Subject: new Hopf listings From: Mark Hovey Date: 04 Mar 2000 10:31:18 -0500 Sorry for the delay; I seem to be getting old and tired. 6 new papers this time. Mark Hovey New papers uploaded to hopf between 1/29/00 and 3/4/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Morava/amrrrfls A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space Authors: Matthew Ando mando@math.uiuc.edu Jack Morava jack@math.jhu.edu We show that the fixed-point formula in an equivariant complex-oriented cohomology theory $E$, applied to the free loop space of a manifold $X$, may be viewed as a (renormalized) Riemann-Roch formula for the quotient of the group law of $E$ by a free cyclic subgroup. If $E$ is $K$-theory, this explains how the elliptic genus associated to the Tate elliptic curve emerges from Witten's analysis of the fixed-point formula in $K$-theory. In general this quotient is not representable, but we show that its torsion subgroup is. In the case that $E$ is the Borel theory associated to the Lubin-Tate theory $E_n$, this leads to a description of the functor represented by $E_n[[q]], analogous to the relationship between the Tate curve and $K$-theory. For a more general equivariant $E$, we show that the formal products which arise in this discussion may be naturally viewed as Thom classes for Thom prospectra as considered by Cohen-Jones-Segal. These prospectra seem to define interesting models for the physicists' space of `small' loops on $X$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Crespo-Saumell/aqfh Title: Non-simply connected $H$-spaces with finiteness conditions Authors: Carlos Broto, Juan A. Crespo and Laia Saumell e-mail addresses: broto@mat.uab.es, chiqui@crm.es, and laia@mat.uab.es This article is concerned with homotopy properties of $H$-spaces $X$ that are reflected in the module of indecomposables $QH^*(X;\F_p)$. It is shown that mod $p$ $H$-spaces $X$ of finite type with finite transcendence degree mod $p$ cohomology and locally finite $QH^*(X;\F_p)$ are $B\Z/p$-null spaces, Eilenberg-MacLane spaces $K(\padic,2)$, $K(\Z/p^r,1)$, and extensions of those. If we restrict attention to $H$-spaces with noetherian mod $p$ cohomology algebra, then we are left with finite mod $p$ $H$-spaces and Eilenberg-MacLane spaces. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fisher/bous Title: A Proof of an Exponent Conjecture of Bousfield Author: Michael J. Fisher Email: mjf7@lehigh.edu Abstract: Let p be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the p-exponent of the spectrum Phi SU(n) is (n-1) + nu_p((n-1)!) for n >= 2. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grodal/limsub Title: Higher limits via subgroup complexes Author: Jesper Grodal Email: jg@math.mit.edu Abstract: We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with values in a G-local coefficient system. We examine when smaller complexes can be used e.g., taking only p-radical subgroups, p-centric subgroups, elementary abelian p-subgroups or various subcollections thereof. Since the subgroup complexes are finite complexes, and often rather small, this provides concrete, computable formulas for these higher limits, generalizing earlier work of especially Jackowski-McClure- Oliver. It also gives a conceptual explanation of high dimensional vanishing results previously established in more indirect ways. As an application we look at the special case where all the higher limits vanish, as for example is the case for group cohomology. If F is a functor on the orbit category our formulas for the higher limits in this case yield five different expressions of F(G) in terms of values of F on proper subgroups. Two of these are `classical' namely Webb's exact sequence of Mackey functors and a formula for calculating stable elements, previously obtained using Alperin's fusion theorem. Examining this case also leads to improvements of sharpness results of homology decompositions due to Dwyer and others. Central to many of the proofs are properties of the Steinberg chain complex of a finite group G, as well as other concepts from the emerging Lie theory for arbitrary finite groups of Alperin, Webb, and others. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhang-Woo/forgetnew1 Title: Phantom maps and Forgetful maps Authors: Jianzhong Pan Institute of Math.,Academia Sinica ,Beijing China and Department of Mathematics Education , Korea University , Seoul , Korea email: pjz62@hotmail.com Moo Ha Woo Department of Mathematics Education , Korea University , Seoul , Korea ABSTRACT: In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so- called Forgetful map. We show that we can insert the Forgetful map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. Remark:This is an upgraded version of a preprint which has been on the archive. A problem in Theorem2.8 has been corrected following a suggestion from K.Iriye. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Karoubi/A_descent_theorem Max KAROUBI A descent theorem in topological K-theory karoubi@math.jussieu.fr Let A be a Banach algebra and A' its complexification. In this paper we show that the homotopy fixed point set of K(A'), the topological K-theory space of A', under complex conjugation is just K(A), the topological K-theory space of A. This result generalizes the well known fact that BO is BU^hZ/2. The proof uses in an essential way Atiyah's KR theory and the Clifford algebra definition of higher K-groups. ---------------------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.