Subject: new Hopf listings Date: 07 Oct 2002 08:36:40 +0100 From: nsthov01@newton.cam.ac.uk (M.A. Hovey) To: dmd1@lehigh.edu The web form for submission to Hopf is a big success! So much so that I will have to send out these letters more frequently. There are 17 new papers this time! So I will break this letter up into two parts. This first part contains 9 new papers this time, from Anton, Bendersky-DavisD, Blanc-Peschke, 3 from Cisinski, Devinatz, Ferland-Lewis, and Hovey. Mark Hovey New papers appearing on hopf between 09/11/02 and 10/07/02, part 1 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/morava Title of Paper: On Morava K-theories of an S-arithmetic group Author: Marian F. Anton AMS Classification numbers: 55N20,19F27,11F75 Address of Author: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK Email address of Author: Marian.Anton@imar.ro Text of Abstract: We completely describe the Morava K-theories with respect to the prime p for the etale model of the classifying space of the general linear group GL(m) over the ring Z[u,1/p] when p is an odd regular prime and u a primitive p-th root of unity. For p=3 and m=2 (and conjecturally in the stable range) these K-theories are the same as those of the classifying space itself. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/SON v1-periodic homotopy groups of SO(n) Martin Bendersky and Donald M. Davis 55Q52, 55T15, 57T20 Hunter College, CUNY, NY, NY 10021 Lehigh University, Bethlehem, PA 18015 Abstract We compute the 2-primary v1-periodic homotopy groups of the special orthogonal groups SO(n). The method is to calculate the Bendersky-Thompson spectral sequence, a K*-based unstable homotopy spectral sequence, of Spin(n). The E2-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres. The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly [log_2(2n/3)] copies of Z/2. As the spectral sequence converges to the v1-periodic homotopy groups of the K-completion of a space, one important part of the proof is that the natural map from Spin(n) to its K-completion induces an isomorphism in v1-periodic homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Peschke/BlancPeschke1 Authors: David Blanc and George Peschke Title: The plus construction, Postnikov towers and universal central module extensions. Given a connected space $X$, we consider the effect of Quillen's plus construction on the homotopy groups of $X$ in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequence \ $AX\to X\to X^+$, \ we explain the contribution of \ $\pi_nX$ \ to \ $\pi_nX^+$, \ $\pi_{n+1}X^+$ \ and \ $\pi_nAX$, \ $\pi_{n+1}AX$ \ explicitly in terms of the low dimensional homology of $\pi_nX$ regarded as a module over $\pi_1X$. \ Key ingredients developed here for this purpose are universal $\Pi$-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/der2 Images directes cohomologiques dans les categories de modeles Denis-Charles Cisinski AMS Classification numbers 55U3, 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski@math.jussieu.fr Abstract Show that every complete model category M admits homotopy limits, and more generaly that every functor between small categories has a cohomological direct image in M (that is a homotopy right Kan extension). Furthermore, we study the local behavor of such constructions. For this purpose, we introduce Grothendieck's notion of derivator. Derivators correspond to the intuition of ``a homotopy complete category'' without speaking about models. Forthcoming papers will show that this setting is rich enough to define classical homotopy theory by a simple universal property. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/propuni Proprietes universelles et extensions de Kan derivees Denis-Charles Cisinski AMS Classification numbers 55U3, 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski@math.jussieu.fr Abstract We show that for all small category A, the derivator associated to the homotopy theory of presheaves in categories (or in simplicial sets) on A is the solution of a universal problem (and a similar statement about the pointed versions of such derivators is proved). When A is the final category, this shows that the derivator HOT associated to the classical homotopy theory is canonically endowed with a monoidal structure, and that every derivator admit a canonical action of HOT. As every model category defines a derivator, Hovey's homotopy coherence conjectures are then a consequence of these constructions. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/winfax Le localisateur fondamental minimal Denis-Charles Cisinski AMS Classification numbers 55U40 Institut de Mathématiques de Jussieu Université Paris 7 Case 7012 2 place Jussieu 75251 Paris cedex 05 France E-mail: cisinski@math.jussieu.fr Abstract Basic localizors were introduced by Grothendieck in Pursuing Stacks. These are classes of arrows in the category Cat of small categories satisfying nice properties of descent (like Quillen's theorem A).For example, every cohomology theory defines a basic localizor. In particular, classical weak equivalences (i.e. those induced from the simplicial weak equivalences from th nerve functor) form a basic localizor. In this paper, we show Grothendieck's conjecture that Cat's usual weak equivalences are the smallest basic localizor. This gives in particular a combinatorial/algebraic way to define classical homotopy theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/LHSspectral Title: A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra Author: Ethan S. Devinatz AMS Subject Classification: 55N20, 55P43, 55T15 Address: Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195 e-mail: devinatz@math.washington.edu Abstract: Let H and K be closed subgroups of the n th Morava stabilizer group with H normal in K. We construct a spectral sequence of the expected form connecting the homotopy of the continuous homotopy H fixed points of the Landweber exact spectrum E_n with the homotopy of the continuous K fixed points of E_n. These continuous homotopy fixed point spectra are the spectra constructed by Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in an appropriate category of module spectra. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Ferland-Lewis/FerlandLewis Title: The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G = \mathbb{Z}/p$ Authors: Kevin K. Ferland and L. Gaunce Lewis, Jr. AMS Classification numbers: Primary 55M35, 55N91, 57S17; Secondary 14M15 55P91 Addresses: Department of Mathematics, Bloomsburg University, Bloomsburg, PA 17815 and Department of Mathematics, Syracuse University, Syracuse NY 13244-1150 email: kferland@bloomu.edu lglewis@syr.edu Abstract: It is well known that the homology of a CW-complex with cells only in even dimensions is free. The equivariant analog of this result for generalized $G$-cell complexes is, however, not obvious, since \roG-graded homology cannot be computed using cellular chains. We consider $G = \mathbb{Z}/p$ and study $G$-cell complexes constructed using the unit disks of finite dimensional $G$-representations as cells. Our main result is that, if $X$ is a $G$-complex containing only even-dimensional representation cells and satisfying certain finiteness assumptions, then its \roG-graded equivariant ordinary homology \HoeX{G}{X}{A} is free as a graded module over the homology \HoPt of a point. This extends a result due to the second author about equivariant complex projective spaces with linear $\mathbb{Z}/p$-actions. Our new result applies more generally to equivariant complex Grassmannians with linear $\mathbb{Z}/p$-actions. Two aspects of our result are particularly striking. The first is that, even though the generators of \HoeX{G}{X}{A} are in one-to-one correspondence with the cells of $X$, the dimension of each generator is not necessarily the same as the dimension of the corresponding cell. This shifting of dimensions seems to be a previously unobserved phenomenon. However, it arises so naturally and ubiquitously in our context that it seems likely that it will reappear elsewhere in equivariant homotopy theory. The second unexpected aspect of our result is that it is not a purely formal consequence of a trivial algebraic lemma. Instead, we must look at the homology of $X$ with several different choices of coefficients and apply the Universal Coefficient Theorem for \roG-graded equivariant ordinary homology. In order to employ the Universal Coefficient Theorem, we must introduce the box product of \roG-graded Mackey functors. We must also compute the $RO(G)$-graded equivariant ordinary homology of a point with an arbitrary Mackey functor as coefficients. This, and some other, basic background material on \roG-graded equivariant ordinary homology is presented in a separate part at the end of the paper. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/comodule Author: Mark Hovey Title: Homotopy theory of comodules over a Hopf algebroid Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma ). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the basic properties of Stable(Gamma), showing that it is a compactly generated tensor triangulated category. ---------------------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://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 or Internet Explorer) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There is a web form for submitting papers to Hopf on this site as well. You can also use ftp, explained below. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/new-html/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive. ------- End of forwarded message -------