever g: Z --> X is a map with cat(Z) < N + 1. We write E(f) > N - 1 in this case. A map has infinite essential category weight ( E(f) = \infty ) if there is no such N. The appendix to this paper contains a brief summary of the main results on essential category weight. It is not hard to see that any map with E(f) = \infty is a phantom map. We give examples to show that the reverse is not always true: there are phantom maps f with E(f) = 1. We also show that in some cases, all the phantom maps f: X --> Y have E(f) = \infty. We are able to adapt many of the results of the theory of phantom maps to give us results about maps with E(f) = \infty. Finally, we use the connections between essential category weight and phantom maps to answer a question (asked by McGibbon) about phantom maps, ----------------- The math department Unix system at Wesleyan was hacked into and had to be totally reinstalled. My web pages are thus down, and may well continue to be down for another week or so. In any case, I anticipate the URL of my home page will change, hopefully to something more reasonable. Nine new papers this time. Mark Hovey New papers uploaded to hopf between 11/7/98 and 12/3/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Karoubi/AK C.R. Acad. Sci. Paris, t. 326 ,Série I, p. 13­17 (1998) Algčbre/Algebra (Topologie algébrique/Algebraic Topology) PERIODIC CYCLIC COHOMOLOGY OF GROUP RINGS Alejandro ADEM et Max KAROUBI A.A. : Mathematics Department, University of Wisconsin, Van Vleck Hall Madison, WI 53706, ETATS­UNIS ; e.mail : adem---math.wisc.edu M. K. : Université Paris 7 ­ Mathématiques, UMR 9994 du CNRS 2, place Jussieu 75251 Paris Cedex 05, FRANCE ; e.mail : karoubi---math.jussieu.fr Abstract. We generalize previous results ([2], [3], [4], etc.) relative to the cyclic homology and cohomology of the group algebra of G. In many cases, we express them in terms of the (co)homology of the discrete groups Z(u) = Z(u)/C(u), where runs through the set of conjugacy classes of G and where Z(u) (resp. C(u)) denotes the centralizer of u (resp. the cyclic group generated by u). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Berrick/berrictp The plus-construction as a localization A J Berrick Department of Mathematics National University of Singapore Kent Ridge 119260 Singapore berrick---math.nus.edu.sg To appear in Algebraic K-Theory and its Applications, Proc Symp ICTP, Trieste, World Scientific (Singapore). AMS Classification numbers. 16S34, 18E35, 19-02, 19D06, 20H25, 55P60 Abstract . An initial survey contrasts two points of view in the historical development of the theory of localization. The first, starting with inversion of elements in a ring, leads to quotient categories and indirectly to the Q-construction. The second considers idempotent functors. This leads to the Berrick-Casacuberta description of the plus-construction on X as the idempotent functor that is nullification of X with respect to an acyclic space W. Focus on the case X = BGLR produces new results, including the classification of perfect normal subgroups of GLR. When R is a group ring AG, links are obtained between these perfect normal subgroups and the A-representability of the group G. A final section studies the relationship between the plus-construction on BGLR and acyclicity of the space W. This prompts some general questions on the K-theory of rings. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fresse/SimpAlgOp Title: On the homotopy of simplicial algebras over an operad Author: Benoit Fresse Address: Laboratoire J.A.Dieudonne Universite de Nice-Sophia-Antipolis et CNRS Parc Valrose F-06108 Nice Cedex 02 E-mail: fresse---math.unice.fr Abstract: According to a result of H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra. Comments: This article is a revised version of the paper: ``Cartan operations for a simplicial algebra over an operad''. In the former version, the proof of the following claim contains a serious defect. Let $M$ be a representation of the symmetric group. Let $V$ be a simplicial vector space. The transfer $$(M\otimes V^{\otimes n})_{S_n}\,\rightarrow\,(M\otimes V^{\otimes n})^{S_n}$$ is homotopy injective. The new construction does not depend on this property. The paper is to appear in the Transactions of the AMS. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grieder/geomrepr Title: Geometric Representation Theory and G-Signature Author: Ralph Grieder Math. Sub. Class. 91: Primary 57S25, 20C15, 57R85; Secondary 20F38, 58G10, 11R29. Department of Mathematics, Northwestern University, Evanston, IL 60208 ralph---math.nwu.edu Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of the representation theory of finite groups. We will prove that B_G consists only of copies of Z and Z/2Z. Furthermore we will show that there is a surjection from the G-equivariant cobordism group of surface diffeomorphisms to B_G. We will define a G-signature which is related to the G-signature of Atiyah and Singer and prove that this new G-signature is injective on the copies of Z in B_G. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/compress1 Embedding, compression and fiberwise homotopy theory. by John R. Klein Wayne State University klein---math.wsu.edu This paper establishes a `Cairns-Hirsch' type result for Poincare embeddings: it gives criteria in the metastable range for a Poincare embedding M x I ---> X x I between n-dimensional Poincare spaces to arise from a Poincare embedding M --> X. This result has quite a few applications. To mention a few: 1) a Poincare embedding theorem for spheres in the middle dimension. 2) a Poincare analogue of Levine's embedding theorem. 3) a Poincare version of the Whitney embedding theorem (settling a question of Levitt) 4) the existence of diagonal Poincare embeddings (in the 1-connected case). Included .eps files: pic1a-comp.eps pic2-comp.eps pic3.eps 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/haef Poincare duality embeddings and fiberwise homotopy theory. by John R. Klein Wayne State University klein---math.wsu.edu This paper establishes an embedding theorem for finite complexes mapping to Poincare spaces. The theorem is the Poincare version of the `embedded thickening theorem' of C.T.C Wall. The theorem says that a (2k - n + 2)-connected map f: K^k --> X^n (from a finite complex of dimension k to an n-dimensional Poincare space) is the underlying map of a Poincare embedding, provided also that k < n - 2. This paper will appear in Topology. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/smale Poincare Immersions. by John R. Klein Wayne State University klein---math.wsu.edu This paper establishes a Poincare variant of the fundamental theorem of immersion theory. It has been already accepted for publication in Forum Mathematicum. Given a map f:M^n --> X^n with M and X n-dimensoinal Poincare duality spaces (with or without boundary). One says that f *immerses* if f x id : M x D^j ---> X x D^j is the underlying map of a Poincare embedding for sufficiently large j. Theorem A of this paper says that f immerses if and only if the pullback of the Spivak normal fibration of X is stable fiber homotopy equivalent to the Spivak normal fibration of M. Also included is a new homotopy theoretic proof (using equivariant duality) of the existence and uniqueness theorems for the Spivak fibration. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/survey-apr10 Poincare Duality Spaces by John R. Klein Wayne State University klein---math.wsu.edu This is a survey paper on Poincare spaces. Among the included topics are classification (low dimensional, highly connected), Poincare embeddings, Poincare surgery, the finite H-space problem. This paper will eventually appear in a volume dedicated to C.T.C Wall's 60th Birthday (edited by S. Cappell, A. Ranicki and J. Rosenberg). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moyaux/artLBRC4 Title of Paper: Two lower bounds for the relative L.S. category. Author: Pierre-Marie MOYAUX. AMS Classification numbers:55M30, 57R70. Addresse of the author: Pierre-Marie MOYAUX; Universite de Lille 1; U.F.R. de Mathematiques & U.R.A 751 au CNRS; 59655 Villeneuve D'Ascq, France. Email address of the author: moyaux---gat.univ-lille1.fr Included EPS or PS files : none. Text of Abstract: We prove that $ \sigma ^{p+1}cat(X) +1 \leq cat(X,X \times S^{p}) $ and that $e(X,X\times S^{p})=e(X)+1$, where $ \sigma ^{p+1}cat$ is the $ \sigma-$category of Vandembroucq and $e$ is the Toomer invariant. The proof is based on an extension to a relative setting of Milnor's construction of the classifying space of a topological group. -------------- The Wesleyan math department has a new web server! My new URL is http://www.math.wesleyan.edu/~mhovey If you have a link to my page, please change it to this new URL One new paper this time. Mark Hovey New papers uploaded to hopf between 12/3/98 and 12/16/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/algi Title: Algebraic invariants for homotopy types Author: David Blanc AMS class.: Primary 55S45; Secondary 55Q35, 55P15, 18G10, 18G55 Address: Univ. of Haifa 31905 Haifa, Israel e-mail: blanc---math.haifa.ac.il ABSTRACT: We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types. ---------------- --------------- A blizzard of papers this time, 15 in all. One of them is the errata to my book on model categories--if you see any other errors in it, please let me know. Don't miss the orthogonal spectra of Mandell and May--they combine the best features of S-modules and symmetric spectra, it seems to me. Mark Hovey New papers uploaded to hopf between 12/16/98 and 1/7/99: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Browder-Pakianthan/pakianat Title: Cohomology of Uniformly Powerful p-groups. Authors' names: William Browder and Jonathan Pakianathan AMS Classifications: Primary 20J06, 17B50 Secondary 17B56 institutions: Princeton University and University of Wisconsin, Madison. email: browder---math.princeton.edu and pakianat---math.wisc.edu Abstract: In this paper, the cohomology of p-central, powerful, p-groups with a certain extension property are studied. Such groups naturally correspond to Lie algebras and the paper exploits this relation to calculate their Fp-cohomology as a module over the Steenrod algebra. For example, a formula for the Bockstein based on the structure constants of the Lie algebra is obtained. Then the first few terms of the Bockstein spectral sequence are calculated and expressed in terms of the corresponding Lie algebra cohomologies. This is then used to study the integral cohomology of these p-groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Butowiez-TurnerP/umc Title: Unstable Multiplicative Cohomology Operations Authors: Jean-Yves Butowiez and Paul Turner Email: jbutowiez---lemel.fr and pt---maths.abdn.ac.uk Abstract: We investigate the relationship between multiplicative unstable cohomolgy operations G^0(-) to E^0(-) and formal group laws for a certain important class of theories. As an application we study additive multiplicative idempotents. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/DavisD/e8 From Representation Thoery to Homotopy Groups Donald M. Davis Lehigh University dmd1---lehigh.edu http://www.lehigh.edu/~dmd1/dmd1.html Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. In this paper, we apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989. The method is completely different than that used by the author in previous papers. There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, we obtain a set of generators of K(E8) on which we have a nice formula for all Adams operations. Bousfield's theorem (and considerable Maple computation) allows us to obtain from this the v1-periodic homotopy groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/model-err Errata for the book Model Categories AMS Mathematical Surveys and Monographs vol. 63 by Mark Hovey hovey---member.ams.org The title is self-explanatory. No wrong theorems so far, but one wrong lemma, an incorrect proof, and a couple of incorrect definitions. Please send further errors to me at the above address. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May/mmnov14 Orthogonal spectra and S-modules M.A. Mandell MIT mandell---math.mit.edu J.P. May University of Chicago may---math.uchicago.edu There are two general approaches to the construction of symmetric monoidal categories of spectra, one based on an encoding of operadic structure in the definition of the smash product and the other based on the categorical observation that categories of diagrams with symmetric monoidal domain are symmetric monoidal. The first was worked out by Elmendorf, Kriz, and the authors in the theory of ``S-modules''. The second was worked out in the case of symmetric spectra by Hovey, Shipley, and Smith and, in a general topological setting, by Schwede, Shipley, and the authors. A comparison between symmetric spectra and S-modules was given by Schwede. Orthogonal spectra are intermediate between symmetric spectra and S-modules: they are defined in the same diagrammatic fashion as symmetric spectra, but, as with S-modules, their stable weak equivalences are just the maps that induce isomorphisms on homotopy groups. We prove that the categories of orthogonal spectra and S-modules are Quillen equivalent and that this equivalence induces Quillen equivalences between the respective categories of ring spectra, of modules over a ring spectrum, and of commutative ring spectra. The equivalence is given by a functor that is closely related to an older and more intuitive functor from orthogonal spectra to S-modules, and a comparison between the two leads to a precise understanding in terms of a category of Thom spaces of the relationship between the definitions of orthogonal spectra and of S-modules. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May-Schwede-Shipley/mmss1nov14 Diagram spaces, diagram spectra, and FSP's M.A. Mandell MIT mandell---math.mit.edu J.P. May University of Chicago may---math.uchicago.edu S. Schwede Universitet Bielefeld, Germany schwede---mathematik.uni-bielefeld.de B. Shipley Purdue University and University of Chicago bshipley---math.purdue.edu Working in the category $T$ of based spaces, we give the basic theory of diagram spaces, diagram spectra, and functors with smash product. For a small topological category $D$, a $D$-space is just a continuous functor $D >--> T$. There is an external smash product that takes a pair of $D$-spaces to a $(D x D)$-space. If $D$ is symmetric monoidal, there is an internalization of this smash product that makes the category $DT$ of $D$-spaces a symmetric monoidal category. This allows the definition of monoids R in $DT$, modules over monoids R, and, when R is commutative, monoids in the category of R-modules. These structures are defined in terms of the internal smash product, but they all have more elementary descriptions in terms of the external smash product. A monoid R is a symmetric monoidal functor $D >--> T$, and the external version of an R-module is a $D$-spectrum over R. We show that there is a new category $D_R$ such that a $D_R$-space has the same structure as a $D$-spectrum over R. When R is commutative, the external version of a monoid in the category of R-modules is a $D$-FSP (functor with smash product) over R. We are especially interested in functors relating categories such as these as $D$ varies. With R taken as a canonical sphere diagram space, examples include Symmetric spectra, as defined by Jeff Smith. Orthogonal spectra, a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category. Gamma-spaces, as defined by Graeme Segal. $W$-spaces, an analogue of Gamma-spaces with finite sets replaced by finite CW complexes in the domain category. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May-Schwede-Shipley/mmss2nov14bis Model categories of diagram spectra M.A. Mandell MIT mandell---math.mit.edu J.P. May University of Chicago may---math.uchicago.edu S. Schwede Universitet Bielefeld, Germany schwede---mathematik.uni-bielefeld.de B. Shipley Purdue University and University of Chicago bshipley---math.purdue.edu In this sequel to our paper ``Diagram spaces, diagram spectra, and FSP's", we construct and compare model structures on the categories of prespectra, symmetric spectra, orthogonal spectra, Gamma-spaces, and $W$-spaces defined there. With the caveat that Gamma-spaces are always connective, the homotopy categories associated to all of these model categories are equivalent to the classical stable homotopy category. In all cases, there is a levelwise model structure, in which the weak equivalences and fibrations are defined levelwise. Actually, it is often convenient or necessary to modify this by considering some but not all levels. There is then a stable model structure in which the cofibrations are the cofibrations in the level model structure and the weak equivalences are the stable weak equivalences. In the cases of prespectra, orthogonal spectra, Gamma-spaces, and W-spaces, stable weak equivalences are just maps whose associated maps of prespectra induce isomorphisms of homotopy groups. In the case of symmetric spectra, a stable weak equivalence f: X >--> Y is a map such that f^*:[Y,E] >--> [X,E] is an isomorphism for all symmetric Omega-spectra E, where the brackets refer to the levelwise homotopy category. Modulo the caveat about Gamma-spaces, the model categories of prespectra, symmetric spectra, orthogonal spectra, Gamma-spaces, and $W$-spaces are Quillen equivalent and thus have equivalent homotopy categories. In favorable cases, the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. Prespectra do not form a symmetric monoidal category, this being the main reason for interest in the other categories. In all other cases, the respective categories of ring spectra are model categories and, with the caveat about Gamma-spaces, they are all Quillen equivalent and thus have equivalent homotopy categories. A similar statement holds for module spectra over ring spectra. The categories of commutative symmetric ring spectra and commutative orthogonal ring spectra are model categories and are Quillen equivalent. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/boardman The hare and the tortoise J.P. May The University of Chicago may---math.uchicago.edu This is a little tribute to Mike Boardman, for inclusion in the proceedings honoring his 60th birthday. It describes some of the history behind his definition of the stable homotopy category, sketches how his original construction worked, and describes its relationship to the recent ``all frills attached'' construction due to Elmendorf, Kriz, Mandell, and myself. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/history STABLE ALGEBRAIC TOPOLOGY, 1945-1966 J. P. MAY Contents 1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 5. Steenrod operations, K(pi; n)'s, and characteristic classes 8 6. The introduction of cobordism 10 7. The route from cobordism towards K-theory 12 8. Bott periodicity and K-theory 14 9. The Adams spectral sequence and Hopf invariant one 15 10. S-duality and the introduction of spectra 18 11. Oriented cobordism and complex cobordism 21 12. K-theory, cohomology, and characteristic classes 23 13. Generalized homology and cohomology theories 25 14. Vector fields on spheres and J(X) 28 15. Further applications and refinements of K-theory 31 16. Bordism and cobordism theories 34 17. Further work on cobordism and its relation to K-theory 37 18. High dimensional geometric topology 40 19. Iterated loop space theory 42 20. Algebraic K-theory and homotopical algebra 43 21. The stable homotopy category 45 References 50 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/Melpaper Equivariant orientations and Thom isomorphisms J.P. May University of Chicago may---math.uchicago.edu Despite a great deal of work, notably by Costenoble and Waner, there is still not a fully satisfactory theory of orientations and Thom isomorphisms of equivariant bundles. Working in a given RO(G)-graded cohomology theory, one wants somehow to grade Thom classes on fiber representations. A G-space B is G-connected if each of its fixed point subspaces B^H is non-empty and connected. Over such base spaces, one can fix the same fiber representation for all fibers, and work of Lewis and myself gives a satisfactory theory. In any approach to more general base spaces, one must parametrize changes of fiber representation as one moves around B on the equivariant fundamental groupoid pi(B), which depends on all components of all fixed point spaces and all paths connecting them. Costenoble and Waner package the complexity in a generalization of RO(G)-graded cohomology. I propose an alternative. Giving up the idea that an orientation should be a single cohomology class, I propose that orientations should be compatible collections of cohomology classes in the cohomologies of the Thom H-spaces of the pullbacks of the given bundle to the ``H-connected covers'' of B. This allows one to quote rather than generalize the theory of Lewis and myself. The H-connected covers introduced for this purpose should have other uses. As in the case of orientations, they provide a substitute in the equivariant world for the standard first step in so many nonequivariant arguments, namely: ``We may assume without loss of generality that X is connected''. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/symplectic TITLE: The Invariants of the Symplectic Groups AUTHOR: Mara D. Neusel AG Invariantentheorie, Germany EMAIL: mdn---sunrise.uni-math.gwdg.de maramara---steenrod.mast.queensu.ca mneusel---cfgauss.uni-math.gwdg.de neusel---math.umn.edu These are lecture notes of the talks "Invarianten klassischer Gruppen III/IV", held at the Oberseminar Algebraische Topologie und Invariantentheorie, University of G\"ottingen, Germany, winter semester 1998/9. In this notes we study the invariant rings of the symplectic groups in odd characteristic in their tautological representation, and try to make the original paper by Carlisle and Kropholler more readable and understandable, i.e., the only new thing is the expository, in particular that/how and where the Steenrod algebra is used is my contribution. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rognes/whdiff Two-primary algebraic K-theory of pointed spaces John Rognes Department of Mathematics University of Oslo P.O. Box 1053, Blindern Norway rognes---math.uio.no We compute the mod 2 cohomology of Waldhausen's algebraic K-theory spectrum A(*) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh^{DIFF}(*). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions * <= 18, and up to extensions in dimensions 19 <= * <= 21. As applications we show that the linearization map L : A(*) -> K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher-Waldhausen map hw : G/O -> Omega Wh^{DIFF}(*) does not admit a four-fold delooping, and there is a 2-complete spectrum map M : Wh^{DIFF}(*) \to Sigma g/o_{oplus} which is precisely 9-connected. Here g/o_{oplus} is a spectrum whose underlying space has the 2-complete homotopy type of G/O. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/dona Title:``The sectional category of spherical fibrations'' Author: Don Stanley AMS-classification number: 55P50 Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley---math.fu-berlin.de Abstract: We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a ceratin homology class in twice the dimension of the sphere is -1 times a square. Otherwise secat is two. We apply out results to construct a fibration $p$ such that $secat(p)=2$ and $genus(p)=\infty$. We also observe that secat, unlike cat, can decrease in a field extension of $\mathbb Q$. 14. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/ls4 Title:``On the Lusternik-Schnirelmann category of maps''(revised version) Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley---math.fu-berlin.de Abstract: We give conditions when $cat(f \times g)N be a homeomorphism of closed piecewise linear manifolds. Then the normal invariant of h is trivial provided 3-dimensional homology of M has no 2-torsion. The goal of this paper is to give a relative simple proof of this theorem in a particular case of manifolds M such that the fundamental and homology group of M are free abelian groups. Motivation: Kirby--Siebenmann proved that TOP/PL is the Eilenberg--Mac Lane space K(Z/2,3) and it actually solves the Hauptvermutung for manifolds. The proof by Kirby--Siebenmann uses a special case of the Normal Invariant Homeomorphism Theorem when M is the product of a torus with a sphere. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/ThomAdjoint On an adjoint functor to the Thom functor Yuli B. Rudyak We construct a right adjoint functor to the Thom functor, i.e., to the functor which assigns the Thom space T\xi to a vector bundle \xi. --------------- Oops! I somehow left off Clarence's paper! Here it is. Mark Hovey New papers uploaded to hopf between 2/14/99 and 3/22/99, bis. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Wilkerson/t-lannes Title: The T-functor of J. Lannes Author: Clarence W. Wilkerson Classification: 55P99 (55M35 55R35 55S10 55T15) Address: Dept. Math., Purdue University, West Lafayette, IN 47907-1395 E-mail: wilker---math.purdue.edu This is a preliminary version of a two page expository article on the Lannes' T-functor for upcoming the Kluwer Encylopedia of Mathematics. Please send comments/corrections/improvements to wilker---math.purdue.edu 03/05/99 ------------ 5 new papers this time, including a classic that has never been publicly available before (Mike Boardman's conditionally convergent spectral sequences). Mark Hovey New papers uploaded to hopf between 3/22/99 and 4/8/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bisson-Pengelley-Williams/stablops Stabilizing the lower operations for mod 2 cohomology Terrence P. Bisson Canisius College, Buffalo, NY 14208 bisson---canisius.edu David J. Pengelley New Mexico State University Las Cruces, NM 88003} davidp---nmsu.edu Frank Williams New Mexico State University Las Cruces, NM 88003} frank---nmsu.edu Primary 55S99; Secondary 16W30, 16W50, 55S10, 57T05 In {Bisson-Joyal} and {Pengelley-Williams} we studied a bialgebra K which underlies both the Steenrod algebra and the Dyer-Lashof algebra. Its elements act as lower-indexed operations in both the mod 2 cohomology of spaces and the mod 2 homology of infinite loop spaces. The algebra K can be defined explicitly by generators and relations {Pengelley-Williams}, or it can be defined as the algebra of operations in the theory of Q-modules {Bisson-Joyal}. In {Pengelley-Williams} a connection between K and the Steenrod algebra A of stable cohomology operations was established by means of a sheared algebra bijection between A and a new algebra K^{(\infty )}, which is a stabilized version of K. In {Bisson-Joyal} the extended Milnor Hopf algebra M is used (among other purposes) to define a convolution algebra containing both K and A. In this paper we establish a connection between these two approaches. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Boardman/ccspseq Title: Conditionally Convergent Spectral Sequences Author: J. Michael Boardman Address: Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore MD 21218-2686 MSC Classification: 55T05 E-mail: boardman---math.jhu.edu Abstract: Convergence criteria for spectral sequences are developed that apply more widely than the traditional concepts. In the presence of additional conditions that depend on data internal to the spectral sequence, they lead to satisfactory convergence and comparison theorems. The techniques apply to whole-plane as well as half-plane spectral sequences. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/hs Author: Louis F. McAuley Title: A Proof of the Hilbert-Smith Conjecture E-mail: louis---math.binghamton.edu The Hilbert-Smith conjecture is that if G is a locally compact group which acts effectively on a compact connected n-manifold M as a topological transformation group, then G is a Lie group. If G is not a Lie group, then G contains a group isomorphic to a p-adic group A_p which acts effectively on M. It is shown in this paper that A_p can not act effectively on M and, consequently, the Hilbert-Smith Conjecture is true. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/finob THE FINITENESS OBSTRUCTION FOR LOOP SPACES Author: Dietrich Notbohm AMS class.: 57Q12, 55R35, 55R10 Address: Mathematisches Institut Universität Göttingen Bunsenstr. 3-5 37073 Göttingen Germany e-mail: notbohm---cfgauss.uni-math.gwdg.de For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite $CW$-complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite $H$-spaces, that are $H$-spaces whose homology looks like the homology of a finite $CW$-complex. In this paper we prove this conjecture for loop spaces. In particular, this shows that every quasi finite loop space is actually homotopy equivalent to a finite $CW$-complex. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Simpson/giraudH Title: A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi Author: Carlos Simpson Address: CNRS UMR 5580, Laboratoire Emile Picard, Universite Paul Sabatier, 31062 Toulouse CEDEX, France Email: carlos---picard.ups-tlse.fr Abstract: Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small $1$-category $C$ and a morphism $g:C\rightarrow A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A\rightarrow \widehat{C}$, such that $i$ admits a left homotopy-adjoint $\psi$. We call a Segal category $A$ which satisfies these equivalent conditions, an {\em $\infty$-pretopos}. Note that (i) implies that $A$ admits all small homotopy limits too. If furthermore there exists $C\rightarrow A$ as in (iii) such that the adjoint $\psi$ preserves finite homotopy limits, then we say that $A$ is an ``$\infty$-topos''. ----------- I am happy to announce that Wesleyan has offered me a tenure-track job, with the tenure decision in the third year. Algebraic topology's longest running soap opera ends at last! In case you hadn't heard, the second-longest running soap opera is also over; John Palmieri is going to the University of Washington, with the same deal as mine at Wesleyan. Matthew and Amy Ando are going to the University of Illinois. A wonderful spring! 7 new papers this time. Note that papers by Larry Smith are currently separated into two directories--LSmith and SmithL. According to Clarence, these will be merged into SmithL. Mark Hovey New papers uploaded to hopf between 4/8/99 and 4/23/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dehon-Lannes/DL Title : Sur les espaces fonctionnels dont la source est le classifiant d'un groupe de Lie compact commutatif Authors : Fran\c{c}ois-Xavier Dehon et Jean Lannes Address : Centre de Math\'ematiques, UMR 7640 du CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France E-mail : dehon---math.polytechnique.fr lannes---math.polytechnique.fr Abstract : We show in this paper how the acquired knowledge on the mapping spaces with source the classifying space of $\mathbb{Z}/p$ (\cite{La2}, \cite{DS}) and the use of unstable $\mathrm{MU}$-resolutions give results on the mapping spaces with source the classifying space of a finite abelian $p$-group or a torus if the target space is required to have a torsion free p-adic cohomology. We prove among other things that the set of homotopy classes of maps from the classifying space $X$ of a torus to some simply connected space $Y$ whose ordinary homology is null in odd degrees and a finite dimensional free abelian group in each even degree, and whose rational cohomology is polynomial, identifies with the set of maps from the K-theory of $Y$ to the K-theory of $X$ which preserve the $\lambda$-ring structure. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Green-Leary-Schuster/gls Title: The subring of group cohomology constructed by permutation representations Authors: David J. Green (green---math.uni-wuppertal.de) Ian J. Leary (ijl---maths.soton.ac.uk) Bj"orn Schuster (schuster---math.uni-wuppertal.de) Date: 21 April 1999 Status: Submitted for publication Abstract: Each permutation representation of a finite group $G$ can be used to pull cohomology classes back from a symmetric group to $G$. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of $G$. We also investigate the effect of restricting to special types of permutation representations, such as $GL_n(F_p)$ acting on flags of subspaces. 1991 Mathematics Subject Classification: 20J06 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lueck-Meintrup/lm Title: The Type of the Classifying Space of a Topological Group for the Family of Compact Subgroups Authors: Wolfgang Lueck and David Meintrup classification number: 55R35 Addresses: Wolfgang Lueck and David Meintrup Institut fuer Mathematik und Informatik Westfaelische Wilhelms-Universtitaet Einsteinstr. 62, 48149 Muenster, Germany e-mail: lueck---math.uni-muenster.de, meintrd---math.uni-muenster.de http://wwwmath.uni-muenster.de/math/u/lueck EPS or PS files: none Text of Abstract: Let G be a locally compact topological group. We investigate the type of the classifying space of G for the family of compact subgroups. We give criteria for this space to have a d-dimensional G-CW-model, a finite G-CW-model or a G-CW-model of finite type. Essentially we reduce these questions to discrete groups and to the homological algebra of the orbit category of discrete groups with respect to certain families of subgroups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/stable `Stable homotopy of algebraic theories' to appear in Topology Stefan Schwede Fakultaet fuer Mathematik Universitaet Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de ABSTRACT: The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several examples of algebraic theories the parameterizing ring spectrum can be identified with something familiar: for the theory of sets we obtain the standard model of the sphere spectrum; the theories of monoids and groups give different, but stably equivalent models for the sphere spectrum; for sets with an action of a fixed groups one gets the spherical group ring; the theory of modules over a fixed ring leads to the Eilenberg-MacLane ring spectrum. For many other algebraic theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to Andre-Quillen homology via certain spectral sequences. We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/t-fun Title: Lannes T-Functor and Invariants of Pointwise Stabilizers Author: Larry Smith AMSCodes: 13A50 Invariant Theory, 55S10 Steenrod Algebra Email: larry---sunrise.uni-math.gwdg.de This is a PostScript file!! Abstract: Let rho : G ---> GL(n, F) be a representation of a finite group G over the field F. The group G acts on the algebra of polynomial functions F[V] on V via rho and the subalgebra of polynomials invariant under this action is denoted by F[V]^G. If U subseteq V = F^n is a linear subspace then the pointwise stabilizer of U is denoted by G_U. In this note we examine the relation between F[V]^G (the subalgebra of invariant polynomials) and F[V]^{G_U} when F is a Galois field. We do so using the T-functor introduced by J. Lannes. What we show is that a wide variety of properties of F[V]^G are inherited by F[V]^{G_U}. For example, among other things: --- we reprove a result of H. Nakajima that F[V]^{G_U} is a polynomial algebra when F[V]^G is; --- we show that the Cohen-Macaulay property is inherited by F[V]^{G_U} from F[V]^G; --- and, when F[V]^G is a complete intersection, then so is F[V]^{G_U}. We apply the T-functor to study degree bounds for generators of rings of invariants, show how the T-functor relates to the transfer homomorphism Tr^G : F[V] ---> F[V]^G, and give an application in the modular case. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/talgebra Title: On Lannes T-Functor Author: Larry Smith AMSCodes: 55S10 Steenrod Algebra Email: larry---sunrise.uni-math.gwdg.de This is a PostScript file!! Abstract: Let K denote the category of unstable algebras over the Steenrod algebra A^*. In connection with the study of the cohomology of function spaces J. Lannes introduced a remarkable functor T_U : K wigglyrightarrow K depending on a finite dimensional vector space U over the prime field. The purpose of this note is to prove some basic facts concerning how T_U relates to many of the standard properties of commutative algebras, such as Noetherean, polynomial, Cohen-Macaulay, etc. Some of the results proved here are already known, and some not. In all cases the proofs are new. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/variety Title: Variations on a Theroem of Haynes R. MIller and a Functor of Jean Lannes Author: Larry Smith AMS Codes: 55S10 Steenrod Algebra, 13A50 Invaraint Theory Email: larry---sunrise.uni-math.gwdg.de This is a PostScript file!! Abstract: Recent advances in modular invariant theory have often made use of Steenrod operations and the T-functor introduced by Jean Lannes. Many key properties of this functor depend on a Theorem of Haynes Miller. These results have been proved by a mixture of algebraic and topological methods for the full algebra of cohomology operations, and hence are only proven for the prime field F_p. Until now, for odd primes, it is not the algebra of cohomology operations that enters invariant theory, but the subalgebra of reduced powers. Deriving from the known results, those needed for invariant theory is sometimes not so obvious. This is a technical manuscript, providing proofs, over an arbitrary Galois field, of those key properties of unstable algebras over the Steenrod algebra that are essential to modular invariant theory. Being technical, it goes without saying that we assume a familiarity with some version of the Steenrod algebra, be it topological, as in the classical book of Steenrod and Epstein, or algebraic as in my book Polynomial Invariants of Finite Groups, AK Peters Ltd. 1996. ---------------- 11 new papers this time. Mark Hovey New papers uploaded to hopf between 4/24/99 and 5/17/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Boardman-Kramer-Wilson/boardman-kramer-wilson Title: The Periodic Hopf Ring of Connective Morava K-Theory Authors: J. Michael Boardman Richard L. Kramer W. Stephen Wilson Address: Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 Email: jmb---math.jhu.edu wsw---math.jhu.edu} Abstract: Let $K(n)_*(-)$ denote the $n$-th periodic Morava K-theory for any fixed odd prime $p$. Let $\underline{k(n)}_{\:*}$ denote the $\Omega$-spectrum of the $n$-th connective Morava K-theory. We give a calculation of the Hopf ring $K(n)_*\underline{k(n)}_{\:*}$, the main result of the second author's thesis. This is a new, shorter, easier proof. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Boardman-Wilson/boardman-wilson-easy-split Title: Unstable splittings related to Brown-Peterson cohomology Authors: J. Michael Boardman W. Stephen Wilson Address: Johns Hopkins University Baltimore, Maryland 21218 Email: jmb---math.jhu.edu wsw---math.jhu.edu Abstract: A new and relatively easy proof of various unstable splittings associated with Brown-Peterson cohomology is presented. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Jones-Segal/holmorse Stability for holomorphic spheres and Morse theory by Ralph L. Cohen, John D.S. Jones, and Graeme B. Segal AMS Classification numbers: 57R19, 58fF09, 32H02 Addresses of Authors: Cohen: Dept. of Mathematics, Stanford University, Stanford, Ca. 94305 Jones: Dept. of Mathematics, University of Warwick, Coventry, England Segal: Dept. of Pure Math. and Math. Statistics, Cambridge University, Cambridge, England Email addresses of Authors: Cohen: ralph---math.stanford.edu Jones: jdsj---maths.warwick.ac.uk Segal: G.B.Segal---dpmms.cam.ac.uk In this paper we study the question of when does a closed, simply connected, integral symplectic manifold $(X, \omega)$ have the "stability property" for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorphic maps from a sphere to $X$, becomes homotopy equivalent to the space of all continuous maps, lim_k Hol_k(S^2, X) = \Omega^2 X Here "=" means homotopy equivalent. The "degree k" is the evaluation of the integral cohomology class represented by the symplectic form on the map S^2 --> X. We describe this limit as a kind of group completion of Hol(S^2, X). We conjecture that this stability property holds if and only if an evaluation map $E: lim_k Hol_k(S^2, X) ---> X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category of the symplectic action functional on the universal cover of the loop space, LX, has a classifying space that realizes the homotopy type of LX. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cole-Greenlees-Kriz/AThom The universality of equivariant complex bordism \author{Michael Cole} \address{Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-0001} \email{mmcole---math.lsa.umich.edu} \author{J.P.C.Greenlees} \address{School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} \author{I.Kriz} \address{Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003} \email{ikriz---math.lsa.umich.edu} We show that if $A$ is an abelian compact Lie group, all $A$-equivariant complex vector bundles are orientable over a complex orientable equivariant cohomology theory. In the process, we calculate the complex orientable homology and cohomology of all complex Grassmannians, and thereby establish that complex orientability corresponds to the existence of a map from $MU$ to the spectrum as in the non-equivariant case. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/LAm \title{Multiplicative equivariant formal group laws.} \author{J.P.C.Greenlees} \address{School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} The notion of an $A$-equivariant formal group law for a compact abelian Lie group $A$ was introduced to study complex oriented $A$-equivariant formal group laws, but has some intrinsic algebraic interest. The theorem that the coefficient ring of equivariant complex bordism is the universal ring for equivariant formal group laws establishes that the definition is the correct one. We shall be concerned here with a very special class of equivariant formal group laws: the multiplicative ones, which appear to play a privileged role amongst all equivariant formal group laws. However our principal motivation for considering this case is its importance in understanding equivariant K-theories, and its close relationship to representation theory. The universal ring for multiplicative equivariant formal group laws is shown to be closely related to the Rees ring of the representation ring at the augmentation ideal, but only equal to it if the group is topologically cyclic. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees-Lyubeznik/ringlct \title{Rings with a local cohomology theorem and applications to cohomology rings of groups.} \author{J.P.C.Greenlees} \address{Department of Pure Mathematics, Hicks Building, Sheffield, S3 7RH, UK.} \email{j.greenlees---sheffield.ac.uk} \author{G.Lyubeznik} \address{Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA} \email{gennady---math.umn.edu} It has recently emerged that the rings of coefficients of equivariant cohomology theories very often have remarkable duality properties. It is the purpose of the present paper to formulate the duality purely algebraically in a particularly favourable case (including the ordinary cohomology rings of discrete or profinite virtual duality groups and classifying spaces of compact Lie groups), and to investigate its ring theoretic implications. We formulate a purely algebraic form of this duality, and investigate its consequences. It is obvious that a Cohen-Macaulay ring of this sort is automatically Gorenstein, and that its Hilbert series therefore satisfies a functional equation, and our main result is a generalization of this to rings with depth one less than their dimension: this proves a conjecture of Benson and Greenlees. Structural counterparts of this are also proved, showing that these rings are very well behaved in codimension 0 and 1. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Iwase/coH-Ganea Title of Paper: Co-H-spaces and the Ganea conjecture Authors: Norio Iwase AMS Classification numbers: Primary 55P45 Addresses of Authors: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan. Email addresses of Authors: iwase---math.rc.kyushu-u.ac.jp Text of Abstract: A non-simply connected co-H-space $X$ is, up to homotopy, the total space of a fibrewise-simply connected pointed fibrewise co-Hopf fibrant $j : X \to B\pi_1(X)$, which is a space with a co-action of $B\pi_1(X)$ along $j$. We construct its homology decomposition, which yields a simple construction of its fibrewise localisation. Our main goal is the construction of a series of co-H-spaces, each of which cannot be split into a one-point-sum of a bunch of circles and a simply connected space, thus disproving the Ganea conjecture. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kashiwabara-Wilson/kashiwabara-wilson-cohomology Title: The Morava K-theory and Brown-Peterson cohomology of spaces related to BP Authors: Takuji Kashiwabara W. Stephen Wilson Addresses: Institut Fourier, Universit\'{e} de Grenoble I, U.M.R. au C.N.R.S., B. P. 74, 38402 Saint-Martin-d'H\`{e}res CEDEX France Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 and Department of Mathematics Kyoto University Kyoto 606-8502 Japan Emails: Takuji.Kashiwabara---ujf-grenoble.fr wsw---math.jhu.edu Abstract: We calculate the Morava K-theory of the spaces in the Omega spectra for BP. They fit into an exotic array of short and long exact sequences of Hopf algebras. We apply this to calculate the p-adically completed Brown-Peterson cohomology, as well as all of the intermediary cohomology theories, E, of these spaces. We give two descriptions of the answer, both of which turn out to be surprisingly nice. One part of our first description is just the image in the E cohomology of the corresponding space in the Omega spectrum for BP, which is as big as it could possibly be and which we show how to calculate. The other part is just the E cohomology of several copies of Eilenberg-MacLane spaces, something which is already known. Our second description is inductive and gives us a new way of looking at the Brown-Peterson cohomology of Eilenberg-MacLane spaces. The Brown-Comenetz dual of BP shows up in our calculations and so we take up the study of this spectrum as well. It was already known that the Morava K-theory of the spaces in the Omega spectrum for the Brown-Comenetz dual of BP made it look like a product of Eilenberg-MacLane spaces and we find, somewhat to our surprise, that the same is true for the BP cohomology. In order to state our answers we set up the foundations for the category of completed Hopf algebras. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LewisG/PrjFlt WHEN PROJECTIVE DOES NOT IMPLY FLAT, AND OTHER HOMOLOGICAL ANOMALIES L. GAUNCE LEWIS, JR. Department of Mathematics, Syracuse University, Syracuse NY 13244-1150 E-mail address: gaunce---ichthus.syr.edu Abstract. The category MG of Mackey functors for a group G carries a symmetric monoidal closed structure. The box product providing this structure encodes the Frobenius axiom, which describes the interaction of induction and multiplication in Mackey functor rings. Mackey functors are of interest in equivariant homotopy theory since good equivariant cohomology theories are Mackey functor valued. In this context, the box product is useful not only because it encodes the interaction between induction and the cup product, but also because of the role it plays in the not yet fully understood universal coefficient and K"unneth formulae. This role makes it important to know whether projective objects in MG are flat, and whether the box product of projective objects in MG is projective. In the most familiar symmetric monoidal abelian categories, the tensor product obviously interacts appropriately with projective objects. However, the box product for MG need not be so well behaved. For example, if G is O(n), projectives need not be flat in MG and the box product of projective objects need not be projective. This misbehavior complicates the search for full strength equivariant universal coefficient and K"unneth formulae. These questions about the interaction of the box product with projective objects can be regarded as compatibility conditions which may be satisfied by a symmetric monoidal closed category M. The primary purpose of this article is to investigate these, and related, conditions. Our focus is on functor categories whose monoidal structures arise in a fashion described by Day. Conditions are given under which such a structure interacts appropriately with projective objects. Further, examples are given to show that, when these conditions aren't met, this interaction can be quite bad. These examples were not fabricated to illustrate the abstract possibility of misbehavior. Rather, they are drawn from the literature. In particular, MG is badly behaved not only for the groups O(n), but also for the groups SO(n), U(n), SU(n), Sp(n), and Spin(n). Similar misbehavior occurs in two categories of global Mackey functors which are widely used in the study of classifying spaces of finite groups. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Real/cupi I attach here the zip dvi.file of the paper ``A combinatorial method for computing Steenrod Squares'' which will be published this summer in JPAA. I would like that this paper would be included in Hopf archive. I think that this paper will completely answer the question raised by Prof. McClure in Algebraic Topology Discussion List about cup-i products. If you have problems with these files, please do not hesitate to contact me. Best regards, Pedro Pedro Real Dpto de Matematica Aplicada I Fac. de Informatica y estadistica Univ. de Sevilla Avda. Reina Mercedes s/n 41012 Sevilla Tfno: 34-95-4556921 Fax: 34-95-4557878 e-mail: real---cica.es 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WilsonWS/wilson-hopf-rings-survey Title: Hopf rings in algebraic topology Author: W. Stephen Wilson Address: Johns Hopkins University Baltimore, Maryland 21218 Email: wsw---math.jhu.edu Abstract: These are colloquium style lecture notes about Hopf rings in algebraic topology. They were designed for use by non-topologists and graduate students but have been found helpful for those who want to start learning about Hopf rings. They are not ``up to date,'' nor are then intended to be, but instead they are intended to be introductory in nature. Although these are ``old'' notes, Hopf rings are thriving and these notes give a relatively painless introduction which should prepare the reader to approach the current literature. ---------------- 5 new papers this time. Mark Hovey New papers uploaded to hopf between 5/17/99 and 6/16/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arone/LoopStiefel The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits Gregory Arone University of Chicago arone---math.uchicago.edu We prove that the Mitchell-Richter filtration of the space of loops on complex Stiefel manifolds stably splits. The result is obtained as a special case of a more general splitting theorem. Another special case is H. Miller's splitting of Stiefel manifolds. The proof uses the theory of orthogonal calculus developed by M. Weiss. The argument is inspired by an old argument of Goodwillie for a different, but related, general splitting result. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dwyer-Palmieri/ohkawas-theorem Title: Ohkawa's theorem: there is a set of Bousfield classes Authors: William G. Dwyer and John H. Palmieri AMS Classification numbers: 55P42, 55P60, 55U35 Addresses of authors: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 Email addresses: dwyer.1---nd.edu, palmieri---member.ams.org Abstract: We give a short, elementary proof of Ohkawa's theorem: there is a set of Bousfield classes. We also discuss Ohkawa classes, which are used in the proof, and provide a weaker equivalence relation on spectra than Bousfield classes do. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grove-Ziller/groveziller Title: Curvature and Symmetry of Milnor Spheres Authors: Karsten Grove and Wolfgang Ziller E-Mail: kng---math.umd.edu , wziller---math.upenn.edu In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively curved metrics on these exotic spheres. A further consequence will be that there are infinitely many almost free actions by SO(3) on S^7, i.e. all isotropy groups are finite. These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do not extend to the disc D^8. We also construct infinitely many such actions on the 15 exotic 7-spheres mentioned above. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/smash Title: Smashing subcategories and the telescope conjecture - an algebraic approach Author: Henning Krause Status: To appear in: Invent. Math. Address: University of Bielefeld, Germany E-mail: henning---mathematik.uni-bielefeld.de Abstract: We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scherer/cellular TITLE: "Cellular approximations using Moore spaces" AUTHORS: Jose L. Rodriguez Universitat Autonoma de Barcelona 08193 Bellaterra, Spain jlrodri---mat.uab.es http://mat.uab.es/jlrodri Jerome Scherer Centre de Recerca Matematica 08193 Bellaterra, Spain jerome---crm.es AMS classification number: 55P60 Localization and completion ABSTRACT: For a two-dimensional Moore space $M$ with fundamental group $G$, we identify the effect of the cellularization $CW_M$ and the fiber $\ov P_M$ of the nullification on an Eilenberg--Mac Lane space $K(N,1)$, where $N$ is any group: both induce on the fundamental group a group theoretical analogue, which can also be described in terms of certain universal extensions. We characterize completely $M$-cellular and $M$-acyclic spaces, in the case when $M=M(\Z/p^k,1)$. --------------- I missed a few papers this morning, so here are four more. By the way, in the latest issue of Notices of the AMS, there is an interview with outgoing DMS director (i.e. the math division of the NSF) D.J. Lewis. In it he says "And one that should be under the same strain is Geometry/Topology. But quite frankly, over the last three years that program made too many very small grants, and so the strain is hidden." He also says later "We're under terrific pressure to increase the size of our grants. If we did what the [National Science] Board wants us to do, we would fund 800 people instead of 1,400." I completely disagree with the tenor of these remarks! Perhaps I am wrong about this, but it seems to me that doubling the size of each grant while giving half as many total grants would have a completely negative effect on mathematics! I don't understand why the NSF is so stupid as to want to do this--it seems like very poor management and not very cost-effective. Perhaps somebody can explain this to me, or perhaps together we can complain about it. It follows from my opinion that I think what Prof. Lewis says should be taken as an indication of the excellent job Ralph Krause has been doing! I am very sorry he is leaving. Even with the best will in the world, his successor will probably not be in a position to resist the pressure to cut the numbers of grants as effectively as Ralph was able to do. One more completely annoying statement that appears in the interview: "We fund proposals, not individuals". I think that in practice, how good a job the NSF is doing can be measured by how much the section heads manage to get around this stated policy. Following this policy would mean Andrew Wiles would not have been funded while he was working on Fermat. It is the exact opposite of how I would run things. Mark Hovey New papers uploaded to hopf between 5/17/99 and 6/16/99, part 2. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/brown Title: Brown representability and flat covers Author: Henning Krause Status: Submitted Address: University of Bielefeld, Germany E-mail: henning---mathematik.uni-bielefeld.de Abstract: We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat cover can be constructed in a canonical way. The proof also shows that Brown representability for objects and morphisms is a consequence of Brown representability for objects and isomorphisms. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/idempotent Title: Decomposing thick subcategories of the stable module category Author: Henning Krause Status: Math. Ann. 313 (1999), 95-108 Address: University of Bielefeld, Germany E-mail: henning---mathematik.uni-bielefeld.de Abstract: Let stmod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of stmod kG. It is shown that every thick tensor-ideal C of stmod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C=\coprod_{i\in I}C_i into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module E_C into indecomposable modules. If C=C_W is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring H^*(G,k), then the decomposition of C reflects the decomposition W=\bigcup_{i=1}^nW_i of W into connected components. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH-Reichenbach/endofiniteness Title: Endofiniteness in stable homotopy theory Author: Henning Krause and Ulrike Reichenbach Status: Submitted Address: University of Bielefeld, Germany E-mail: henning---mathematik.uni-bielefeld.de reichenb---mathematik.uni-bielefeld.de Abstract: We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/strickland_chapx Chern approximations for generalised group cohomology Neil P. Strickland University of Sheffield N.P.Strickland---sheffield.ac.uk Let G be a finite group, and let E be a generalised cohomology theory, subject to certain technical conditions. We study a certain ring C(E,G) that is the best possible approximation to E^0BG that can be built using only knowledge of the complex representations of G. There is a natural map C(E,G) -> E^0BG, whose image is the subring of E^0BG generated over E^0 by all Chern classes of such representations. There is ample precedent for considering this subring in the parallel case of ordinary cohomology. However, although the generators of this subring come from representation theory, the same cannot be said for the relations; one purpose of our construction is to remedy this. We also also develop a kind of generalised character theory which gives good information about the rationalisation of C(E,G). In the few cases that we have been able to analyse completely, either C(E,G) is rationally different from E^0BG for easy character-theoretic reasons, or we have C(E,G)=E^0BG. Rather than working directly with rings, we will study the formal schemes X(G)=spf(E^0BG) and XCh(G)=spf(C(E,G)). Suitably interpreted, our main definition is that XCh(G) is the scheme of homomorphisms from the Lambda-semiring R^+(G) of complex representations of G to the Lambda-semiring scheme of divisors on the formal group associated to E. ---------------- Ah, summertime! When everyone gets a chance to finish those projects. 11 new papers this time, including--brace yourself for this one--a new version of Hopkins-Kuhn-Ravenel!!! Mark Hovey New papers uploaded to hopf between 6/16/99 and 7/16/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbounov-Malikov-Schechtman/chiral Title: Gerbes of chiral differential operators Authors: Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman Already submitted to xxxLANL math.AG/9906117 V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA;\ vgorb\---ms.uky.edu F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA;\ fmalikov\---mathj.usc.edu V.S.: Department of Mathematics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK;\ vs\---maths.gla.ac.uk In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Schechtman and Vaintrob, and are canonically defined for an arbitrary $X$. One can try to define a purely even counterpart of $\Omega^{ch}_X$, a sheaf of graded vertex algebras $\CO^{ch}_X$, called a {\it chiral structure sheaf}. The obstraction to its existence turns out to admit a very simple expression in terms of characteristic classes of $X$, namely it is expressed in terms of the second component of Chern character of the tangent bundle of $X$. From a different viewpoint, one can regard the above result as a geometric interpretation of the second component of the Chern character. In particular, it provides a geometric criterion for a Calabi-Yau manifold to be a $BU\langle 6\rangle$-manifold: those are precisely the manifolds which admit the above mentioned sheaf $\CO^{ch}_X$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hopkins-Kuhn-Ravenel/hkr Generalized group characters and complex oriented cohomology theories Michael J. Hopkins Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 mjh---math.mit.edu Nicholas J. Kuhn Department of Mathematics, University of Virginia, Charlottesville, VA 22903 njk4x---virginia.edu Douglas C. Ravenel Department of Mathematics, University of Rochester, Rochester, NY 14627 drav---math.rochester.edu AMS classification numbers: Primary 55N22; Secondary 20C99, 55N91, 55R35 Though it seems a shame to mess with an undergraound cult classic, this July 1999 preprint is intended to replace earlier versions dating from 1989 and 1992. It is also intended to get those of you who regularly bug us about this off our case. Cheers, Nick Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E^*, the assignment G ---> E^*(BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E^*, using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented theories: the abelian subgroups of G serve as a detecting family for E^*(BG), modulo torsion dividing the order of G. When E^* is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes E^*(BG) tensored with the rationals. The domain of the characters is G(n,p), the set of n--tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E_n^--theory, etc. The nth Morava K--theory Euler characteristic for BG is computed to be the number of G--orbits in G(n,p). For various groups G, including all symmetric groups, we prove that K(n)^*(BG) is concentrated in even degrees. Our results about E^*(BG) extend to theorems about E^*(EG\times_G X), where X is a finite G--CW complex. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/IsaksenD/prospace Title: A Model Structure on the Category of Pro-Simplicial Sets Author: Daniel C. Isaksen Address: Department of Mathematics University of Chicago Chicago, IL 60637, USA Email: dci---math.uchicago.edu Abstract: We study the category pro-SS of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SS so that it is possible to do homotopy theory in this category. This model structure is related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several other descriptions of weak equivlences, including a cohomological characterization. An appendix contains dual constructions for ind-spaces. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Asymptotics Title: Cobordism of symplectic manifolds and asymptotic expansions Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack---math.jhu.edu AMS classification numbers: 55N22, 58Z05, 81S10 Abstract: The cobordism ring defined by manifolds with symplectic structure, in the sense of V.L. Ginzburg [which is NOT the cobordism ring of the Thom spectrum MSp] is shown to be isomorphic to the cobordism ring defined by almost-complex manifolds together with a complex line bundle, and as well to a cobordism ring defined by prequantized manifolds in the sense of Kostant and others. The author uses this as a hook upon which to hang some far-fetched speculations about formal group laws in topology and asymptotic expansions in physics. This paper is to appear in the Mathematical Publications of the Steklov Institue, in a volume dedicated to S.P. Novikov. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Invols Title: Cobordism of involutions revisited, revisited Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack---math.jhu.edu AMS classification numbers: 55-03, 55N22, 55N91 Abstract: This is the writeup of a talk at the 1998 AMS Winter meeting, on Mike Boardman's early work on the Conner-Floyd five-halves conjecture; it will appear in Contemporary Mathematics 239. The main point is that Boardman's technical innovations [in the case of unoriented geometric bordism of involutions] foreshadow recent work by Greenlees and Kriz, on equivariant homotopy-theoretic bordism. Attention is also drawn to related old work of Quillen on the use of the language of residues in algebraic topology. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Sympquestions Title: Questions about cobordism of symplectic and toric manifolds Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack---math.jhu.edu AMS classification numbers: 55N22, 14M25, 58Z05 Abstract: This note contains no results. It is a kind of footnote to an earlier paper, about V.L. Ginzburg's cobordism ring of manifolds with symplectic structure. Toric manifolds have such structure, and the purpose of this note is to raise some natural questions about the symplectic cobordism classes of such manifolds. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Topgrav Title: Topological gravity in dimensions two and four Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack---math.jhu.edu AMS classification numbers: 55P, 58D, 83C Abstract: This is a writeup of a talk at the Utrecht conference on Operads in June 1999: the main observation is that the category with d-manifolds as objects, and (d+1)-dimensional cobordisms as morphisms, is naturally a two-category, with diffeomorphisms as the two-morphisms. The corresponding topological category obtained by replacing the morphism categories with their classifying spaces has deep connections with Riemannian geometry; its monoidal representations are the physicists' theories of topological gravity. Five examples are sketched, four corresponding to d=1 and one to d=3, and the paper concludes with a remark about adjoint structures on such categories. A mistake in some earlier papers on 2D gravity [posted in this Archive] is noted. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak-Tralle/MasseyThom On Thom spaces, Massey products and non-formal symplectic manifolds Yuli Rudyak and Aleksy Tralle July 6, 1999 We suggest a simple general method of constructing of non-formal manifolds. In particular, we construct a large family of non-formal symplectic manifolds. Here we detect non-formality via non-triviality of rational Massey products. In fact, we analyze the behaviour of Massey products of closed manifolds under the blow-up construction. In this context Thom spaces play the role of a technical tool which allows us to construct non-trivial Massey products in an elegant way. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Salvatore/conflab Title: Configuration spaces with summable labels Author: Paolo Salvatore AMS Classification numbers: 55R35; 55S15; 57N65 xxx preprint math.AT/9907073 Address: University of Bonn Beringstrasse 1 53115 Bonn Germany e-mail: salvator---math.uni-bonn.de Let M be an n-manifold, and let A be a space with a partial sum behaving as an n-fold loop sum. We define the space C(M;A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I^n,dI^n;A) is an n-fold classifying space of C(I^n;A), and for n=1 it is homeomorphic to the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M;A) is homotopic to the mapping space Map(M,C(I^n,\de I^n;A)). 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/moz2 Real Equivariant Bordism and Stable Transversality Obstructions for $\ints/2$ by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps---math.brown.edu In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $\Omega^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques of \cite{Si1}. Because we are working in the real setting only with $\ints/2$, these techniques simplify greatly. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/mugcomp Computations of Complex Equivariant Bordism Rings by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps---math.brown.edu This paper is a significantly revised version of a previous submission to the Hopf archive. In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of $S^1$ actions on stably complex four-manifolds with precisely three fixed points, answering a question of Bott. --------------- I have decided that the addresses of the authors and the Math Subject Classifications, while certainly useful in the archive itself, just get in the way in these mailings. So I have deleted them from these abstracts. Let me know if you think this is the wrong thing to do. 8 new papers this time. Mark Hovey New papers uploaded to hopf between 7/16/99 and 8/19/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Stanley/gan8 Title:"The cone length of a product of co-H-spaces and a problem of Ganea" Authors: Martin Arkowitz and Donald Stanley Abstract: It is proved that the cone length or strong category of a product of two co-H-spaces is less than or equal to two. This yields the following positive solution to a problem of Ganea. Let $\alpha \in \pi_{2p}(S^3)$ be an element of order p, p a prime $\geq 3$, and let $X(p)=S^3\cup_{\alpha}e^{2p+1}$. Then $X(p)\times X(p)$ is the mapping cone of some map $\phi:Y \rightarrow Z$, where $Z$ is a suspension. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Devinatz-Hopkins/homotopy-fixed-point (Note: you need the file approx.ps as well as the dvi file to print this). Title: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups Author: Ethan S. Devinatz and Michael J. Hopkins Text of Abstract: Let G be a closed subgroup of the semi-direct product of the nth Morava stabilizer group with the Galois group of the field extension of degree n of the field of p elements. We construct a "homotopy fixed point spectrum" whose homotopy fixed point spectral sequence involves the continuous cohomology of G. These spectra have the expected functorial properties and agree with the Hopkins-Miller fixed-point spectra when G is finite. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Stabilization of Model Categories by Mark Hovey This is an update of a previous paper on the archive. Recall that the idea of this paper is to construct spectra and symmetric spectra starting from an arbitrary model category. There was a mistake in the previous version; I asserted that, whenever symmetric spectra and spectra could both be defined, they were the same up to a chain of Quillen equivalences. There is a simple argument showing that this must be false without some hypothesis. This mistake has now been fixed and the correct hypothesis added. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey-Palmieri/galois Galois theory of thick subcategories in modular representation theory by Mark Hovey and John Palmieri Suppose B is a finite-dimensional cocommutative Hopf algebra over a field k. Define a thick subcategory to be a full subcategory of the category of finite-dimensional B-modules that is closed under summands and, if two out of three modules in a short exact sequence are in it, so is the third. Define a thick subcategory to be tensor-closed if it is closed under tensoring with any finite-dimensional module. The classification of these tensor-closed thick subcategories, analogous to the Hopkins-Smith classification of thick subcategories in the stable homotopy category, has been carried out for B=k[G], where G is a finite group and k is an algebraically closed field of positive characteristic, by Benson-Carlson-Rickard. A similar classification has been obtained by the current authors when B is a finite subalgebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of Z/2. In the present paper, we eliminate the annoying requirement that the field be algebraically closed. We show that, if the expected classification of tensor-closed thick subcategories holds for B tensor L, where L is a normal extension field of k, then it holds for B as well. The proof involves importing the basic ideas of Galois theory into axiomatic stable homotopy theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Joyal-Tierney/JT-chap-01 Title: An Introduction to Simplicial Homotopy Theory Authors: Andre Joyal and Myles Tierney Abstract: This is a preliminary version of the first chapter of a book on simplicial homotopy theory. It introduces simplicial sets, and supplies the basic background material, anodyne extensions, fibrations, homotopy between maps, etc, leading to a new, combinatorial proof of the existence of the classical Quillen model structure. It finishes with Milnor's Theorem showing that the category of Kan complexes and homotopy classes of maps is equivalent to the category of CW-complexes and homotopy classes of maps. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Leary/ijltork Title: A torsion projective class for a group algebra Author: Ian J. Leary Abstract: For a certain cyclic-by-finite group, we construct an element of order two in the algebraic $K_0$ of the rational group ring, whose image in $K_0$ of the complex group ring is zero. Non-triviality of the element is established using topological methods. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Leary-Nucinkis/ijlbeanCW Title: Every CW-complex is a classifying space for proper bundles Authors: Ian J. Leary and Brita E. A. Nucinkis Abstract: We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by an action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Snaith/vpsnaithBP Title: Hurewicz images in BP and related homology theories Author: Victor Snaith In this paper $BP$-theory is used to give a proof that there exists a stable homotopy element in $\pi_{2^{n+1} - 2}^{S}( {\bf R}P^{\infty})$ with non-zero Hurewicz image in $ju$-theory if and only if there exists an element of $\pi_{2^{n+1} - 2}^{S}( S^{0})$ which is represented by a framed manifold of Arf invariant one. -------------- 5 new papers this time. Mark Hovey New papers uploaded to hopf between 8/19/99 and 8/31/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Crossley-Whitehouse/hiconj Title: Higher Conjugation Cohomology in Commutative Hopf Algebras Authors: M. D. Crossley and Sarah Whitehouse Abstract Text: The dual Steenrod algebra can be expressed as the homotopy of a smash product of two copies of the Eilenberg-MacLane spectrum, and the conjugation arises by permutation of the two factors. This can be generalized to an action of the symmetric group $\Sigma_n$ acting on an n-1-fold tensor product of copies of the dual Steenrod algebra; this action was described by the second author in [6], purely in terms of the Hopf algebra structure. So, formally, one has a similar action for any commutative Hopf algebra. In this paper, we study the cohomology ring $H^*(\Sigma_n; A^{\otimes n-1})$, where $A$ is a graded commutative Hopf algebra. We show that for a certain class of Hopf algebras the cohomology ring is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. Then, by choosing a sufficiently simple coproduct, we are able to deduce significant information about the $\Sigma_n$ invariants of $A^{\otimes n-1}$, including dimensions and algebra structure. In particular, we give a complete solution to the "conjugation invariants" problem for the mod $p$ dual Steenrod algebra when $p>2$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbounov-Malikov-Schechtman/gerbes Title: Gerbes of chiral differential operators Authors: Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman This is an improved version of the paper we have submitted earlier. In particular we have computed the "conformal anomaly": the obstruction to the existence of a globally defined Virasoro field. In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Schechtman and Vaintrob, and are canonically defined for an arbitrary $X$. One can try to define a purely even counterpart of $\Omega^{ch}_X$, a sheaf of graded vertex algebras $\CO^{ch}_X$, called a {\it chiral structure sheaf}. The obstraction to its existence turns out to admit a very simple expression in terms of characteristic classes of $X$, namely it is expressed in terms of the second component of Chern character of the tangent bundle of $X$. The obstruction to the existence of a globally defined Virasoro field $L(z)$; it is given by the first Chern class $c_1(X)/2$, cf. Theorem 9.1. In particular, Theorem 3, provides a geometric criterion for a manifold to admit a $BU\langle 6\rangle$-structure: those are precisely the manifolds which admit the above mentioned sheaf $\CO^{ch}_X$ and for which the conformal anomaly vanishes. If such a manifold is Calabi-Yau (i.e. has the trivial canonical bundle) then $\CO^{ch}_X$ is a sheaf of {\it conformal} vertex algebras, cf. Corollary 9.3. From a different viewpoint, one can regard the above result as a geometric interpretation of the second component of the Chern character. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GrayB/associativity Associativity in two-cell complexes Brayton Gray Let P be the mapping cone of an element in an even stem in the homotopy groups of spheres localized at an odd prime. Generalizing the case of a mod p^r Moore space, we show that the smash square of P splits as a wedge of two iterated suspensions of P. Furthermore, this can be done in a unique way satisfying certain identities, and if p>3, one of these identities is an associativity condition. There are two consequences: 1) If E is a (commutative) associative ring spectrum, then E^P is as well when localized at p>3. 2) A Samelson product can be defined in homotopy with coefficients in P which will satisfy all the usual identities including the Jacobi identity if p>3. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GrayB/twocells On the homotopy groups of 2-cell complexes Brayton Gray In 1978, Cohen, Moore, and Neisendorfer gave a decomposition of the loops on a mod p^r Moore space when p>2. This decomposition involved an atomic factor T^(2n+1) which was encompassed in a fibration sequence with other terms whose homotopy was better understood. This paper considers the case when the mod p^r Moore space is replaced by the mapping cone P of an element in an even stem. Exactly the same results are obtained when the attaching map is divisible by p, or the dimension of P is even. The first obstruction to such a result is displayed in general, and the example of beta_1 is presented. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhong-Woo/forget Title: Phantom maps and Forgetable maps Authors: Pan Jianzhong and Moo Ha Woo In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so-called Forgetable map. We show that we can insert the Forgetable map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. --------------- 4 new papers this time. Mark Hovey New papers uploaded to hopf between 8/31/99 and 9/20/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/sheaves Title: Model category structures on chain complexes of sheaves Author: Mark Hovey Abstract: In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on injective resolutions for an arbitrary Grothendieck category, as has apparently also been done by Morel. In particular, this works for sheaves on a ringed space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. However, this injective model structure is not well suited to studying the derived tensor product, so we investigate other model structures. The most successful of these is the flat model structure on complexes of sheaves over a ringed space. This is based on flat resolutions, and is compatible with the tensor product. As a corollary, we get model categories of differential graded algebras of sheaves and differential graded modules over a given differential graded algebra of sheaves. This is the author's first attempt to understand sheaves, so comments from those more experienced with the subject are welcome. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/hs Author: Louis F. McAuley Title: A Proof of the Hilbert-Smith Conjecture E-mail: louis---math.binghamton.edu The Hilbert-Smith conjecture is that if G is a locally compact group which acts effectively on a compact connected n-manifold M as a topological transformation group, then G is a Lie group. If G is not a Lie group, then G contains a group isomorphic to a p-adic group A_p which acts effectively on M. It is shown in this paper that A_p can not act effectively on M and, consequently, the Hilbert-Smith Conjecture is true. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/beta Title of Paper: On the action of $\beta_1$ in the stable homotopy of spheres at the prime $3$ Author: Katsumi Shimomura Text of Abstruct: The element $\beta_1$ is the generator of the stable homotopy group $\pi_{10}(S^0)$. Here $S^0$ denotes the $3$-localized sphere spectrum. Toda showed that $\beta_1^5\neq 0$ and $\beta_1^6=0$. Here we generalize it to $\beta_1^4\beta_{9t+1}\neq 0$ and $\beta_1^5\beta_{9t+1}= 0$ for $\beta_{9t+1}\in\pi_{144t+10}(S^0)$ with $t\ge 0$. In particular, $\beta_1^4\beta_{10}\neq 0$ and $\beta_1^5\beta_{10}= 0$ for $\beta_{10}$ shown to exist by Oka. This is proved by determining subgroups of $\pi_*(L_2S^0)$, where $L_2$ denotes the Bousfield localization functor with respect to $v_2^{-1}BP$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/L2V-0- Title of Paper: The homotopy groups of the $L_2$-localized mod 3 Moore spectrum Author: Katsumi Shimomura Text of Abstruct: Let $L_2$ denote the Bousfield localization with respect to the 2nd Johnson-Wilson spectrum $E(2)$. The homotopy groups $\pi_*(L_2V(0))$ of the mod 3 Moore spectrum $V(0)$ are determined by using the results on $\pi_*(L_2V(1))$, where $V(1)$ denotes the Toda-Smith spectrum. As an application, we show that $\beta_s\in \pi_*(L_2S^0)$ if and only if s=0,1,2,3,5,6 mod 9. ---------------- 8 new papers this time, including a disproof of the chromatic splitting conjecture by Shimomura and Wang. Mark Hovey New papers uploaded to hopf between 9/20/99 and 10/25/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/marspin Title: The 1-line of the K-theory Bousfield-Kan spectral sequence for Spin(2n+1) Authors: Martin Bendersky and Donald M. Davis mbenders---shiva.math.hunter.cuny.edu ddavis---math.jhu.edu Abstract: For X a simply-connected finite H-space, there is a Bousfield-Kan spectral sequence which converges to the homotopy groups of the K-completion of X. When X=Spin(2n+1), we expect that these homotopy groups equal the v1-periodic homotopy groups of X in dimension greater than n^2. In this paper, we accomplish two things: (1) We prove that, for any X, the 1-line of the spectral sequence is determined explicitly by K-theory and Adams operations. (2) For X=Spin(2n+1), we make an explicit computation of this 1-line. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/ARtriangle Title: Auslander-Reiten theory via Brown representability Author: Henning Krause E-mail: henning---mathematik.uni-bielefeld.de Abstract: We develop an Auslander-Reiten theory for triangulated categories which is based on Brown's representability theorem. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McClure-SmithJH/deligne_conj A solution of Deligne's conjecture. James E. McClure and Jeffrey H. Smith mcclure---math.purdue.edu jhs---math.purdue.edu ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action of an operad equivalent to the little 2-cubes. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Menichi/Cohomology_Fiber Title: On the Cohomology algebra of a fiber Author: Luc Menichi Luc.Menichi---univ-angers.fr Abstract: Let $f:E\rightarrow B$ be a fibration of fiber $F$. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between $H^{*}(F;\mathbb{F}_p)$ and $\mbox{Tor}^{C^{*}(B)}(C^{*}(E),\mathbb{F}_p)$. Generalizing the rational case proved by Sullivan, Anick proved that if $X$ is a finite $r$-connected CW-complex of dimension $\leq rp$ then the algebra of singular cochains $C^{*}(X;\mathbb{F}_p)$ can be replaced by a commutative differential graded algebra $A(X)$ with the same cohomology. Therefore if we suppose that $f:E\hookrightarrow B$ is an inclusion of finite $r$-connected CW-complexes of dimension $\leq rp$, we obtain an isomorphism of vector spaces between the algebra $H^{*}(F;\mathbb{F}_p)$ and $\mbox{Tor}^{A(B)}(A(E),\mathbb{F}_p)$ which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin, we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^{*}(F;\mathbb{F}_p)$ is a divided powers algebra and $p^{th}$ powers vanish in the reduced cohomology $\tilde{H}^{*}(F;\mathbb{F}_p)$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/decomposition HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS D. Notbohm For a prime $p$, a homology decomposition of the classifying space $BG$ of a finite group $G$ consists of a functor $F :\BD --->>> \spaces$ from a small category into the category of spaces and a map $\hcl{} F --->>> BG$ from the homotopy colimit to $BG$ which induces an isomorphism in mod-$p$ homology. Associated to a modular representation $G --->>> Gl(n;\fp)$ we construct a family of subgroups closed under conjugation, which gives rise to three different homology decompositions, the so called subgroup, centralizer and normalizer decomposition. For an action of $G$ on a $\fp$-vector space $V$, the collection consist of the isotropy groups of all nontrivial proper subspaces of $V$ with nontrivial $p$-Sylow subgroup. These decomposition formulas connect the modular representation theory of $G$ with the homotopy theory of $BG$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura-Wang/L2S0 The homotopy groups $\pi_*(L_2S^0)$ at the prime 3 Katsumi Shimomura and Xiangjun Wang katsumi---math.kochi-u.ac.jp xwang---math.kochi-u.ac.jp The homotopy groups $\pi_*(L_2S^0)$ of the $L_2$-localized sphere are determined by studying the Bockstein spectral sequence. The results indicate also the homotopy groups $\pi_*(L_{K(2)}S^0)$ and we see that the fiber of the localization map $L_2S^0_3\to L_{K(2)}S^0$ is homotopic to $\Sigma^{-2}L_1S^0_3$, while Hopkins' chromatic splitting conjecture says that it has three summands. [Editor: this requires the font "min10" to print correctly. I don't have this font, so the ps, lj, and pdf files may have small imperfections] This is a slightly revised version of the one received originally Sept 1. (but not announced on this list because of font difficulties) 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Tsukiyama/forget8 We consider the forgetful map from the group of equivariant self equivalences to the group of non-equivariant self equivalences. A sufficient condition for this forgetful map being a monomorphism is obtained. Several examples are given. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WarnerG/warner-book Topics in Topology and Homotopy Theory Garth Warner warner---math.washington.edu eps file "top-homotopy.eps" included This book is a systematic account of the homotopical foundations of algebraic topology. The depth of coverage is substantial and I have made a point to include material which is ordinarily not included. Here is a sample of what is taken up. (1) Nilpotency and its role in homotopy theory. (2) Bousfield's theory of the localization of spaces and spectra. (3) Homotopy limits and colimits and their applications. (4) The James construction, symmetric products, and the Dold-Thom theorem. (5) Brown and Adams representability in the setting of triangulated categories. (6) Operads and the May-Thomason theorem on the uniqueness of infinite loop space machines. (7) The plus construction and theorems A and B of Quillen. (8) Hopkins' global picture of stable homotopy theory. (9) Model categories, cofibration categories, and Waldhausen categories. (10) The Dugundji extension theorem and its consequences. ---------------- Six new papers this time. I expect you all to be holed up at home writing papers while the world falls apart around us Jan. 1! Mark Hovey New papers uploaded to hopf between 10/25/99 and 12/8/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen/derived Derived categories and projective classes Dan Christensen jdc---math.jhu.edu Abstract: An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show how more general forms of homological algebra also fit into Quillen's framework. Specifically, any set of objects in a complete and cocomplete abelian category A generates a projective class on A, which is exactly the information needed to do homological algebra in A. The main result is that if the generating objects are "small" in an appropriate sense, then the category of chain complexes of objects of A has a model category structure which reflects the homological algebra of the projective class. The motivation for the work is the construction of the "pure derived category" of a ring R. Pure homological algebra has applications to phantom maps in the stable homotopy category and the (usual) derived category of a ring, and these connections will be described. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Keller-Neeman/purity Failure of Brown Representability in Derived Categories Dan Christensen, Bernhard Keller and Amnon Neeman jdc---math.jhu.edu, keller---math.jussieu.fr, neeman---wintermute.anu.edu.au Abstract: Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GreenDJ-Minh/ogawa Title: Almost all extraspecial p-groups are Swan groups Authors: David John Green Pham Anh Minh E-mail: green---math.uni-wuppertal.de paminh---bdvn.vnmail.vnd.net Abstract: Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p^3 and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer N_G(P). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ha-Strom/gray5 THE GRAY FILTRATION ON PHANTOM MAPS L^E MINH HA AND JEFFREY STROM ha---math.wayne.edu jeffrey.strom---dartmouth.edu Abstract This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition holds for all k, we recover McGibbon and Roitberg's theorem. There is a dual result, and a theorem on phantom maps into spheres which holds one dimension at a time as well. Finally, we examine the set of phantom maps whose Gray in- dex is infinite. The main theorem is a partial verification of our conjecture that if X and Y are nilpotent and of finite type, then every phantom map f : X -! Y must have finite index. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/louis-hs Author: Louis F. McAuley Title: A Proof of the Hilbert-Smith Conjecture E-mail: louis---math.binghamton.edu The Hilbert-Smith conjecture is that if G is a locally compact group which acts effectively on a compact connected n-manifold M as a topological transformation group, then G is a Lie group. If G is not a Lie group, then G contains a group isomorphic to a p-adic group A_p which acts effectively on M. It is shown in this paper that A_p can not act effectively on M and, consequently, the Hilbert-Smith Conjecture is true. (This is a revised version of a previously announced paper). 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strom/hiorder HIGHER ORDER PHANTOM MAPS JEFFREY STROM jeffrey.strom---dartmouth.edu Abstract For each ordinal number ff, we define phantom maps of order ff. We construct universal phantom maps out of X with order ff, and show that under easily verifiable condi- tions, every one of these universal phantom maps is essential. ---------------- I thought the last message was my last this century, but suddenly there are three more papers. Mark Hovey New papers uploaded to hopf between 12/8/99 and 12/9/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Benson-KrauseH/pureinj Title: Pure injectives and the spectrum of the cohomology ring of a finite group Authors: David Benson and Henning Krause E-mail: djb---byrd.math.uga.edu henning---mathematik.uni-bielefeld.de Abstract: For a finite group and a field of prime characteristic, we study certain pure injective representations in terms of the spectrum of the group cohomology ring. This includes a complete classification of all representations which arise as a direct summand of a (possibly infinite) product of syzygies of the trivial representation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pengelley-Peterson-Williams/globstr (That's "Global Structure", not "g lobster" :) ) A global structure theorem for the mod two Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras David J. Pengelley davidp---nmsu.edu Franklin P. Peterson fpp---math.mit.edu Frank Williams frank---nmsu.edu The Dickson algebra W_{n+1} of invariants in a polynomial algebra over F_2 is an unstable algebra over the mod 2 Steenrod algebra A, or equivalently, over the Kudo-Araki-May algebra K of ``lower'' operations. We prove that W_{n+1} is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyze the interplay of actions over A and K to characterize unstable cyclic modules with trivial action by the subalgebra A_{n-2} on a fundamental class in degree (2^n)-a, where a is a nonnegative integer. This involves a new family of left ideals I_a in K, which play the role filled by the ideals generated by A_{n-2} in the Steenrod algebra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pengelley-Williams/limalg Limits of algebras with shifting and a relationship between the mod two Steenrod and Dyer-Lashof algebras David J. Pengelley davidp---nmsu.edu Frank Williams frank---nmsu.edu We provide a construction, refined from an inverse limit, that produces the mod 2 Steenrod and Dyer-Lashof algebras from each other. In fact, the construction relates various subalgebras and quotients of the universal Steenrod algebra of operations for H-infinity ring spectra. We also describe how the construction transforms the axiomatic properties of homogeneous pre-Koszul algebras and Poincare-Birkhoff-Witt algebras. ---------------- And yet another two more. This is the 100th announcement I have made! Mark Hovey New papers uploaded to hopf between 12/9/99 and 12/10/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dugger/smod Title: Replacing Model Categories with Simplicial Ones Author: Daniel Dugger Email: ddugger---math.purdue.edu Abstract: We show that model categories of a very large class can be replaced up to Quillen equivalence by simplicial model categories. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Giambalvo-Peterson/DicksonAGen A-Generators for Ideals in the Dickson Algebra by V. Giambalvo and F. P. Peterson vince---math.uconn.edu fpp---math.mit.edu The Dickson Algebra $D_q$ on $q$-variables is the algebra of invariants of the action of the mod-2 general linear group on a polynomial algebra in $q$-variables. We study the structure of certain ideals in this algebra as a module over the Steenrod Algebra $\A$, and develop methods to determine which elements are hit by Steenrod operations. This allows us to display a very small set of $\A$-generators for these ideals and show that the set is minimal in some cases. We include a minimal set of generators for $D_5$. --------------- Papers are suddenly appearing by the truckload! 6 more this time. Mark Hovey New papers uploaded to hopf between 12/10/99 and 12/14/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc_algi Title: Algebraic invariants for homotopy types Author: David Blanc e-mail: blanc---math.haifa.ac.il ABSTRACT: We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types. Note: to appear in Math. Proc. Camb. Phil. Soc. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc-steenrod Title: Realizing coalgebras over the Steenrod algebra Author: David Blanc e-mail: blanc---math.haifa.ac.il ABSTRACT: We describe algebraic obstruction theories for realizing an abstract coalgebra K_* over the mod p Steenrod algebra as the homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K_*. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cole/specmod Title: A model structure for inclusion prespectra in which the fibrant objects are the spectra Author: Michael Cole e-mail: matmzc---hofstra.edu ABSTRACT:We construct a Quillen closed model structure on the category of inclusion prespectra of Lewis and May for which an object is fibrant if and only if it is a spectrum. The spectrification functor from inclusion prespectra to spectra serves as a fibrant replacement functor. The homotopy category associated to this model structure is equivalent to the stable category. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rosu/ioanidellc Title: Equivariant Elliptic Cohomology and Rigidity Author: Ioanid Rosu, ioanid---math.mit.edu Equivariant elliptic cohomology with complex coefficients was defined axiomatically by Ginzburg, Kapranov and Vasserot and constructed by Grojnowski. We give an invariant definition of S^1-equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rosu-Knutson/ioanidkt Title: Equivariant K-theory and Equivariant Cohomology Author: Ioanid Rosu, with an appendix by Allen Knutson and Ioanid Rosu ioanid---math.mit.edu allenk---math.berkeley.edu For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/ScottJA/ls-bss Algebraic Structure of the Loop Space Bockstein Spectral Sequence Jonathan A. Scott e-mail: scott---math.toronto.edu Abstract: Let X be a finite, n-dimensional, r-connected CW complex. We prove the following theorem: If p > n/r - 1 is an odd prime, then the loop space homology Bockstein spectral sequence modulo p is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.