There are 8 new papers this time, from BauerT, Blanc-Markl, Casacuberta-Gutierrez, Dugger-Hollander-Isaksen, Dugger-Isaksen, Maltsiniotis, Toen-Vezzosi, and ZhengQb. Note that papers sent by e-mail take much longer to appear on the archive than papers submitted by ftp. If ftp is an option, it will be quicker for you and make Clarence's life much easier if you use it. Mark Hovey New papers appearing on hopf between 06/29/02 and 07/18/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BauerT/pcfm Title: p-compact groups as framed manifolds author: Tilman Bauer Address: Department of Mathematics, Rm. 2-492, Massachusetts Institute of Technology, Cambridge (MA) 02139 E-mail: tilman@mit.edu We describe a natural way to associate to any p-compact group an element of the p-local stable stems, which, applied to the p-completion of a compact Lie group G, coincides with the element represented by the manifold G with its left-invariant framing. To this end, we construct a d-dimensional sphere SG with a stable G- action for every d-dimensional p-compact group G, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with SG . 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Markl/blanc-markl Title: Higher Homotopy Operations Authors: David Blanc and Martin Markl Posted to xxx.lanl.gov as math.AT/0207082 DB: Dept. of Mathematics, Univ. of Haifa, 31905 Haifa, Israel blanc@math.haifa.ac.il MM: Mathematical Inst. of the Academy, Zitna, 115 67 Prague 1, Czech Republic markl@math.cas.cz Abstract: We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Casacuberta-Gutierrez/hloc_modspc Title of Paper: Homotopy Localizations of Module Spectra Authors: Carles Casacuberta and Javier J. Gutierrez AMS Classification numbers: 55P42, 55P43, 55P60. Adresses of Authors: Carles Casacuberta Departament d'Algebra i Geometria Universitat de Barcelona, Gran Via 585 E-08007 Barcelona, Spain Javier J. Gutierrez Departament de Matematiques Universitat Autonoma de Barcelona E-08193 Bellaterra, Spain e-mail: casac@mat.ub.es jgutierr@mat.uab.es Text of Abstract: We prove that stable homotopical localizations preserve ring spectrum structures and module spectrum structures under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-Mac Lane spectrum HZ. More generally, we describe the main features of localizations of HZ-modules (i.e., stable GEMs), motivated by similar results in unstable homotopy. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Hollander-Isaksen/hypspre Title: Hypercovers and simplicial presheaves Authors: Daniel Dugger Sharon Hollander Daniel C. Isaksen AMS subject classification: 55U35, 18F20 Addresses: Department of Mathematics, Purdue University ddugger@math.purdue.edu Department of Mathematics, University of Chicago sjh@math.uchicago.edu Department of Mathematics, University of Notre Dame isaksen.1@nd.edu Abstract: We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects can be explicitly identified in terms of a hypercover descent condition. Another is a very simple approach to change-of-site functors. In an appendix, we discuss how this hypercover localization compares to the more naive process of localizing at the Cech complexes; the two are not the same in general, but agree in some cases of interest. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/wesp Title: Weak equivalences of simplicial presheaves Authors: Daniel Dugger Daniel C. Isaksen AMS subject classification: 55U35, 18F20 Addresses: Department of Mathematics, Purdue University ddugger@math.purdue.edu Department of Mathematics, University of Notre Dame isaksen.1@nd.edu Abstract: The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic results about the local homotopy theory of simplicial presheaves. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Maltsiniotis/Groth-homot-th Title: La théorie de l'homotopie de Grothendieck Authors: G. Maltsiniotis, with two appendices by D.-C. Cisinski AMS Classification Numbers: 18F20, 18G30, 18G50, 18G55, 55P10, 55P15, 55P60 Addresses: Université Paris 7 Denis Diderot Case Postale 7012 2, place Jussieu F-75251 PARIS CEDEX 05 Email addresses: maltsin@math.jussieu.fr cisinski@math.jussieu.fr Abstract: This paper is an introduction to the homotopy theory of Grothendieck as developed in "Pursuing Stacks". The aim is to study "Elementary modelizers" i.e. presheaf categories modelizing the homotopy types, thus generalizing the theory of simplicial sets. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Toen-Vezzosi/agmod-I-fin-web Title: Homotopical Algebraic Geometry I: Topos theory Authors: Bertrand Toen and Gabriele Vezzosi AMS Classification: 14A20; 18G55; 55P43; 55U40; 18F10. Submitted to the xxx.lanl archive as math.AG/0207028 Addresses: Bertrand Toen, Laboratoire J. A. Dieudonn\'e, UMR CNRS 6621, Universit\'e de Nice Sophia-Antipolis, France; Gabriele Vezzosi, Dipartimento di Matematica, Universit\`a di Bologna, Italy. E-mail addresses: toen@math.unice.fr vezzosi@dm.unibo.it ABSTRACT: This is the first of a series of papers devoted to the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this paper we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of $\infty$-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove that for an S-site T, there is a model category of stacks over T, generalizing Joyal-Jardine structure on simplicial presheaves on a Grothendieck site. We also shows, as an analog of the relation between topologies and localizing subcategories of the categories of presheaves, that there is a bijection between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Then we study the notion of model topos due to C. Rezk, and relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that Dwyer-Kan simplicial localization provides a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites. As an application, we propose a definition of \'etale K-theory of ring spectra. An appendix gives an alternative approach to the theory which uses Segal categories. We define Segal topologies, Segal sites, stacks over Segal sites and Segal topoi. The existence of internal Hom's in this context allows us to define the Segal category of geometric morphisms between Segal topoi. An application to the reconstuction of a space via its Segal category of stacks is given. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhengQb/extgroup Title of the Paper: A Subspace of Ext$_A(Z_p,Z_p)$ Author: Zheng Qibing AMS Classification Number: 55 18G Address of Author Zheng Qibing Department of Mathematics Nankai University Tianjin, 300071, P.R.China Email Address of Author: zhengqb@eyou.com Abstract In this paper, we compute the cohomology of some Hopf algebras and find a subspace of the cohomology of the Steenrod algebra that includes the representative of the Greek letter families.