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 Künneth Formula for Topological Joins. Author: Göran 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 Künneth 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 Université de Nantes, Département de mathématiques 2 rue de la Houssinière BP 92208 Nantes Cedex 03 France laurent.piriou---math.univ-nantes.fr Lionel Schwartz Université Paris 13 Institut Galilée LAGA UMR 7539 du CNRS Av. J. B. Clément 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 Técnico 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 Fröbenius 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 Möbius 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 København. 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 Département de Mathématiques 2, rue de le Houssinière - BP 92208 44322 NANTES Cédex 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 théorie 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: Université 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 Théories 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 Mathématiques de Jussieu Université 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 résolvons génériquement le problème ``hit'' (posé en 1986 par Franklin P. Peterson) par la découverte en degrés génériques d'un système générateur minimal explicite pour l'algèbre polynomiale comme module sur l'algèbre de Steenrod mod 2. Cette solution implique en particulier un résultat de J. Repka-P. Selick, une partie de celui de M. C. Crabb-J. R. Hubbuck et nous permet en même temps de vérifier une conjecture due à M. Kameko. Ce système générateur sera appliqué à l'étude du transfert algébrique de W. M. Singer et de la représentation modulaire du groupe linéaire général. 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 Löh) Address: Einsteinstr. 62, 48143 Münster, 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 Università 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-dime