Subject: new Hopf listings From: Mark Hovey Date: 16 Jun 2000 09:31:36 -0400 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. ---------------------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. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to Purdue seminars, and other math related things on this page as well. 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/pub/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.