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. ---------------