Subject: new Hopf listings From: Mark Hovey Date: 01 Oct 2005 10:20:39 -0400 There are 8 new papers this time, from Arkowitz-Lupton, DavisDaniel(2), DavisD-Sun, Felix-Lupton, Henn, Kreck-Lueck, and Lupton-Phillips-Schochet-SmithSB. Mark Hovey New papers appearing on hopf between 9/5/05 and 10/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Lupton/ArkLupActions Homotopy Actions, Cyclic Maps and their Duals Martin Arkowitz and Gregory Lupton MSC 2000 55Q05, 55M30, 55P30 M.Arkowitz@Dartmouth.edu G.Lupton@csuohio.edu Abstract: An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1. Note: Appeared in Homology, Homotopy and Applications, vol. 7(1) (2005), 169-184. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/galois Title: Rognes's theory of Galois extensions and the continuous action of G_n on E_n Author: Daniel G. Davis E-mail: dgdavis@math.purdue.edu Address: Purdue University Abstract: Let us take for granted that $L_{K(n)}S0 \rightarrow E_n$ is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that "E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This inactive manuscript arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available. The author thanks Paul Goerss for his encouragement. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg Title: Attempting to construct homotopy orbits for profinite groups Author: Daniel G. Davis E-mail: dgdavis@math.purdue.edu Address: Purdue University Abstract: This note gives a heuristic argument for how one might like to define X_{hG}, for G profinite; it represents a first step in attempting to do this. The argument is not shown to work, and though the heuristic seems plausible, the author does not know how to complete the critical Definition 4.2. Also, the proof of Theorem 5.2 is incomplete. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Sun/DavisSun A number-theoretic approach to homotopy exponents of SU(n) Donald M. Davis and Zhi-Wei Sun dmd1@lehigh.edu zwsun@nju.edu.cn AMS Classifications: 55Q52, 57T20, 11A07, 11B65, 11S05 Abstract: We use methods of combinatorial number theory to prove that, for all n and p, some homotopy group pi_i(SU(n)) contains an element of order p^{n-1+ord_p([n/p]!)}, where ord_p(m) denotes the exponent of p in m. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Felix-Lupton/FelLupEval Title: Evaluation Maps in Rational Homotopy Authors: Yves Felix and Gregory Lupton e-mail: felix@math.ucl.ac.be and G.Lupton@csuohio.edu AMS MSC2000: 55P62, 55Q05 arXiv: math.AT/0509632 Abstract: Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X, X;1) -> X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational homotopy groups is injective. This result has strong consequences: if the image in rational homotopy groups of ev is trivial, then the generalized evaluation map is null-homotopic after rationalization; unless X satisfies a very strong splitting condition, any generalized evaluation map induces the trivial homomorphism in rational cohomology; the map Gamma_ev is rationally a homotopy monomorphism and a generalized evaluation map may be written as a composition of a homotopy epimorphism and this homotopy monomorphism. We include illustrative examples and prove numerous subsidiary results of interest. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn/kn-res-ded Title: On finite resolutions of K(n)-local spheres Author: Hans-Werner Henn e-mail address: henn@math.u-strasbg.fr Abstract: For odd primes p we construct finite resolutions of the trivial module Z_p for the n-th Morava stabilizer group by (direct summands of) permutation modules with respect to finite p-subgroups. Furthermore we discuss the problem of realizing these resolutions by finite resolutions of the K(n)-local sphere via spectra which are (direct summands of) wedges of homotopy fixed point spectra for the action of these finite p-subgroups on the Lubin-Tate spectrum E_n. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kreck-Lueck/kreck+lueck0905 Title of Paper: Topological rigidity for non-aspherical manifolds Author(s): Matthias Kreck and Wolfgang Lueck AMS Classification number: 57N99, 57R67. xxx_archive: math.GT/0509238 kreck@mathi.uni-heidelberg.de lueck@math.uni-muensetr.de The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N ---> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N ---> M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for S^k x S^d, for sphere bundles over aspherical closed manifolds of dimension less or equal to 3 and for 3-manifolds with torsionfree fundamental groups. We show that this rigidity is inherited under connected sums in dimensions greater or equal to 5. We also classify manifolds of dimension 5 or 6 whose fundamental group is the one of a surface and whose second homotopy group is trivial. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-Phillips-Schochet-SmithSB/RationalTaylor Title: Banach Algebras and Rational Homotopy Theory Authors: Gregory Lupton, N.Christopher Phillips, Claude L.~Schochet and Samuel B. Smith AMS MSC (2000): 46J05, 46L85, 55P62, 54C35, 55P15, 55P45 arXiv number: math.AT/0509269 g.lupton@csuohio.edu claude@math.wayne.edu smith@sju.edu Abstract: Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of $GL_n (A)$, the group of invertible $n \times n$ matrices with coefficients in A in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of ``last columns'' of $GL_n (A).$ We construct a natural isomorphism \[ {\check{H}}^s (Max(A); Q) \cong \pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q \] for $n > (1/2) s + 1$ which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F(X,G) for G a Lie group or, more generally, a rational H-space. ---------------------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 Web browser 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 should submit 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 at math.purdue.edu telling him what you have uploaded. The largest archive of math preprints is at http://arxiv.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 the arXiv, send e-mail to math@arxiv.org with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.