Subject: new Hopf listings From: Mark Hovey Date: 09 Nov 2000 05:59:37 -0500 Ten new papers this time. Mark Hovey New papers appearing on hopf between 10/2/00 and 11/8/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Luptin-Murillo/Subgps ofHEs Title: Subgroups of the Group of Self-Homotopy Equivalences Authors: Martin Arkowitz, Gregory Lupton and Aniceto Murillo. Classification Nos. (1991): Primary 55P10; Secondary 55P62, 55Q05. Addresses: Department of Mathematics, Dartmouth College, Hanover NH 03755 U.S.A. Department of Mathematics, Cleveland State University, Cleveland OH 44115 U.S.A. Departmento de Algebra, Geometria y Topologia, Universidad de Malaga, Ap. 59, 29080 Malaga, Spain e-mail Addresses: Martin.Arkowitz@Dartmouth.edu Lupton@math.csuohio.edu Aniceto@agt.cie.uma.es Abstract: Denote by $\mathcal{E}(Y)$ the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex $Y$. We give a selection of results about certain subgroups of $\mathcal{E}(Y)$. We establish a connection between the Gottlieb groups of $Y$ and the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of $Y$, denoted by $\mathcal{E}_{\#}(Y)$. We give an upper bound for the solvability class of $\mathcal{E}_{\#}(Y)$ in terms of a cone decomposition of $Y$. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of $\mathcal{E}(Y)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent. 2. 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 October, 2000 This is the final version, to appear in JPAA. There are several significant notational changes, and many minor corrections in this version. (Rest of abstract elided, since it has appeared twice already.) 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/torsext2 Cotorsion theories, model category structures, and representation theory by Mark Hovey mhovey@wesleyan.edu AMS Classification: 20C05,20J05,18E30,18G35, 55U35 We make a general study of Quillen model structures on abelian categories. Given a proper class P of short exact sequences on an abelian cateory A, we define what it means for a model structure to be compatible with P. We then give a complete characterization of model structures compatible with P. This characterization is in terms of cotorsion theories, which were introduced by Salce and have been much studied recently by Enochs and coauthors. We apply the general method to construct a stable category of $K[G]$-modules where $K$ is a principal ideal domain and $G$ is a finite group. This is a compactly generated triangulated category that generalizes the well-known stable category of $k[G]$-modules, where $k$ is a field. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/brown-II A Brown representability theorm via coherent functors Author: Henning Krause Address of Author: University of Bielefeld, Germany Email address of Author: henning@mathematik.uni-bielefeld.de Abstract: We discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-Shipley/telescope Title: A telescope comparison lemma for THH (to appear in Topology and its Applications) Authors: Mike Mandell and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Mike Mandell 5734 University Ave. Chicago, IL 60637 USA Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: mandell@math.uchicago.edu bshipley@math.purdue.edu Abstract: The usual telescope or sequential homotopy colimit construction of the underlying infinite loop space must be replaced for symmetric spectra by a homotopy colimit over the category of finite sets and injections. Here we show that for convergent symmetric spectra this modified homotopy colimit agrees with the usual telescope construction. This sharpens B\"okstedt's original lemma because no connectivity conditions are necessary here. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede/2local `The stable homotopy category has a unique model at the prime 2' Stefan Schwede Fakultaet fuer Mathematik Universitaet Bielefeld 33615 Bielefeld, Germany schwede@mathematik.uni-bielefeld.de ABSTRACT: In a closed model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences. But passage to the homotopy category loses information and in general the `homotopy theory' can not be recovered from the homotopy category. We show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory, at least 2-locally. We prove a uniqueness theorem which says that there is only one model structure (up to so called Quillen equivalence) underlying the stable homotopy category of 2-local spectra. This theorem is a 2-local strenghtening of a result with B. Shipley, given in `A uniqueness theorem for stable homotopy theory', in that we use only the triangulated structure of the stable homotopy catgory. The earlier result with Shipley works integrally, but needs additional structure, namely the action of the ring of stable homotopy groups of spheres. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Schwede-Shipley/unique Title: A uniqueness theorem for stable homotopy theory Authors: Stefan Schwede and Brooke Shipley AMS Classification numbers: 55U35 55P42 Addresses: Stefan Schwede Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: schwede@mathematik.uni-bielefeld.de bshipley@math.purdue.edu Abstract: In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres, $\pi^s$. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of $\pi^s$. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space $QS^0$ to the endomorphism space is a weak equivalence. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/monoid.unique Title: Monoidal uniqueness of stable homotopy Author: Brooke Shipley AMS Classification numbers: 55U35 55P42 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley@math.purdue.edu Abstract: We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences produced here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, $\mathcal{W}$-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shipley/rational.circle Title: An algebraic model for rational $S^1$-equivariant stable homotopy theory Author: Brooke Shipley AMS Classification numbers: 55P62 55P91 55P42 55N91 18E30 Address: Brooke Shipley 1395 Math. Bldg. Purdue University West Lafayette, IN 47907 USA Email addresses: bshipley@math.purdue.edu Greenlees defined an abelian category $A$ whose derived category is equivalent to the rational $S^1$-equivariant stable homotopy category whose objects represent rational $S^1$-equivariant cohomology theories. We show that in fact the model category of differential graded objects in $A$ ($dgA$) models the whole rational $S^1$-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between $dgA$ and the model category of rational $S^1$-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The new ingredients here are certain Massey product calculations and the work on rational stable model categories from "Classification of stable model categories" and "Equivalences of monoidal model categories" with Schwede; see http://www.math.purdue.edu/~bshipley/ 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Tamanoi/orbifold Title: Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory Author: Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz Santa Cruz, CA 95064 Email: tamanoi@math.ucsc.edu Abstract: We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mutually commuting elements (of order powers of $p$) in the wreath product G~S_n in terms of corresponding numbers of G. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K-theories of symmetric products of a G-manifold M. AMS Classification Numbers: 55N20, 55N91, 57S17, 57D15, 20E22 ---------------------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.