There are 7 new papers this time. This is a good time to remind you that people decide whether to download your paper based on your abstract. It is therefore crucial that there be an abstract and that it be readable by humans. It is not enough to just e-mail Clarence a dvi file; you must also e-mail him an abstract, under separate cover, with minimal TeX symbols. Mark Hovey New papers appearing on hopf between 5/16/01 and 6/1/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Kitchloo/BrKi Classifying spaces of Kac-Moody groups Carles Broto and Nitu Kitchloo broto@mat.uab.es nitu@math.nwu.edu We study the structure of classifying spaces of Kac-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes' T-functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kac-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kac-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kac-Moody groups, and centralizers of finite p-subgroups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative (This is the final version, to appear in Math Proc Camb Phil Soc) Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc@julian.uwo.ca hovey@member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category of chain complexes of objects of A has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/hopfalgebroids Morita theory for Hopf algebroids and presheaves of groupoids Mark Hovey Wesleyan University Middletown, CT mhovey@wesleyan.edu 5/17/01 AMS classification nos: 14L05, 14L15, 16W30, 18F20, 18G15, 55N22 Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel change of rings theorems in algebraic topology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/ainf Author: Andrey Lazarev Title: Spaces of multiplicative maps between highly structured ring spectra. We uncover a somewhat unexpected connection between spaces of multiplicative maps between A-infinity ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory MU canonically lifts to an A-infinity map MU-->MU. This implies, in particular, that the Brown-Peterson spectrum BP splits off MU as an A-infinity ring spectrum. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/tower Towers of MU-algebras and the generalized Hopkins-Miller theorem Author: A.Lazarev Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK. email A.Lazarev@bristol.ac.uk AMS classification number 55N22 Our results are of three types. First we describe a general procedure of adjoining polynomial variables to A-infinity-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of MU, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable K(n)-localizations of such spectra. This generalizes the famous Hopkins-Miller theorem and gives strengthened versions of various splitting theorems. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localb The algebraic K-theory spectrum of a 2-adic local field by Stephen A. Mitchell mitchell@math.washington.edu (There was no abstract with this paper, so I made one up. If you don't like it, Steve, send in one!) A local field F of characteristic 0 is a finite extension of the L-adic rationals of finite degree d, where L is a prime. When L is odd, Dwyer and the author determined the homotopy type of the etale K-theory spectrum of F, but their methods fail when L=2 and -1 is not a square in F. The purpose of this paper is to study this remaining case. The recent work on the Lichtenbaum-Quillen conjecture at 2 by Rognes and Weibel allows the author to get from the etale K-theory of F to the 2-adic completion of the algebraic K-theory of F. The result essentially says that, rather than a splitting as you get in the odd primary case, there is some room for a few non-trivial extensions (which are completely determined). This is a generalization of Rognes' calculation of the 2-adic K-theory of the 2-adic rationals. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YauD/catcocat Title: Clapp-Puppe Type Lusternik-Schnirelmann (Co)category in a Model Category Donald Yau AMS Classification: Primary 55M30; Secondary 55P30, 55U35 math.AT/0104267 Department of Mathematics MIT, 2-230 77 Massachusetts Avenue Cambridge, MA 02139 USA donald@math.mit.edu We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given.