Now that we seem to have survived Y2K, the show must go on. 4 new papers this time. Mark Hovey New papers uploaded to hopf between 12/14/99 and 1/2/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/charact Title: An algebraic geometric realization of the Chern character Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph---math.stanford.edu and plfilho---math.tamu.edu Text of abstract Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map $ch : K^{-i}_{hol}(X) \to \prod_* L^*H^{2*-i}(X)\otimes Q$ from the "holomorphic $K$-theory of $X$ to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic $K$-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from $X$ into $BU$. It has been further studied by the authors in a companion paper. Holomorphic $K$-theory sits between algebraic $K$-theory and topological $K$-theory in the same way that morphic cohomology sits between motivic cohomology and ordinary cohomology. Our constructions provide a bridge between these two worlds. We also realize Chern classes in the case where $X$ is smooth, and establish a universal relation between the Chern character and the Chern classes. For this we use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/holo-k-th Title: Holomorphic $K$-theory, algebraic co-cycles, and loop groups Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph---math.stanford.edu and plfilho---math.tamu.edu Text of abstract In this paper we study the ``holomorphic $K$-theory" of a projective variety. This theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we show that there is a rational isomorphism between holomorphic $K$-theory and the appropriate "morphic cohomology", defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the ``symmetrized loop group" $\om U(n)/\Sigma_n$ where the symmetric group $\Sigma_n$ acts on $U(n)$ via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians $BU(k)$ by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic $K$-theory by inverting the Bott class, then it is rationally isomorphic to topological $K$-theory. Finally we produce explicit obstructions to periodicity in holomorphic $K$ - theory, and show that these obstructions vanish for generalized flag manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/ab Classifying subcategories of modules Mark Hovey mhovey---wesleyan.edu In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We claim that these wide subcategories are analogous to thick subcategories of the derived category D(R). Indeed, let C_0 denote the wide subcategory generated by R; C_0 is the collection of all finitely presented modules precisely when R is coherent. When R is a quotient of a regular commutative coherent ring by a finitely generated ideal, we classify wide subcategories of C_0. In fact, they are on 1-1 correspondence with thick subcategories of small objects of D(R). The proof relies heavily on Thomason's thick subcategory theorem for D(R). We also classify wide subcategories closed under arbitrary coproducts; these are analogous to localizing subcategories of D(R). In this case, we must assume that R is Noetherian, where we use Neeman's classification of localizing subcategories of D(R). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lawson-Lima-Filho-Michelsohn/alg-cycles1 Title: Algebraic Cycles and the Classical Groups - Part I, Real Cycles Authors : H. Blaine Lawson, Jr. and Paulo Lima-Filho and Marie-Louise Michelsohn Email addresses: blaine---math.sunysb.edu, plfilho---math.tamu.edu, mmichelsohn---math.sunysb.edu The groups of algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real structure, it is natural to ask for the properties of the groups of real algebraic cycles on P(V). Similarly, if V carries a quaternionic structure, one can define quaternionic algebraic cycles and ask the same question. In this paper and its sequel the homotopy structure of these cycle groups is completely determined. It turns out to be quite simple and to bear a direct relationship to characteristic classes for the classical groups. It is shown, moreover, that certain functors in K-theory extend directly to these groups. It is also shown that, after taking colimits over dimension and codimension, the groups of real and quaternionic cycles carry E_{\infty}-ring structures, and that the maps extending the K-theory functors are E_{\infty}-ring maps. In fact this stabilized space is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed at the representations R^{n,n} for n \geq 0. This gives a wide generalization of the results in [BLLMM] on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a simple quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. These calculations yield an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory. --------------- 7 new papers this time. Mark Hovey New papers uploaded to hopf between 1/2/00 and 1/25/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Anderson-DavisJ/MacPherson Title: Mod 2 Cohomology of Combinatorial Grassmannians Authors: Laura Anderson and James F. Davis Abstract: Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes. The homotopy groups of this poset are related to the image of the J-homomorphism from stable homotopy theory. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dwyer-Wilkerson/center-calc/center-calc Centers and Coxeter elements by W. G. Dwyer and C. W. Wilkerson dwyer.1---nd.edu wilker---math.purdue.edu Abstract: Suppose that $G$ is a connected compact Lie group. We show that simple numerical information about the Weyl group of $G$ can be used to obtain bounds, often sharp, on the size of the center of $G$. These bounds are obtained with the help of certain Coxeter elements in the Weyl group. Variants of the method use generalized Coxeter elements and apply to $p$-compact groups; in this case a splitting theorem emerges. The Lie group results are mostly known, but our arguments have a conceptual appeal. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey-Palmieri/quillen Stably thick subcategories of modules over Hopf algebras by Mark Hovey and John Palmieri hovey---member.ams.org and palmieri---member.ams.org We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional cocommutative Hopf algebra B. Our method is based on that of Benson-Carlson-Rickard, who classify such subcategories when B=kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of Z/2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/bdi4 ON THE 2-COMPACT GROUP DI(4) Author: D. Notbohm Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group $DI(4)$. The mod-2 cohomology of the associated classifying space $BDI(4)$ realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space $BDI(4)$ and that the maximal torus normalizer determines the isomorphism type of $DI(4)$ as 2-compact group. We also calculate the set of homotopy classes of self maps of $BDI(4)$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/orthogonal A UNIQUENESS RESULT FOR ORTHOGONAL GROUPS AS 2-COMPACT GROUPS D. Notbohm Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizer are isomorphic. It is conjectured that this result generalizes to p-compact groups. Here, we prove the generalization for orthogonal groups $O(n)$, the special orthogonal groups $SO(2k+1)$ and the spinor groups $Spin(2k+1)$ considered as 2-compact groups. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SchwartzL/FK Author: Lionel Schwartz Title: La filtration de Krull de la categorie U et la cohomologie des espaces Jan. 6, 2000 The present paper gives a proof of a conjecture of N. Kuhn : if the mod 2 cohomology of a space has finite Krull filtration in the category of unstable modules, it has to be a locally finite unstable module. Some technical assumptions are required. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WuJ/newwgroup Title of Paper: A braided simplicial group Author(s): Jie Wu Email address of Authors: matwuj---nus.edu.sg Text of Abstract: By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general homotopy group of the 3-sphere is the fixed set of the pure braid group action on a certain combinatorially described group. We also give a certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere. Comments are welcome. --------------- This seems a good time to remind you that if you have submitted a paper to Hopf and it does not appear on this list, it is NOT because Clarence has rejected it. Hopf is not an automated archive, so sometimes it takes a while for papers to be moved into the appropriate spot. On the other hand, it is always possible that Clarence or I have made a mistake, so it doesn't hurt to send e-mail reminding us. 5 new papers this time. Mark Hovey New papers uploaded to hopf between 1/25/00 and 1/29/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/homotopy-ext On spaces of self homotopy equivalences of p-completed classifying spaces of finite groups and homotopy group extensions By Carles Broto and Ran Levi Fix a prime p. A mod-p homotopy group extension of a group $\pi$ by a group G is a fibration with base space $B\pi^\wedge_p$ and fibre $BG^\wedge_p$. In this paper we study homotopy group extensions for finite groups. We observe that there is a strong analogy between homotopy group extensions and ordinary group extensions. The study involves investigating the space of self homotopy equivalences of a p-completed classifying space. In particular we show that under the appropriate assumption on $G$, the identity component of this space is homotopy equivalent to $BZ(G)$, the classifying space of the centre of $G$. We proceed by studying the group of components. We show that this group maps into a group of natural equivalences of a certain functor with kernel and cokernel, which are computable in terms of the first and second derived functors of the inverse limit for a certain diagram of abelian groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/simp-models Combinatorial models for iterated loop spaces By: Fred Cohen and Ran Levi The objective of this paper is to provide free simplicial group models for the functors $\Omega^n X$ and $\Omega^n\Sigma^{n+k}X$. The models are based on classical constructions in simplicial homotopy theory. Specifically, Milnor's functor F, Kan's loops group functor G and the Moore loop space construction $\Omega$ are used to produce these models. The models are given in terms of free groups with specific generators and the formulas defining the simplicial operators are given. The utility of these models is that in them certain maps can be written explicitly in a relatively easy way. To illustrate this a null homotopy of the commutator map on a double loop space is given. Similar ideas are used to give a model for pointed mapping spaces out of a Riemann surface. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/stunted-proj ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES By: Fred Cohen and Ran Levi Let $X_n$ denote the infinite stunted projective space ${\Bbb R}P^\infty/{\Bbb R}P^{n-1}$. In this note we study the homotopy type of this family of spaces. In particular we show that for $n=2 $ and 4, the space $X_n$ splits after looping once and for $n=3$ after looping four times and passing to connected covers. In each case the factors are loop spaces on naturally occuring finite complexes. These result generalise to higher values of $n$, but in those cases without a splitting result. The splittings enable us to carry out a calculation of low dimensional homotopy and loop space homology for these spaces, which complements a computer calculation of Sergeraert and Smirnov. A number of interesting related facts and questions is also discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McGibbon-Strom/numphant Numerical invariants of phantom maps C. A. McGibbon and Jeffrey Strom Wayne State University and Dartmouth College Two numerical homotopy invariants of phantom maps, the Gray index G(f) and the essential category weight E(f), are studied. The possible values of these invariants are determined. In certain cases bounds on these values are given in terms of rational homotopy data. Examples are provided showing that the Gray index can take any positive finite value. For certain cases it is shown that every essential phantom f: X --> Y has finite Gray index. However it is also shown that there exist spaces, e. g. CP^\infty, which are the domains of essential phantoms with infinite index. The same type of analysis is carried out on the essential category weight of a phantom map. If the loop space of X is homotopy equivalent to a finite complex, then every phantom f: X --> Y has E(f) = \infty. However, in certain other cases it is shown that E(f) is strictly less than the rational Lusternik-Schnirelmann category of the domain. A homotopy classification of phantoms f: K(Z, n)--> S^m is given along with the values of E(f). The invariants G and E provide decreasing filtrations on the set of homotopy classes of phantoms from X to Y. A third filtration on this set is introduced for certain special targets. When the rational cohomology of the domain X is finitely generated, this filtration enables one to reduce the search for essential phantoms (into finite type targets) to a finite list of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dickson Dickson Invariants hit by the Steenrod Squares BY K. F. Tan and Kai Xu Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$ by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in attacking the question in the general case. (This paper contains some tedious computations which will be dropped in the simplified version that will be written later.) ---------------- Sorry for the delay; I seem to be getting old and tired. 6 new papers this time. Mark Hovey New papers uploaded to hopf between 1/29/00 and 3/4/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Morava/amrrrfls A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space Authors: Matthew Ando mando---math.uiuc.edu Jack Morava jack---math.jhu.edu We show that the fixed-point formula in an equivariant complex-oriented cohomology theory $E$, applied to the free loop space of a manifold $X$, may be viewed as a (renormalized) Riemann-Roch formula for the quotient of the group law of $E$ by a free cyclic subgroup. If $E$ is $K$-theory, this explains how the elliptic genus associated to the Tate elliptic curve emerges from Witten's analysis of the fixed-point formula in $K$-theory. In general this quotient is not representable, but we show that its torsion subgroup is. In the case that $E$ is the Borel theory associated to the Lubin-Tate theory $E_n$, this leads to a description of the functor represented by $E_n[[q]], analogous to the relationship between the Tate curve and $K$-theory. For a more general equivariant $E$, we show that the formal products which arise in this discussion may be naturally viewed as Thom classes for Thom prospectra as considered by Cohen-Jones-Segal. These prospectra seem to define interesting models for the physicists' space of `small' loops on $X$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Crespo-Saumell/aqfh Title: Non-simply connected $H$-spaces with finiteness conditions Authors: Carlos Broto, Juan A. Crespo and Laia Saumell e-mail addresses: broto---mat.uab.es, chiqui---crm.es, and laia---mat.uab.es This article is concerned with homotopy properties of $H$-spaces $X$ that are reflected in the module of indecomposables $QH^*(X;\F_p)$. It is shown that mod $p$ $H$-spaces $X$ of finite type with finite transcendence degree mod $p$ cohomology and locally finite $QH^*(X;\F_p)$ are $B\Z/p$-null spaces, Eilenberg-MacLane spaces $K(\padic,2)$, $K(\Z/p^r,1)$, and extensions of those. If we restrict attention to $H$-spaces with noetherian mod $p$ cohomology algebra, then we are left with finite mod $p$ $H$-spaces and Eilenberg-MacLane spaces. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fisher/bous Title: A Proof of an Exponent Conjecture of Bousfield Author: Michael J. Fisher Email: mjf7---lehigh.edu Abstract: Let p be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the p-exponent of the spectrum Phi SU(n) is (n-1) + nu_p((n-1)!) for n >= 2. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grodal/limsub Title: Higher limits via subgroup complexes Author: Jesper Grodal Email: jg---math.mit.edu Abstract: We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with values in a G-local coefficient system. We examine when smaller complexes can be used e.g., taking only p-radical subgroups, p-centric subgroups, elementary abelian p-subgroups or various subcollections thereof. Since the subgroup complexes are finite complexes, and often rather small, this provides concrete, computable formulas for these higher limits, generalizing earlier work of especially Jackowski-McClure- Oliver. It also gives a conceptual explanation of high dimensional vanishing results previously established in more indirect ways. As an application we look at the special case where all the higher limits vanish, as for example is the case for group cohomology. If F is a functor on the orbit category our formulas for the higher limits in this case yield five different expressions of F(G) in terms of values of F on proper subgroups. Two of these are `classical' namely Webb's exact sequence of Mackey functors and a formula for calculating stable elements, previously obtained using Alperin's fusion theorem. Examining this case also leads to improvements of sharpness results of homology decompositions due to Dwyer and others. Central to many of the proofs are properties of the Steinberg chain complex of a finite group G, as well as other concepts from the emerging Lie theory for arbitrary finite groups of Alperin, Webb, and others. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhang-Woo/forgetnew1 Title: Phantom maps and Forgetful maps Authors: Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing China and Department of Mathematics Education , Korea University , Seoul , Korea email: pjz62---hotmail.com Moo Ha Woo Department of Mathematics Education , Korea University , Seoul , Korea ABSTRACT: In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so- called Forgetful map. We show that we can insert the Forgetful map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. Remark:This is an upgraded version of a preprint which has been on the archive. A problem in Theorem2.8 has been corrected following a suggestion from K.Iriye. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Karoubi/A_descent_theorem Max KAROUBI A descent theorem in topological K-theory karoubi---math.jussieu.fr Let A be a Banach algebra and A' its complexification. In this paper we show that the homotopy fixed point set of K(A'), the topological K-theory space of A', under complex conjugation is just K(A), the topological K-theory space of A. This result generalizes the well known fact that BO is BU^hZ/2. The proof uses in an essential way Atiyah's KR theory and the Clifford algebra definition of higher K-groups. ---------------- 6 new papers this time. Mark Hovey New papers uploaded to hopf between 3/4/00 and 4/9/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/comp1 Compositions in the v1-periodic homotopy groups of spheres Martin Bendersky and Donald M. Davis mbenders---shiva.hunter.cuny.edu, dmd1---lehigh.edu 21 pages, completed March 7, 2000, submitted to Forum Mathematicum Abstract Let p_i in pi_{n+8i-1}(S^n) denote an element which suspends to a generator of the image of the stable 2-primary J-homomorphism. We determine the image of the composite p_j o p_k in v1-periodic homotopy v_1^{-1} pi_{n+8i+8j-2}(S^n). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson. Odd-primary analogues are also obtained. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org April, 2000 This is a revised version. The basic idea is to automate the passage from unstable to stable homotopy theory, so that it applies in particular to the A^1 category of Voevodsky. So if we start with a model category C and a left Quillen endofunctor G of C, we want to make a new model category, the stabilization of C, where G becomes a Quillen equivalence. The simplest way to do this is with ordinary spectra. Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms. In particular, this holds for the A^1 category. If C has a tensor product, and G is given by tensoring with a cofibrant object K, then we also can construct symmetric spectra. The localization techniques apply here as well, so we get a stable model structure of symmetric spectra without having to assume anything like the Freudenthal suspension theorem. In particular, this is a new construction of the stable model structure on simplicial symmetric spectra. Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general. Also new to this revision is a much more careful comparison between symmetric spectra and ordinary spectra when both are defined. Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity. Under some additional technical hypotheses (which again are satisfied in the A^1 category), we construct a zigzag of Quillen equivalences between symmetric spectra and ordinary spectra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Oliver-Segev/2dim Fixed point free actions on $Z$-acyclic 2-complexes by Bob Oliver and Yoav Segev E-mail: bob---math.univ-paris13.fr, yoavs---math.bgu.ac.il We show that a finite group has an "essential" fixed point free action on an acyclic 2-complex if and only if it is one of the simple groups in the following list: - $PSL_2(2^k)$ for $k\ge2$, - $PSL_2(q)$ for $q\equiv3,5$ (mod 8) and $q\ge5$, - $Sz(2^k)$ for odd $k\ge3$. More precisely, for any finite group $G$, and any 2-dimensional acyclic $G$-CW complex $X$ without fixed points, there is a normal subgroup $H$ in $G$ such that $G/H$ is in the above list, and such that the $G$-action on $X$ looks "essentially" like the $G/H$-action which we construct. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-simpl-alg-proper Title: Every homotopy theory of simplicial algebras admits a proper model Author: Charles Rezk rezk---math.nwu.edu Abstract: We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory, which is allowed to be multi-sorted. The results have applications to the construction of localization model category structures. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Scheerer-Stanley-Tanre/Qcat Fibrewise construction applied to Lusternik-Schnirelmann category by Hans Scheerer, Donald Stanley and Daniel Tanr\'e scheerer---math.fu-berlin.de Don.Stanley---agat.univ-lille1.fr Daniel.Tanre---agat.univ-lille1.fr Abstract: In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted by Qcat(X). It is obtained by applying a base-point free version of Q = Omega-infinity Sigma-infinity fibrewise to the Ganea fibrations. We prove cat(X) >= Qcat(X) >= scat(X), where scat(X) denotes Y. Rudyak's strict category weight. However, Qcat(X) approximates cat(X) better, because e.g. in the case of a rational space Qcat(X)=cat(X) and scat(X) equals the Toomer invariant. We show that Qcat(X x Y) <= Qcat(X)+Qcat(Y). The invariant Qcat is designed to measure the failure of the formula cat(X x S^r)=cat(X)+1. In fact for 2-cell complexes Qcat(X)< cat(X) if and only if cat(X x S^r) <= cat(X) for some r >= 1. We note that the paper is written in the more general context of a functor L from the category of spaces to itself satisfying certain conditions; L= Q, Omega^n Sigma^n, Sp^infinity or L_f are just particular cases. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dicknew (This is a revised version) Dickson Invariants hit by the Steenrod Squares BY K. F. Tan and Kai Xu Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$ by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in attacking the question in the general case. (This paper contains some tedious computations which will be dropped in the simplified version that will be written later.) --------------- 9 new papers this time, including the Mahowald-Ravenel-Shick paper returning the telescope conjecture to the community. Mark Hovey New papers uploaded to hopf between 4/9/00 and 6/4/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Carlson-Karagueuzian-Milgram/hs The Cohomology of the Sylow 2-Subgroup of the Higman-Sims Group A. Adem Mathematics Department University of Wisconsin Madison WI 53706 J. F. Carlson Mathematics Department University of Georgia Athens GA 30602 D. B. Karagueuzian Mathematics Department University of Wisconsin Madison WI 53706 R. James Milgram Mathematics Department Stanford University Stanford CA 94305 Abstract In this paper we compute the mod 2 cohomology of the Sylow 2-subgroup of the Higman--Sims group HS, one of the 26 sporadic simple groups. We obtain its Poincare series as well as an explicit description of it as a ring with 17 generators and 79 relations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adem-Pakianathan/adpak On the Cohomology of Central Frattini Extensions Alejandro Adem and Jonathan Pakianathan Mathematics Department University of Wisconsin Madison, Wisconsin, 53706 adem---math.wisc.edu, pakianat---math.wisc.edu Abstract In this paper we provide calculations for the mod p cohomology of certain p-groups, using topological methods. More precisely, we look at p-groups G defined as central extensions 1-> V -> G ->W ->1 of elementary abelian groups such that the mod p reduction of G/[G,G] is W and the defining k-invariants span the entire image of the Bockstein. We show that if p>dim V-dim W+1, then the mod p cohomology of G can be explicitly computed as an algebra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ausoni-Rognes/tcl_us Title: Algebraic K-theory of topological K-theory Author: Christian Ausoni Author2: John Rognes Email: ausoni---math.ethz.ch Email2: rognes---math.uio.no Abstract: Let l_p = BP<1>_p be the p-complete connective Adams summand of topological K-theory, and let V(1) be the Smith-Toda complex. For p>3 we explicitly compute the V(1)-homotopy of the algebraic K-theory spectrum of l_p. In particular we find that it is a free finitely generated module over the polynomial algebra P(v_2), except for a sporadic class in degree 2p-3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is actually made in the V(1)-homotopy of the topological cyclic homology of l_p. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dwyer-Greenlees/CompleteTorsion Complete modules and torsion modules by W. G. Dwyer and J. P. C. Greenlees Suppose that $R$ is a ring and that $A$ is a chain complex over $R$. Inside the derived category of differential graded $R$-modules there are naturally defined subcategories of $A$-torsion objects and of $A$-complete objects. Under a finiteness condition on $A$, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case $R=Z$ and $A=Z/p$. Addresses: University of Notre Dame, Notre Dame, IN 46556, USA dwyer.1---nd.edu School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK j.greenlees---sheffield.ac.uk 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kuhn/kuhnsplit Stable Splittings and the Diagonal Nicholas J. Kuhn Department of Mathematics, University of Virginia, Charlottesville, VA 22903 njk4x---virginia.edu AMS classification numbers: Primary 55P35; Secondary 55P42 Many approximations to function spaces admit natural stable splittings, with a typical example being the stable splitting of a space C_d(X) approximating Omega^d Sigma^d X. With an eye towards understanding cup products in the cohomology of such function spaces, we describe how the diagonal interacts with the stable splitting. The description involves group theoretic transfers. In an appendix independent of the rest of the paper, we use ideas from Goodwillie calculus to show that such natural stable splittings are unique, and discuss three different constructions showing their existence. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mahowald-Ravenel-Shick/telconj Title: The triple loop space approach to the telescope conjecture Authors: Mark Mahowald, Doug Ravenel, Paul Shick Addresses: Northwestern University, University of Rochester, John Carroll University email: mark---math.mwu.edu, drav---harpo.cc.rochester.edu, shick---jcu.edu AMS Classification: 55 Abstract: The purpose of this paper is to describe an unsuccessful attempt to prove that the telescope conjecture is false for all $n \ge 2$ and all primes $p$. At the time it was originally proposed over 20 years ago, the telescope conjecture appeared to be the simplest and most plausible statement about the relationship between two different localization functors. We hope that the present paper will show that this is no longer the case. We will set up a spectral sequence converging to the homotopy of one of the two localizations (the geometrically defined telescope) of a certain spectrum, and it will be apparent that only a bizarre pattern of differentials would lead to the known homotopy of the localization defined in terms of $BP$-theory. While we cannot exclude such a pattern, it is certainly not favored by Occam's razor. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mahowald-Ravenel-Shick/temss Title: The Thomified Eilenberg-Moore spectral sequence Authors: Mark Mahowald, Doug Ravenel, Paul Shick Addresses: Northwestern University, University of Rochester, John Carroll University email: mark---math.mwu.edu, drav---harpo.cc.rochester.edu, shick---jcu.edu AMS Classification: 55 Abstract: We construct a generalization of the Eilenberg-Moore spectal sequence, which in some interesting cases turns out to be a form the Adams spectral sequence. We apply the spectral sequence to give a new construction of the $Z /p$-equivariant Adams spectral sequence of Greenlees. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert This one has an abstract in .dvi form, so I do not include it. The title is A Proof of the Hilbert-Smith Conjecture by Louis F. McAuley (The Hilbert-Smith conjecture is the one about a topological group having to be a Lie group under certain conditions). ---------------- 7 new papers this time. Sometimes there is a considerable delay between the time the author puts a paper on Hopf and the time it is announced. This delay is sometimes at my end, and sometimes at Clarence's end. I believe the delay on Clarence's end is longer when the author e-mails him the paper, as Clarence then has to do more work. I believe this is the reason that some of the papers announced this time were actually submitted sooner than some of the papers announced last time. Mark Hovey New papers appearing on hopf between 6/4/00 and 6/16/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenD-CohenF-Xicotencatl/CCX Title: Lie algebras associated to fiber-type arrangements Authors: Daniel C. Cohen, Frederick R. Cohen, Miguel Xicotencatl math.AT/0005091 Addresses of Authors D. Cohen, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 F. Cohen, Department of Mathematics, University of Rochester, Rochester, NY 14627 M. Xicotencatl, Depto. de Mathematicas, Cinvestav del IPN, Mexico City Max-Plank-Institut fur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany Email address of Authors cohen---math.lsu.edu cohf---math.rochester.edu xico------mpim-bonn.mpg.de Abstract: Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive elements in the homology of the loop space for the complement of the associated subspace arrangement. Furthermore, this last Lie algebra is given by the homotopy groups modulo torsion of the loop space of the complement of the subspace arrangement. Looping further yields an associated Poisson algebra, and generalizations of the "universal infinitesimal Poisson braid relations." 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fausk-Lewis-May/FLMApril20 The Picard Group of Equivariant Stable Homotopy Theory by H. Fausk, L.G. Lewis, Jr, and J.P. May The University of Chicago (Fausk and May) Syracuse University (Lewis) fausk---math.uchicago.edu, lglewis---mailbox.syr.edu, may---math.uchicago.edu April 20, 2000 Let G be a compact Lie group. We describe the Picard group Pic(HoGS) of invertible objects in the stable homotopy category of G-spectra in terms of a suitable class of homotopy representations of G. Combining this with results of tom Dieck and Petrie, which we reprove, we deduce an exact sequence that gives an essentially algebraic description of Pic(HoGS) in terms of the Picard group of the Burnside ring of G. The deduction is based on an embedding of the Picard group of the endomorphism ring of the unit object of any stable homotopy category C in the Picard group of C. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org June, 2000 (This is an updated version; following an idea of Voevodsky, we strengthen our proof that stable homotopy isomorphisms agree with stable equivalences of ordinary spectra so that it applies to one version of motivic homotopy theory. ) The basic idea is to automate the passage from unstable to stable homotopy theory, so that it applies in particular to the A^1 category of Voevodsky. So if we start with a model category C and a left Quillen endofunctor G of C, we want to make a new model category, the stabilization of C, where G becomes a Quillen equivalence. The simplest way to do this is with ordinary spectra. Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms. If C has a tensor product, and G is given by tensoring with a cofibrant object K, then we also can construct symmetric spectra. The localization techniques apply here as well, so we get a stable model structure of symmetric spectra without having to assume anything like the Freudenthal suspension theorem. In particular, this is a new construction of the stable model structure on simplicial symmetric spectra. Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general. Also new to this revision is a much more careful comparison between symmetric spectra and ordinary spectra when both are defined. Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity. Under some additional technical hypotheses (which again are satisfied in the A^1 category), we construct a zigzag of Quillen equivalences between symmetric spectra and ordinary spectra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May/MMM Equivariant orthogonal spectra and S-modules by M.A. Mandell and J.P. May The University of Chicago mandell---math.uchicago.edu may---math.uchicago.edu April 20, 2000 The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of S-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of S-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to S-modules. We then develop the equivariant theory. For a compact Lie group G, we construct a symmetric monoidal model category of orthogonal G-spectra whose homotopy category is equivalent to the classical stable homotopy category of G-spectra. We also complete the theory of S_G-modules and compare the categories of orthogonal G-spectra and S_G-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/PicApril20 Picard groups, Grothendieck rings, and Burnside rings of categories J.P. May The University of Chicago may---math.uchicago.edu For Saunders Mac Lane, on his 90th birthday April 20, 2000 We discuss the Picard group, the Grothendieck ring, and the Burnside ring of a symmetric monoidal category, and we consider examples from algebra, homological algebra, topology, and algebraic geometry. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May-Neumann/MNApril20 On the cohomology of generalized homogeneous spaces by J.P. May and F. Neumann The University of Chicago Georg-August-Universit\"at, G\"ottingen, Germany may---math.uchicago.edu neumann---cfgauss.uni-math.gwdg.de April 20, 2000 We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces G/H, where G is a finite loop space or a p-compact group and H is a ``subgroup'' in the homotopical sense. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Santos/equivariant-D-T A note on the equivariant Dold-Thom theorem by Pedro F. dos Santos Addresses of Author: Department of Mathematics, Texas A&M University, College Station TX-77840 Department of Mathematics, Instituto Superior Tecnico, 1049 Lisboa, Portugal Email: pedfs---math.ist.utl.pt In this note we prove a version of the classical Dold-Thom theorem for the RO(G)-graded equivariant homology functors H^G_*(-;RM), where G is a finite group, M is a discrete Z[G]-module, and RM is the Mackey functor associated to M. In the case where M=Z with the trivial G-action, our result says that, for a G-CW-complex X, and for a finite dimensional G-representation V, there is a natural isomorphism [S^V,Z_0(X)]_G \cong H^G_V(X;RM); where Z_0(X) denotes the free abelian group on X. ---------------- 13 new papers this time. Mark Hovey New papers appearing on hopf between 6/16/00 and 7/16/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/AlAgl-Brown-Steiner/multiplecat Multiple categories: the equivalence of a globular and a cubical approach Fahd A. A. Al-Agl, Ronald Brown, Richard Steiner math.CT/0007009 Fahd A. A. Al-Agl\\Um-Alqura University,\\ Makkah\\Saudi Arabia Ronald Brown, \\ School of Informatics, \\ Mathematics Division, \\ University of Wales,\\ Bangor, Gwynedd LL57 1UT, \\ United Kingdom. Richard Steiner, \\ Department of Mathematics, \\ University of Glasgow, \\University Gardens, \\ Glasgow G12 8QW \\ United Kingdom r.brown---bangor.ac.uk r.steiner---maths.gla.ac.uk We show the equivalence of two kinds of strict multiple category, namely the well known globular omega-categories, and the cubical omega-categories with connections. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Strom/TrivModF Homotopy Classes that are Trivial Mod F Martin Arkowitz (Martin.Arkowitz---Dartmouth.edu) Jeffrey Strom (Jeffrey.Strom---Dartmouth.edu) Dartmouth College If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha _* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_{F}(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_{\Sigma}(X,Y) \subseteq Z_{\M}(X,Y) \subseteq Z_{\S}(X,Y), and we find examples of {\it finite complexes} X and Y for which these inclusions are strict. We are also interested in Z_{F}(X) = Z_{F}(X,X) which under composition has the structure of a semi-group with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semi-group Z_{F}(X) is nilpotent. More generally, the nilpotency of Z_{F}(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_{F}(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arlettaz/Arlettaz-survey Title: Algebraic K-theory of rings from a topological viewpoint Author: Dominique Arlettaz Dominique Arlettaz, Institut de math\'ematiques, Universit\'e de Lausanne, CH-1015 Lausanne, Switzerland dominique.arlettaz---ima.unil.ch Abstract: This paper is a long survey providing the basic definitions of the algebraic K-theory of rings and an overview of the main classical theorems which have been obtained by arguments from algebraic topology (in particular by using methods from stable homotopy theory, group cohomology and Postnikov theory). It will appear in Publicacions Matem\`atiques. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arlettaz-Ausoni-Mimura-Yagita/Arlettaz-A-M-Y Title: Integral cohomology and Chern classes of the special linear group over the ring of integers Author1: Dominique Arlettaz Author2: Christian Ausoni Author3: Mamoru Mimura Author4: Nobuaki Yagita Author1: Dominique Arlettaz, Institut de math\'ematiques, Universit\'e de Lausanne, CH-1015 Lausanne, Switzerland Author2: Christian Ausoni, Departement Mathematik, HG, ETH-Zentrum, 8092 Z\"urich, Switzerland Author3: Mamoru Mimura, Department of Mathematics, Faculty of Science, Okayama University, Okayama, Japan 700 Author4: Nobuaki Yagita, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan E-mail1: dominique.arlettaz---ima.unil.ch E-mail2: ausoni---math.ethz.ch E-mail3: mimura---math.okayama-u.ac.jp E-mail4: yagita---mito.ipc.ibaraki.ac.jp Abstract: This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(Z) over the ring of integers Z. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Casacuberta-Scherer/casasche Homological localizations preserve 1-connectivity by Carles Casacuberta and Jerome Scherer Universitat Autonoma de Barcelona Universite de Lausanne casac---mat.uab.es jerome.scherer---ima.unil.ch To appear in Contemporary Mathematics, Proceedings of the 1999 Arolla Conference on Algebraic Topology. Every generalized homology theory $E$ yields a localization functor $L$ that sends the $E$-equivalences to homotopy equivalences. We prove that if $X$ is any $1$-connected space, then $LX$ is also $1$-connected, for every generalized homology theory $E$. This is deduced from a result by Hopkins and Smith stating that if $K(\Z,2)$ is $E$-acyclic then $E$ is trivial. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dugger/ddpres Title: Combinatorial Model Categories Have Presentations Author: Daniel Dugger Purdue University West Lafayette, IN 47906 Email: ddugger---math.purdue.edu We show that every combinatorial model category can be obtained---up to Quillen equivalence---by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of `generators' and a set of `relations' ---i.e., any combinatorial model category has a presentation. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dugger/dduniv Title: Universal Homotopy Theories Author: Daniel Dugger Address: Purdue University West Lafayette, IN 47906 Email: ddugger---math.purdue.edu Abstract: Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and also discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Goebel-Rodriguez-Shelah/locsimple TITLE: Large localizations of finite simple groups AUTHORS: Ruediger Goebel, Jose L. Rodriguez, and Saharon Shelah R.Goebel---uni-essen.de, jlrodri---mat.uab.es, shelah---math.huji.ac.il ABSTRACT: A group homomorphism $\eta: H\to G$ is called a localization of $H$ if every homomorphism $\varphi : H\to G$ can be `extended uniquely' to a homomorphism $\Phi :G\to G$ in the sense that $\Phi \eta = \varphi$. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation $A_n\to SO_{n-1}(\R)$ of the alternating group $A_n$, which turns out to be a localization for $n$ even and $n\geq 10$. Dror Farjoun asked if there is any upper bound in cardinality for localizations of $A_n$. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group $H$, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space $K(H,1)$ for any non abelian finite simple group $H$. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/finite Equivariant p-adic Homotopy Theory Michael A. Mandell mandell---math.uchicago.edu Let G be a finite group. We show that the cochain functor with coefficients in \FPbar is an equivalence between the p-adic G-equivariant homotopy category of finite type nilpotent G-spaces and a full subcategory of the homotopy category of diagrams of \einf \FPbar-algebras indexed on the orbit category of G. This turns out to be an easy consequence of Elmendorf's Theorem and Kan's work on diagrams in closed model categories plus the equivalence in the nonequivariant context. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity Title: Pretty Good Gravity Author: Jack Morava (not yet on xxx, but will be soon) Address: Dept. of Mathematics, the Johns Hopkins Uniperversity e-mail address: jack---math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. [This is a writeup of a talk at the RIMS Symposium on algebraic geometry and integrable systems related to string theory, June 12-16, 2000.] 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ravenel/first Title of paper: The first Adams-Novikov differential for the spectrum T(m) Author: Douglas C. Ravenel Address of Author: University of Rochester, Rochester, NY 14627 Email address of author: drav---math.rochester.edu Abstract: There are p-local spectra T(m) with $BP_{*}(T(m))=BP_{*}[t_{1},\dots ,t_{m}]$. In this paper we determine the first nontrivial differential in the Adams--Novikov spectral sequence for each of them for p odd. For m=0 (the sphere spectrum) this is the Toda differential, whose source has filtration 2 and whose target is the first nontrivial element in filtration 2p+1. The same goes for m=1, and for larger m the target is $v_2$ times the first such element. The proof uses the Thomified Eilenberg-Moore spectral sequence. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ravenel/micro Title of paper: The microstable Adams-Novikov spectral sequence Author: Douglas C. Ravenel Address of Author: University of Rochester, Rochester, NY 14627 Email address of author: drav---math.rochester.edu Abstract: In the Adams--Novikov spectral sequence one considers Ext groups over the Hopf algebroid $\Gamma =BP_{*}(BP)$. There are spectra $T(m)$ with $BP_{*} (T (m))=BP_{*}[t_{1},...,t_{m}]$, which leads one to replace $\Gamma $ by $\Gamma (m+1)=\Gamma / (t_{1},... ,t_{m})$. The corresponding Ext groups have certain structural features that are independent of $m$. In this paper we set up an algebraic framework for studying the limit as $m \to \infty $. In particular there is an analog of the chromatic spectral sequence in which the Morava stabilizer group gets replaced by an infinitesimal analog, hence the title. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/koszulii I can't read the abstract of this file, but I think this is not Larry's fault. Clarence is out of town though, and I am about to be, so I wanted to announce it now. It has to do with invariant theory of Z/p acting on a polynomial ring F[V]. The detailed abstract will appear next time. --------------- Sorry for the long delay since the last such announcement. One big factor contributing to the delay is e-mail attachments. Clarence has trouble dealing with these, and it also messes up my system. So it would be a big help to us if you could follow the old ftp method, or the newer web browser method, of uploading papers to Hopf. 14 new papers this time, including the abstract of Larry Smith's paper that was announced last time. Mark Hovey New papers appearing on hopf between 7/16/00 and 9/14/00. 0. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/koszulii (This paper was announced last time without abstract. Here is the abstract.) Title of Paper: Invariant Theory and the Koszul Complex Representations of Z/p in Characteristic p Applications Author: Larry Smith AMS Code: 13A50 Invariant Theory Address: Mathematisches Institut Bunsenstrasse 3--5 D 37073 Goettingen Federal republic of germany e-mail: larry---sunrise.uni-math.gwdg.de THIS IS a POstScript file. Summary: We study the ring of invariants $\F[V]^{\Z/p}$\/, and its derived functors $H^i(\Z/p\semicolon \F[V])$\/, of the cyclic group $\Z/p$ of prime order $p$ over a field $\F$ of characteristic $p$\/. We verify a formula of Ellingsrud and Skjelbred \cite{norway} for the homological codimension, show the quotient algebra $\F[V]^{\Z/p}/\Im(\Tr^{\Z/p})$ is Cohen-Macaulay, and that the ideal generated by the elements in the image of the transfer homomorphism, $\Im(\Tr^{\Z/p}) \subset \F[V]^{\Z/p}$\/, is primary of height $n-1$ when $V$ is an $n$-dimensional irreducible representation of $\Z/p$\/. Using our cohomological computations and a previous result \cite{vectors} about permutation representations we are able to obtain an upper bound for the degree of homogeneous forms in a minimal algebra generating set for $\F[V]^{\Z/p}$\/. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Basterra/abwgeec Title: The Witten genus and equivariant elliptic cohomology Authors: Matthew Ando mando---math.uiuc.edu Maria Basterra basterra---math.uiuc.edu Department of Mathematics, The University of Illinois at Urbana-Champaign Abstract: We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of the Witten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Hopkins-Strickland/eswgtc-2/ Elliptic spectra, the Witten genus, and the theorem of the cube. (revised version) M. Ando, M. J. Hopkins, and N. P. Strickland University of Illinois at Urbana-Champaign mando---math.uiuc.edu MIT mjh---math.mit.edu University of Sheffield N.P.Strickland---sheffield.ac.uk This is a revised version of an earlier paper (1998) with the same title. We show that every elliptic spectrum receives a natural MU<6>-orientation. For the elliptic spectrum defined by the Tate curve, this orientation specializes to the Witten genus. The naturality of the orientation implies that the modularity of the Witten genus for MU<6>-manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi-Oliver/blo1 Homotopy equivalences of p-completed classifying spaces of finite groups by Carles Broto, Ran Levi, and Bob Oliver We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category with the following properties. Two p-completed classifying spaces BG_p^\wedge and BG'_p^\wedge have the same homotopy type if and only if the associated categories are equivalent. And the topological group Aut(BG_p^\wedge) of self equivalences is determined by the self equivalences of the associated category. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Davis-Mahowald/eo2/ Nonimmersions of real projective spaces implied by eo2 Robert R. Bruner Wayne State University, Detroit, MI 48202 rrb---math.wayne.edu Donald M. Davis Lehigh University, Bethlehem, PA 18018 dmd1---lehigh.edu Mark Mahowald Northwestern University, Evanston, IL 60201 mark---math.nwu.edu AMS Classifications: 57R42, 55N20 Abstract: Recently Hopkins and Mahowald constructed a new 2-primary ring spectrum eo2, satisfying H^*(eo2)=A//A2. We use eo2 to obtain new results regarding nonimmersions of real projective spaces in Euclidean space. The method is to say enough about eo2-cohomology of a product of real projective spaces to obtain nonexistence of certain axial maps. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Chacholski-Dwyer-Intermont/complication The A-complication of a space W. Chacholski, W. G. Dwyer, and M. Intermont Suppose that A is a pointed CW-complex. We look at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complication of B. Studying the complication leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CW_A(X) of an arbitrary space X. Yale University, New Haven, CT 06520 USA University of Notre Dame, Notre Dame IN 46556 USA Kalamazoo College, Kalamazoo MI, 49006 USA MSC2000: 55P60, 55P99 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fors/AugmHom Title: Augmental Homology Theory and the Künneth Formula for Topological Joins. Author: Göran Fors. AMS Classification numbers: 55N10. Address: Department of Mathematics, University of Stockholm, SE-106 91 Stockholm, Sweden E-mail address: goranf---matematik.su.se We prove topological join versions of the relative Eilenberg-Zilber Theorem and the relative Künneth Formula. We also express the local homology groups for topological joins and products in terms the local homology groups for the factors. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbunov-Malikov-Schechtman/group-all-fedin1 On chiral differential operators over homogeneous spaces Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA;\ vgorb\---ms.uky.edu F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA;\ fmalikov\---mathj.usc.edu V.S.: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France;\ vadik\---ihes.fr The notion of an algebra of chiral differential operators (cdo for short) over a smooth algebraic variety X has been studied by the authors previously. We give a classification of cdo over X in the following cases: X=G is an affine algebraic group; X=G/N or G/P where N is a unipotent subgroup and P is a parabolic subgroup and G is simple (the extension to the case of a semisimple G being straightforward). The above sheaves are constructed using the BRST (or quantum Hamiltonian) reduction of the corresponding cdo's on G. The classification of cdo over homogeneous spaces is exactly reflected in the BRST world: namely the square of the corresponding BRST charge is zero at all levels for G/N, only at the critical level for G/B and is never zero for G/P. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ishiguro/G2 "Classifying spaces and a subgroup of the exceptional Lie group G_2" Kenshi Ishiguro Mathematics subject classification: 55R35 Abstract: We consider a problem on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group, as well as some related problems. A previously obtained necessary condition is shown to be not sufficient. Our counterexample is given by a quotient group \Gamma_2 of a subgroup of the exceptional Lie group G_2 at p=3. The 3-adic K-theory of B\Gamma_2 and BG_2 are isomorphic , though the loop space of the 3-completion of B\Gamma_2 is not a 3-compact group. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/quinn Title: The dualizing spectrum of a topological group Author: John R. Klein AMS subjclass: Primary: 55P91, 55N91, 55P42, 57P10. Secondary: 55P25, 20J05,18G15. Address: Dept. Of Mathematics, Wayne State University, Detroit, MI 48202 e-mail: klein---math.wayne.edu Abstract: To a topological group G, we assign a naive G-spectrum D_G, called the "dualizing spectrum" of G. When the classifying space BG is finitely dominated, we show that D_G detects Poincare duality in the sense that BG is a Poincare duality space if and only if D_G is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a "norm map" which is defined for any G and for any naive G-spectrum E. Applications include: (1) a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of finitely dominated spaces, the total space satisfies Poincare duality if and only if the base and fiber do. (2) An entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincare duality space. (3) A new proof of Browder's theorem that every finite H-space satisfies Poincare duality. (4) We show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. We prove a vanishing result for this theory. In an appendix, we identify the homotopy type of D_G for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. (This paper has already been accepted for publication in Math. Annalen.) 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/chains Cochain Multiplications Michael A. Mandell mandell---math.uchicago.edu Abstract We describe a refinement of the Eilenberg--Steenrod axioms that provides a necessary and sufficient condition for functors from spaces to algebras or E-infty algebras to be naturally quasi-isomorphic to the singular cochain functor. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert (This is a revised version of the author's paper proving the Hilbert-Smith conjecture about certain topological groups being forced to be Lie. The abstract has appeared at least twice before here, so I omit it). MH 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pakianathan-YalcinE/nc Title: On Commuting and Non-Commuting Complexes Authors: Jonathan Pakianathan and Erg\"un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 06A09, 05E25. Addresses: Department of Mathematics University of Rochester N.Y., U.S.A. Department of Mathematics Bilkent University Ankara, Turkey Abstract: In this paper we study various simplicial complexes associated to the commutative structure of a finite group $G$. We define $NC(G)$ (resp. $C(G)$) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets of nontrivial elements in $G$. We observe that $NC(G)$ has only one positive dimensional connected component, which we call $BNC(G)$, and we prove that $BNC(G)$ is simply connected. Our main result is a simplicial decomposition formula for $BNC(G)$ which follows from a result of A. Bj\"orner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary we obtain that if $G$ has a nontrivial center or if $G$ has odd order, then the homology group $H_{n-1}(BNC(G))$ is nontrivial for every $n$ such that $G$ has a maximal noncommuting set of order $n$. We discuss the duality between $NC(G)$ and $C(G)$, and between their $p$-local versions $NC_p(G)$ and $C_p(G)$. We observe that $C_p(G)$ is homotopy equivalent to the Quillen complexes $A_p(G)$, and obtain some interesting results for $NC_p(G)$ using this duality. Finally, we study the family of groups where the commutative relation is transitive, and show that in this case, $BNC(G)$ is shellable. As a consequence we derive some group theoretical formulas for the orders of maximal non-commuting sets. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YalcinE/clpg4 Title: Set Covering and Serre's Theorem on the Cohomology Algebra of a $p$-Group Author: Erg\" un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J06; Secondary: 20D15, 20D60, 51E20. Address: Department of Mathematics Bilkent University Ankara, Turkey Email: yalcine---math.mcmaster.ca Abstract: We define a group theoretical invariant, denoted by $s(G)$, as a solution of a certain set covering problem, and show that it is closely related to $chl(G)$, the cohomology length of a $p$-group $G$. By studying $s(G)$, we improve the known upper bounds for the cohomology length of a $p$-group, and determine $chl(G)$ completely for extra-special $2$-groups of real type. ---------------- Two new papers this time. Mark Hovey New papers appearing on hopf between 9/14/00 and 9/28/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lundell/stircenid1 Title: Stirling and Central Factorial Number Identies Author: Albert T. Lundell Address: Department of Mathematics, Box 395 University of Colorado Boulder, Colorado 80309 E-mail: lundell---euclid.colorado.edu This paper contains many identities related to Stirling numbers and central factorial numbers, with an emphasis toward divisibility properties. The paper is self-contained and contains proofs of the identities. There is a short section relating these numbers to the James numbers U(n,r), i.e., the index of p_*(\pi_{2n-1}(W_{n,r})\subset\pi_{2n-1}(S^{2n-1}), where p:W_{n,r}\arrow S^{2n-1} is the fibration of complex Stiefel manifolds. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Menichi/Free_Loop Title: The cohomology ring of free loop spaces Author: Luc Menichi AMS classification numbers: 55P35, 16E40, 55P62, 57T30, 55U10. address: Universite d'Angers Faculte des Sciences Departement de Mathematiques 2 Boulevard Lavoisier 49045 ANGERS Cedex 01 - FRANCE Luc.Menichi---univ-angers.fr Abstract: Let $X$ be a simply connected space and $\Bbbk$ a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space $HH^{*}S_*(\Omega X)$ is isomorphic to the free loop space cohomology $H^{*}(X^{S^{1}})$. We proved that this isomorphism is compatible with both the cup product on $HH^{*}S_*(\Omega X)$ and on $H^{*}(X^{S^{1}})$. In particular, we explicit the algebra $H^{*}(X^{S^{1}})$ when $X$ is a suspended space, a complex projective space or a finite CW-complex of dimension $p$ such that $\frac {1}{(p-1)!}\in {\Bbbk}$. --------------- Four new papers this time, all from some energetic guy named Greenlees. He maintains a bibliography on Hopf as well, under http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/greenleesbiblio Mark Hovey New papers appearing on hopf between 9/28/00 and 10/2/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/axiomatic Title: ``Tate cohomology in axiomatic stable homotopy theory.'' Author: J.P.C.Greenlees AMS classification numbers: 55U35, 55T99, 55P42, 55P91, 55N91 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: Any smashing localization in an axiomatic stable homotopy theory in the sense of Hovey-Palmieri-Strickland gives rise to a Tate theory. Various known versions of Tate cohomology (for example in commutative algebra, in the cohomology of groups, in equivariant homotopy theory and in chromatic stable homotopy theory) are considered from this point of view. Status: Submitted for publication. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/guanajuato Title: Local cohomology in equivariant topology Author: J.P.C.Greenlees AMS classification numbers: 13D45, 19L41, 20Jxx, 55N91, 55N22, 55P43 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: The article (based on talks at the Guanajuato Workshop on Local Cohomology, December 1999) describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather different families of examples. Both topology and commutative algebra benefit from the connection, and many interesting questions remain open. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees/so3q Title: Rational SO(3)-equivariant cohomology theories Author: J.P.C.Greenlees AMS classification numbers: 55N91, 55P42, 55P62, 55P91 Address: University of Sheffield, UK Email: j.greenlees---shef.ac.uk Abstract: The results of previous work for the circle and O(2) are used to give an explicit algebraic model of the category of rational SO(3)-spectra. This gives a complete classification of rational SO(3)-equivariant cohomology theories. A number of new features appear for the first time for this group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Greenlees-Hopkins-Rosu/ellT Title: Rational S^1-equivariant elliptic cohomology Authors:J.P.C.Greenlees, M.J.Hopkins and I.Rosu AMS Class numbers: 55N34, 55N91, 55P42, 55P62 \address{JPCG: Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.} \email{j.greenlees---sheffield.ac.uk} \address{MJH: Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.} \email{mjh---math.mit.edu} \address{IR: Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.} \email{ioanid---math.mit.edu} Abstract: We give a functorial construction of a rational $S^1$-equivariant cohomology theory from an elliptic curve equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an $S^1$-representation is given by the sheaf cohomology of a suitable line bundle on the curve. The construction is easy: by considering functions on the elliptic curve with specified poles one may write down the representing $S^1$-spectrum in the first author's algebraic model of rational $S^1$-spectra. ---------------- Ten new papers this time. Mark Hovey New papers appearing on hopf between 10/2/00 and 11/8/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Luptin-Murillo/SubgpsofHEs Title: Subgroups of the Group of Self-Homotopy Equivalences Authors: Martin Arkowitz, Gregory Lupton and Aniceto Murillo. Classification Nos. (1991): Primary 55P10; Secondary 55P62, 55Q05. Addresses: Department of Mathematics, Dartmouth College, Hanover NH 03755 U.S.A. Department of Mathematics, Cleveland State University, Cleveland OH 44115 U.S.A. Departmento de Algebra, Geometria y Topologia, Universidad de Malaga, Ap. 59, 29080 Malaga, Spain e-mail Addresses: Martin.Arkowitz---Dartmouth.edu Lupton---math.csuohio.edu Aniceto---agt.cie.uma.es Abstract: Denote by $\mathcal{E}(Y)$ the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex $Y$. We give a selection of results about certain subgroups of $\mathcal{E}(Y)$. We establish a connection between the Gottlieb groups of $Y$ and the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of $Y$, denoted by $\mathcal{E}_{\#}(Y)$. We give an upper bound for the solvability class of $\mathcal{E}_{\#}(Y)$ in terms of a cone decomposition of $Y$. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey---member.ams.org October, 2000 This is the final version, to appear in JPAA. There are several significant notational changes, and many minor corrections in this version. (Rest of abstract elided, since it has appeared twice already.) 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/torsext2 Cotorsion theories, model category structures, and representation theory by Mark Hovey mhovey---wesleyan.edu AMS Classification: 20C05,20J05,18E30,18G35, 55U35 We make a general study of Quillen model structures on abelian categories. Given a proper class P of short exact sequences on an abelian cateory A, we define what it means for a model structure to be compatible with P. We then give a complete characterization of model structures compatible with P. This characterization is in terms of cotorsion theories, which were introduced by Salce and have been much studied recently by Enochs and coauthors. We apply the general method to construct a stable category of $K[G]$-modules where $K$ is a principal ideal domain and $G$ is a finite group. This is a compactly generated triangulated category that generalizes the well-known stable category of $k[G]$-modules, where $k$ is a field. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/brown-II A Brown representability theorm via coherent functors Author: Henning Krause Address of Author: University of Bielefeld, Germany Email address of Author: henning---mathematik.uni-bielefeld.de Abstract: We discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-Shipley/telescope Title: A telescope comparison lemma for THH (to appear in Topology and its Applications) Authors: Mike Mandell and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Mike Mandell 5734 University Ave. Chicago, IL 60637 USA Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: mandell---math.uchicago.edu bshipley---math.purdue.edu Abstract: The usual telescope or sequential homotopy colimit construction of the underlying infinite loop space must be replaced for symmetric spectra by a homotopy colimit over the category of finite sets and injections. Here we show that for convergent symmetric spectra this modified homotopy colimit agrees with the usual telescope construction. This sharpens B\"okstedt's original lemma because no connectivity conditions are necessary here. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/2local `The stable homotopy category has a unique model at the prime 2' Stefan Schwede Fakultaet fuer Mathematik Universitaet Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de ABSTRACT: In a closed model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences. But passage to the homotopy category loses information and in general the `homotopy theory' can not be recovered from the homotopy category. We show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory, at least 2-locally. We prove a uniqueness theorem which says that there is only one model structure (up to so called Quillen equivalence) underlying the stable homotopy category of 2-local spectra. This theorem is a 2-local strenghtening of a result with B. Shipley, given in `A uniqueness theorem for stable homotopy theory', in that we use only the triangulated structure of the stable homotopy catgory. The earlier result with Shipley works integrally, but needs additional structure, namely the action of the ring of stable homotopy groups of spheres. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede-Shipley/unique Title: A uniqueness theorem for stable homotopy theory Authors: Stefan Schwede and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Stefan Schwede Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: schwede---mathematik.uni-bielefeld.de bshipley---math.purdue.edu Abstract: In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres, $\pi^s$. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of $\pi^s$. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space $QS^0$ to the endomorphism space is a weak equivalence. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/monoid.unique Title: Monoidal uniqueness of stable homotopy Author: Brooke Shipley AMS Classification numbers: 55U35 55P42 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley---math.purdue.edu Abstract: We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences produced here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, $\mathcal{W}$-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/rational.circle Title: An algebraic model for rational $S^1$-equivariant stable homotopy theory Author: Brooke Shipley AMS Classification numbers: 55P62 55P91 55P42 55N91 18E30 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley---math.purdue.edu Greenlees defined an abelian category $A$ whose derived category is equivalent to the rational $S^1$-equivariant stable homotopy category whose objects represent rational $S^1$-equivariant cohomology theories. We show that in fact the model category of differential graded objects in $A$ ($dgA$) models the whole rational $S^1$-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between $dgA$ and the model category of rational $S^1$-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The new ingredients here are certain Massey product calculations and the work on rational stable model categories from "Classification of stable model categories" and "Equivalences of monoidal model categories" with Schwede; see http://www.math.purdue.edu/~bshipley/ 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Tamanoi/orbifold Title: Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory Author: Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz Santa Cruz, CA 95064 Email: tamanoi---math.ucsc.edu Abstract: We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mutually commuting elements (of order powers of $p$) in the wreath product G~S_n in terms of corresponding numbers of G. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K-theories of symmetric products of a G-manifold M. AMS Classification Numbers: 55N20, 55N91, 57S17, 57D15, 20E22 --------------- Twelve new papers this time. Mark Hovey New papers appearing on hopf between 11/8/00 and 11/26/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dorabiala/transfer Abstract: The goal of this paper is to show that if a smooth fiber bundle has a compact Lie group as structure group, then the transfer map for the algebraic K-theory of spaces satisfies analogs of the Mackey Double coset formula and Feshbach's sum formula. We also prove a "cut and paste" formula for parametrized Reidemeister torsion. Wojtek Dorabiala 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/local Author: Stephen A. Mitchell Title: The algebraic K-theory spectrum of a 2-adic local field e-mail: mitchell---math.washington.edu We explicitly determine the homotopy type of the 2-completed algebraic K-theory spectrum KF, where F is an arbitrary finite extension of the 2-adic rational numbers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localhom Author: Stephen A. Mitchell Title: The mod 2 homology of the general linear group of a 2-adic local field e-mail: mitchell---math.washington.edu Let F be a finite extension of the 2-adic rational numbers. We compute the mod 2 homology of the general linear group GL(F) as a Hopf algebra over the Steenrod algebra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morton-Strickland/cost-hrk The Hopf Rings for KO and KU Dena S. Cowen Morton and Neil P. Strickland 55N15; 55P43 math.AT/0011125 Department of Mathematics Xavier University Cincinnati OH 45207 USA Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We compute the mod two homology Hopf rings of the spectra KO and KU. The spaces in these spectra are the infinite classical groups and their coset spaces, and their homology was first calculated in the Cartan seminars, but the Hopf ring structure was first determined in the second author's unpublished PhD thesis. The presentation given here serves as an introduction to the first author's much more intricate work on the connective spectrum bo. The Hopf ring viewpoint turns out to be very convenient for understanding the homological effect of various maps between classical groups and fibrations of their connective covers. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Saneblidze-Umble/SUpaper Title: A Diagonal on the Associahedra Authors: Samson Saneblidze and Ronald Umble MSC-class: 57T30; 55U10; 55N20; 55N10 xxx.LANL.gov: math.AT/0011065 Author's Addresses: A. Razmadze Mathematical Institute, M. Aleksidze St., 1, 380093 Tbilisi, Georgia Department of Mathemaitcs, Millersville Univ. of PA, Millersville, PA 17551 Author's e-mail addresses: sane---rmi.acnet.ge ron.umble---millersville.edu ABSTRACT: An associahedral set is a combinatorial object generated by Stasheff associahedra K_n and equipped with appropriate face and degeneracy operators. Associahedral sets are similar in many ways to simplicial or cubical sets. In this paper we give a formal definition of an associahedral set, discuss some naturally occurring examples and construct an explicit geometric diagonal \Delta :C_*(K_n) --> C_*(K_n) \otimes C_*(K_n) on the cellular chains C_*(K_n). The diagonal \Delta, which is analogous to the Alexander-Whitney diagonal on the simplices, gives rise to a diagonal on any associahedral set and leads immediately to an explicit diagonal on the A_\infty operad. As an application of this, we use the diagonal \Delta to define a tensor product in the A_\infty category. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-bce The BP cohomology of elementary Abelian groups Neil P. Strickland 20J06; 55N20; 14L05 math.AT/0011120 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk In this paper we study E^*BV_k, where E=BP is a cohomology theory with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n] (if m=0). We use ideas from the theory of multiple level structures, developed in earlier work of the author with John Greenlees. Our results apply when k is less than or equal to w=n+1-m. If k E^0BG is an isomorphism. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-csi Common subbundles and intersections of divisors Neil P. Strickland 55N20; 14L05; 14M15 math.AT/0011123 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that V_0\cap V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-fsfg Formal schemes and formal groups Neil P. Strickland 14L05; 55N22 math.AT/0011121 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We set up a framework for using algebraic geometry to study the generalised cohomology rings that occur in algebraic topology. This idea was probably first introduced by Quillen and it underlies much of our understanding of complex oriented cohomology theories, exemplified by the work of Morava. Most of the results have close and well-known analogues in the algebro-geometric literature, but with different definitions or technical assumptions that are often inconvenient for topological applications. We merely put everything together in a systematic and convenient way. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-ghd Gross-Hopkins duality Neil P. Strickland 55N20; 55P42; 20E18 math.AT/0011108 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk We give a new and simpler proof of a result of Hopkins and Gross relating Brown-Comenetz duality to Spanier-Whitehead duality in the K(n)-local stable homotopy category. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-kld K(n) local duality for finite groups and groupoids Neil P. Strickland 55P42; 55P60; 55R40 math.AT/0011109 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Included postscript file: st-kld.eps We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the usual inner product on the representation ring when n=1, and to the Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure. In many places we find it convenient to use groupoids rather than groups, and to assist with this we include a detailed treatment of the homotopy theory of groupoids. We also explain some striking formal similarities between our duality and Atiyah-Poincare duality for manifolds. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-pmm Products on MU-modules Neil P. Strickland 55T25 math.AT/0011122 Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK N.P.Strickland---sheffield.ac.uk Included postscript file: st-pmm.eps We use the new categories of spectra and MU-modules constructed by Elmendorf, Kriz, Mandell and May to get improved results about multiplicative structures on spectra such as P(n) and E(n), particularly in the case p=2. ---------------- There are so many new papers this time that I am breaking this post into at least 2 posts. 8 new papers have modification dates in December, and those are announced here. The January ones will be in the next message. Mark Hovey New papers appearing on hopf between 11/26/00 and 12/31/00 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/f4 A stable approach to an unstable homotopy spectral sequence Martin Bendersky Hunter College, CUNY, NY 10021 mbenders---shiva.hunter.cuny.edu Donald M. Davis Lehigh University, Bethlehem, PA 18015 dmd1---lehigh.edu AMS classification: 55T15, 55Q52 Abstract: Recently Bendersky and Thompson introduced a spectral sequence which, for many spaces X, converges to the v1-periodic homotopy groups of X. It is proved that the E2-term of this spectral sequence is given by Ext in the category of stable p-adic Adams modules of PK^1(X;Zphat)/im(psi^p). We compute this spectral sequence when p=2 and X is the exceptional Lie group F4, yielding as a new result the 2-primary v1-periodic homotopy groups of F4. Some new general results about convergence of this spectral sequence are also proved. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc---julian.uwo.ca hovey---member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category of chain complexes of objects of A has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Intermont-JohnsonM/ijxspace Model Structures on the Category of Ex-spaces Michele Intermont Mark W. Johnson Primary: 55R70, 55U35; Secondary: 55P91, 55U40 Department of Mathematics Kalamazoo College Kalamazoo, MI 49006 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 intermon---kzoo.edu johnson.295---nd.edu Abstract: This paper describes several model structures on the categories of ex-spaces and ex-$G$-spaces when $G$ is a compact Lie group. Two of these are of particular interest in that they have expected applications to the study of transfer maps and to parametrized spectra. These two structures are shown to coincide on the collection of Hurewicz fibrations, and an indication is also given, mainly via examples, of how they differ. The last two sections of this paper are mostly expository; they set forth the model category techniques needed to prove the main theorems. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kashiwabara-Wilson/kash-wil The Morava K-theory and Brown-Peterson cohomology of spaces related to BP Takuji Kashiwabara Institut Fourier, Universit\'{e} de Grenoble I, U.M.R. au C.N.R.S., B. P. 74, 38402 Saint-Martin-d'H\`{e}res CEDEX France Takuji.Kashiwabara---ujf-grenoble.fr W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu This is the "final" version of the paper. We calculate the Morava K-theory of the spaces in the Omega spectra for BP. They fit into an exotic array of short and long exact sequences of Hopf algebras. We apply this to calculate the p-adically completed Brown-Peterson cohomology, as well as all of the intermediary cohomology theories, E, of these spaces. We give two descriptions of the answer, both of which turn out to be surprisingly nice. One part of our first description is just the image in the E cohomology of the corresponding space in the Omega spectrum for BP, which is as big as it could possibly be and which we show how to calculate. The other part is just the E cohomology of several copies of Eilenberg-MacLane spaces, something which is already known. Our second description is inductive and gives us a new way of looking at the Brown-Peterson cohomology of Eilenberg-MacLane spaces. The Brown-Comenetz dual of BP shows up in our calculations and so we take up the study of this spectrum as well. It was already known that the Morava K-theory of the spaces in the Omega spectrum for the Brown-Comenetz dual of BP made it look like a product of Eilenberg-MacLane spaces and we find, somewhat to our surprise, that the same is true for the BP cohomology. In order to state our answers we set up the foundations for the category of completed Hopf algebras. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/mandell-taq Topological Andre-Quillen Cohomology and E-infty Andre-Quillen Cohomology Michael A. Mandell mandell---math.uchicago.edu Abstract This paper compares Andre-Quillen cohomology in various categories of E-infty rings. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Piriou-Schwartz/schwartz La filtration du degre sur la cohomologie modulo 2 des 2-groupes abeliens elementaires Laurent Piriou Université de Nantes, Département de mathématiques 2 rue de la Houssinière BP 92208 Nantes Cedex 03 France laurent.piriou---math.univ-nantes.fr Lionel Schwartz Université Paris 13 Institut Galilée LAGA UMR 7539 du CNRS Av. J. B. Clément 93430 Villetaneuse France schwartz---math.univ-paris13.fr Code AMS 55S10 This article considers two filtrations on the mod-$2$ cohomology $H^*E$ of an abelian $2$-groups $E$. The first one is the primitive fitration, recall that $H^*E$ is a Hopf algebra. The second one is a kind of socle or Loewy filtration of $H^*E$ as unstable module. If dimension of $E$ is $1$ the two filtrations are the same, if the dimension is larger than $2$ it is shown that the filtration are, in some sense compatible. There is an analogous statement in ${\cal F}$, the category of functors from the category of finite dimensional ${\bf F}_2$-vector spaces to the category of all ${\bf F}_2$-vector spaces, for the functor $V \mapsto {\rm map}({\rm Hom}(V,E),{\bf F}_2)$. However, it is better to work with unstable modules because the Steenrod algebra allows computation on certain classes, that are central in the proof, given by the representation theory of symmetric groups that are central in the proof. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scherer-Thevenaz/simplegroups Finite simple groups and localization Jose L. Rodriguez, Jerome Scherer and Jacques Thevenaz 20D06, 20D08, 55P60 Departamento de Geometria, Topologia y Quimica Organica Universidad de Almeria E--04120 Almeria Spain Institut de Mathematiques Universite de Lausanne CH--1015 Lausanne Switzerland jlrodri---ual.es, jerome.scherer---ima.unil.ch, jacques.thevenaz---ima.unil.ch The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Weibel/Homotopyends-R TITLE: Homotopy Ends and Thomason Model Categories AUTHOR: Chuck Weibel weibel---math.rutgers.edu AUTHOR ADDRESS: Math. Dept. Rutgers University New Brunswick, NJ 08903 USA AMS CLASSIFICATION: Primary 55U35; Secondary 18F20, 55P05, 55Q05 ABSTRACT: In the last year of his life, Bob Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. In this paper we explain and prove Thomason's results, based on his private notebooks. The first half presents Thomason's ideas about homotopy ends and its generalizations. This material may be of independent interest. Then we define Thomason model categories and give some examples. The usual proof shows that the homotopy category exists. In the last two sections we prove the main theorem: functor categories inherit a Thomason model structure, at least when the original category is enriched over simplicial sets and fibrations are preserved by limits. These are the January papers, of which there are 13. Mark Hovey New papers appearing on hopf between 1/1/01 and 2/3/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker/regquotients On the homology of regular quotients Andrew Baker Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker---maths.gla.ac.uk We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. We easily deduce that for $s>1$, the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker-Lazarev/Rmod-ASS On the Adams Spectral Sequence for $R$-modules Andrew Baker \& Andrej Lazarev Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker---maths.gla.ac.uk Department of Mathematics, University of Bristol, Bristol BS8 1TW, England. A.Lazarev---bris.ac.uk We consider the Adams Spectral Sequence for $R$-modules based on commutative localized regular quotient ring spectra of a commutative $S$-algebra $R$ in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case, and we reduce to algebra involving the cohomology of certain `brave new Hopf algebroids' $E^R_*E$. In order to work out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an $R$ ring spectrum. We show that the Adams Spectral Sequence for $S_R$ based on $E=R/I[X^{-1}]$ converges to the homotopy of the $E$-nilpotent completion which has homotopy \[ \pi_*\hat{\mathrm{L}}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}. \] We also show that $\hat{\mathrm{L}}^R_ES_R$ is equivalent to $\L^R_ES_R$, the Bousfield localization of $S_R$ with respect to $E$-theory. This seems surprising since the spectral sequence collapses at $\E_2$, but $\E_r$ does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree, thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve a construction of the internal $I$-adic tower \[ R/I\la R/I^2\la\cdots\la R/I^s\la R/I^{s+1}\la\cdots \] whose homotopy limit is $\hat{\mathrm{L}}^R_ES_R$. Finally, we describe some examples for the case $R=MU$. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Greenlees/kubg The Connective K-theory of Finite Groups Robert Bruner and John Greenlees MSC2000: Primary 19L41, 19L47, 19L64, 55N15. Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30. Department of Mathematics, School of Mathematics and Statistics, Wayne State University, Hicks Building, Detroit MI 48202-3489, Sheffield S3 7RH, USA. UK. rrb---math.wayne.edu, j.greenlees---sheffield.ac.uk Included graphics files: AdamsA4.eps AdamsBip.eps AdamsC2.eps AdamsC4.eps AdamsC5.eps AdamsD8.eps AdamsQ8.eps AdamsSl23.eps AdamsV2.eps AdamsX.eps ExtIE.eps Extku.eps Extl.eps L.eps Qrank4.eps Qrank4lc.eps T3rank6.eps T3rank6lc.eps Xku.eps rank8.eps string.eps tku2.eps Abstract: This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen, we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. The variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, the interest lying in the way these parts fit together. The main technical obstacle is that the Kunneth spectral sequence does not collapse, so we have to show that it collapses up to isomorphism of varieties. In Chapter 2 we give several families of new complete and explicit calculations of the ring ku^*(BG). In Chapter 3 we consider the associated homology ku_*(BG), as a module over ku^*(BG) by using the local cohomology spectral sequence. This gives new specific calculations, but also illuminating structural information, including remarkable duality properties. Finally, in Chapter 4 we make a particular study of elementary abelian groups V. Despite the group-theoretic simplicity of V, the detailed calculation of ku^*(BV) and ku_*(BV) exposes a very intricate structure, and gives a striking illustration of our methods. Unlike earlier work, our description is natural for the action of GL(V). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/JohnsonM/shfloop Loop Spaces as Sheaves: A Sheaf-Theoretic View of Loop Spaces Mark W. Johnson \address {Department of Mathematics\\ University of Notre Dame\\ Notre Dame, IN 46556} \email{johnson.295---nd.edu} The context of enriched sheaf theory introduced in \cite{thesis} provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current document is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as $n$-fold loop spaces and pointed $G$-spaces. This language also includes the diagram categories of spectra from \cite{MMSS} as well as spectra in the sense of \cite{Lewis}, which will be discussed in some detail. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision Title: Excision for simplicial sheaves on the Stein site and Gromov's Oka Principle Author: Finnur Larusson AMS classification numbers: Primary: 32Q28; secondary: 18F10, 18F20, 18G30, 18G55, 32E10, 32H02, 55U35 arXiv:math.CV/0101103 Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada larusson---uwo.ca ABSTRACT: A complex manifold $X$ satisfies the Oka-Grauert property if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak equivalence for every Stein manifold $S$, where the spaces of holomorphic and continuous maps from $S$ to $X$ are given the compact-open topology. Gromov's Oka principle states that if $X$ has a spray, then it has the Oka-Grauert property. The purpose of this paper is to investigate the Oka-Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka-Grauert property is equivalent to $X$ representing a finite homotopy sheaf on the Stein site. This expresses the Oka-Grauert property in purely holomorphic terms, without reference to continuous maps. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert This is another revised version of the proof of the Hilbert-Smith conjecture. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Looptan Title: The equivariant tangent bundle of a free smooth loopspace Author: Jack Morava AMS classification: 58Dxx; 53C29, 55P91 Address: The Johns Hopkins Uniperversity e-mail: jack---math.jhu.edu ABSTRACT: The space of free loops on a manifold X inherits an action of the circle group \T. A Riemannian metric on X defines an equivariant isomorphism of the complexified tangent bundle of the loopspace with \bT X \otimes (\oplus \C(n)), where \C(n) is the standard one-dimensional representation of \T, and \bT X \otimes \C is an equivariant bundle on the loopspace, nonequivariantly isomorphic to the pullback of the complexified tangent bundle of X along evaluation at the basepoint. On a flat manifold, this analogue of Fourier analysis is quite familiar. [Perhaps this is all nonsense; if so, please let me know.] 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity5 Title: Pretty Good Gravity Author: Jack Morava AMS Classification: 19Dxx, 57Rxx, 83Cxx (not yet on xxx, but will be soon) Address: Dept. of Mathematics, the Johns Hopkins Uniperversity e-mail address: jack---math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. [This is a writeup of a talk at the RIMS Symposium on algebraic geometry and integrable systems related to string theory, June 12-16, 2000.] 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Tate2MU Title: Duality of Tate cohomology of framed circle actions Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: Abstract: The complex Mahowald pro-spectrum \CP^{\infty}_{-\infty} is not, as might seem at first sight, Spanier-Whitehead self-dual; rather, its S-dual is its own double suspension. This assertion makes better sense as a claim about the Tate cohomology spectrum t_{\T}S^0 defined by circle actions on framed manifolds. A subtle twist in some duality properties of infinite-dimensional projective space results, which has consequences [via work of Madsen and Tillmann] for the Virasoro symmetries [discovered by Witten and Kontsevich] of the stable cohomology of the Riemann moduli space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moreno/moreno Author: Guillermo Moreno Title: Alternative elements in the Cayley--Dickson algebras We describe the alternative elements in the Cayley-Dickson algebras for n>3. Also we ``measure'' the failure of these algebras of being a normed algebra in terms of the alternative elements. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk-Schwede-Shipley/simplicial Title: Simplicial structures on model categories and functors Authors: Charles Rezk, Stefan Schwede, Brooke Shipley To appear in American Journal of Mathematics Institute for Advanced Study School of Mathematics Olden Lane Princeton, NJ 08540, USA rezk---ias.edu Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany schwede---mathematik.uni-bielefeld.de Department of Mathematics Purdue University West Lafayette, IN 47907, USA bshipley---math.purdue.edu We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/ks-hgr The homotopy groups $\pi_*(L_nT(m)\wedge V(n-2))$ Katsumi Shimomura Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520 Japan katsumi---math.kochi-u.ac.jp Let $V_{T(m)}(n)$ denote the spectrum such that $BP_*(V_{T(m)}(n))=BP_*/I_{n+1}[t_1,\dots, t_m]$ for the ideal $I_{n+1}=(p,v_1,\dots, v_{n})$. In the title, we write $T(m)\wedge V(n-2)$ as $V_{T(m)}(n-2)$. Ravenel determined the structure of the Adams-Novikov $E_2$-term for the homotopy groups $\pi_*(L_nV_{T(m)}(n-1))$ for $n\le m+2$ and $n3$. Here are the February papers on Hopf, of which there are 9. So far this "monster snowstorm" hasn't amounted to much, but the real action is supposed to be tonight and tomorrow. Mark Hovey New papers appearing on hopf between 2/3/01 and 3/5/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Clarke-Crossley-Whitehouse/KKbases Bases for cooperations in $K$-theory Francis Clarke, M. D. Crossley and Sarah Whitehouse Primary: 55S25; % K-theory operations and generalized cohomology operations Secondary: 19L64, % Computations, geometric applications 11B65. % Binomial coefficients; factorials; q-identities Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, Wales Laboratoire de G\'eom\'etrie-Alg\`ebre, Universit\'{e} d'Artois, 62307 Lens, France F.Clarke---Swansea.ac.uk M.D.Crossley---Swansea.ac.uk whitehouse---euler.univ-artois.fr Gaussian polynomials are used to define bases with good multiplicative properties for the algebra $K_{*}(K)$ of cooperations in $K$-theory and for the invariants under conjugation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Devoto/elbg-disc Title of Paper: On the elliptic cohomology of the classifying space of discrete groups Author: Jorge A. Devoto AMS Classification: 20J06, 55N34 Addresses of authors: Dept.\ de Matem\'aticas, ITBA, Av. E. Madero 399, Buenos Aires, Argentina and Dept.\ de Matem\'aticas, FCEN, Ciudad Univ. (1428) Buenos Aires, Argentina e-mail: jdevoto---itba.edu.ar We study, for $\Gamma$ a discrete group of finite virtual cohomological dimension, the elliptic cohomology of the classifying space $B\Gamma$. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision Title: Excision for simplicial sheaves on the Stein site and Gromov's Oka Principle Author: Finnur Larusson This is an updated version of a paper announced last month, with the same abstract, so the abstract is omitted. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McClure-SmithJH/deligne-conj (This is also an updated version, but the previous version was announced in 10/99, so I include the abstract). A solution of Deligne's Hochschild cohomology conjecture. James E. McClure and Jeffrey H. Smith ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action by an operad that is equivalent to the little 2-cubes operad. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/bertin AUTHOR: Mara D. Neusel TITLE: The Transfer in the Invariant Theory of Modular Permutation Representations (Trente Ans Apr\`es) Pacific Journal of Mathematics -- to appear -- This note investigates the image of the transfer homomorphism for permutation representations of finite groups over finite fields. One obtains a number of results showing that the image of the transfer $\Im (\Tr)$ together with certain Chern classes generate the ring of invariants as an algebra. By a careful analysis of orbit sums one finds the surprising fact that the ideal $\Im (\Tr)$ is a prime ideal for cyclic $p$-groups and determines an upper bound on its height. AMS CODE: 13A50 Invariant Theory KEY WORDS: Polynomial Invariants of Finite Groups, Permutation Representation, Transfer, $p$-Regular Representation neusel.1---nd.edu 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/bertin2 AUTHOR: Mara D. Neusel TITLE: The Transfer in the Invariant Theory of Modular Permutation Representations II (Trente Ans Apr\`es, Bis) Canadian Mathematical Bulletin -- to appear -- In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals. AMS CODE: 13A50 Invariant Theory KEY WORDS: Polynomial Invariants of Finite Groups, Permutation Representation, Transfer neusel.1---nd.edu 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/kokusu AUTHOR: Mara D. Neusel TITLE: The Lasker-Noether Theorem in the Category $U(\H^*)$ (denizin kokusu) Journal of Pure and Applied Algebra -- to appear -- We prove the Lasker-Noether Theorem in the category $U(\H^*)$ of unstable $\H^*\odot \P^*$-modules. Along the way, we generalize Lam's $\J$-functor to the context of modules. AMS CODE: 55S10 Steenrod Algebra, 13A50 Invariant Theory, 13XX Commutative Rings and Algebras, 55XX Algebraic Topology KEY WORDS: Lasker-Noether Theorem, Unstable Modules, Steenrod Algebra, Dickson Algebra, Polynomial Invariants of Finite Groups neusel.1---nd.edu 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/strassen AUTHOR: Mara D. Neusel TITLE: Lots of Degree Bounds or On the Use of the T-Functor in Invariant Theory We introduce a new method employing J. Lannes's $T$-functor to describe homological properties of rings of invariants. We illustrate the power of this method by applying it to the calculation of degree bounds. We find seven bounds: two for special families of representations, two relative bounds, two general degree bounds and a general bound for $p$-groups. AMS CODE: 13A50 Invariant Theory, 55S10 Steenrod Algebra, 55XX Algebraic Topology KEY WORDS: Invariant Theory of Finite Groups, Degree Bounds, $T$-Functor, Integral Closure, $P^*$-inseparable Closure, Cohen-Macaulay, Gorenstein, Depth, Modular Invariant Theory neusel.1---nd.edu 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/uncoma AUTHOR: Mara D. Neusel TITLE: Unstable Cohen--Macaulay Algebras Mathematical Research Letters -- to appear -- We characterize Cohen--Macaulay algebras in the category $K_{fg}$ of unstable Noetherian algebras over the Steenrod algebra via the depth of the $P^*$-invariant ideals. This allows us to transfer the Cohen--Macaulay property to suitable subalgebras. We apply this to rings of invariants of finite groups and to the $P^*$-inseparable closure. AMS CODE: 55S10 Steenrod Algebra, 13XX Commutative Rings and Algebras, 55XX Algebraic Topology} KEY WORDS: Steenrod Algebra, Cohen--Macaulay Algebras, Unstable Algebras, $P^*$-Invariant Prime Ideal Spectrum, $P^*$-Inseparable Closure, Polynomial Invariants of Finite Groups neusel.1---nd.edu The re-organization of Hopf threw me off somewhat, so I might have missed a paper. Let me know if you think I missed yours. Mark Hovey New papers appearing on hopf between 3/5/01 and 5/16/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Aguade-Ruiz/mapsBKtoBK Maps between classifying spaces of Kac-Moody groups by Jaume Aguad\'e and Albert Ru\'iz (aguade---mat.uab.es, cirera---mat.uab.es) Kac-Moody groups are an important generalisation of Lie groups. Roughly speaking, they are like "Lie groups with infinite Weyl groups". Let K be the unitary form of a Kac-Moody group of rank two. In this paper we determine the self maps of BK. Contents: 1. Introduction. 2. Rank two Kac-Moody groups. 3. Relations between global and local maps. 4. Maps into BK^p and representations. 5. Admissible matrices. 6. Groups with the same classifying space. 7. Adams maps. 8. Homotopically trivial self maps. 9. Detecting maps on the maximal torus. 10. [BK,BK]. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Costenoble-May-Waner/CMWfinal Equivariant orientation theory by S.R. Costenoble, J.P. May, and S. Waner subjclass: Primary 55P91; Secondary 18B40, 20L15, 55N25, 55N91, 55P20, 55R91, 57Q91, 57R91 Hofstra University, University of Chicago, and Hofstra University Steven.R.Costenoble---Hofstra.edu, may---uchicago.edu, matszw---hofstra.edu We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal target category of G-vector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/bdi4 (This is a new version of an paper previously announced). ON THE 2-COMPACT GROUP DI(4) Author: D. Notbohm Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group $DI(4)$. The mod-2 cohomology of the associated classifying space $BDI(4)$ realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space $BDI(4)$ and that the maximal torus normalizer determines the isomorphism type of $DI(4)$ as 2-compact group. We also calculate the set of homotopy classes of self maps of $BDI(4)$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Notbohm/orthogonal (This is a new version of a paper previously announced). A UNIQUENESS RESULT FOR ORTHOGONAL GROUPS AS 2-COMPACT GROUPS D. Notbohm Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizer are isomorphic. It is conjectured that this result generalizes to \pcg s. Here, we prove the generalization for orthogonal groups $O(n)$, the special orthogonal groups $SO(2k+1)$ and the spinor groups $Spin(2k+1)$ considered as 2-compact groups. There are 7 new papers this time. This is a good time to remind you that people decide whether to download your paper based on your abstract. It is therefore crucial that there be an abstract and that it be readable by humans. It is not enough to just e-mail Clarence a dvi file; you must also e-mail him an abstract, under separate cover, with minimal TeX symbols. Mark Hovey New papers appearing on hopf between 5/16/01 and 6/1/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Kitchloo/BrKi Classifying spaces of Kac-Moody groups Carles Broto and Nitu Kitchloo broto---mat.uab.es nitu---math.nwu.edu We study the structure of classifying spaces of Kac-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes' T-functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kac-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kac-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kac-Moody groups, and centralizers of finite p-subgroups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative (This is the final version, to appear in Math Proc Camb Phil Soc) Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc---julian.uwo.ca hovey---member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exa