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.