. 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 BPshows 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 BPmade 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/sim plegroups 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. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://www.math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your WWW client (Netscape< Internet Explorer) to the URL listed. 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