Subject: new Hopf listings Date: 17 Sep 2001 14:36:06 -0400 From: Mark Hovey Hope all of your loved ones are alright. There are 9 new papers on Hopf in the last two weeks. Mark Hovey New papers appearing on hopf between 9/2/01 and 9/17/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ahearn-Kuhn/towers Title: Product and other fine structure in polynomial resolutions of mapping spaces Authors: Stephen T. Ahearn and Nicholas J. Kuhn AMS classification: Primary 55P35; Secondary 55P42 Authors addresses: Department of Mathematics, De Pauw University, Greencastle, IN 46135. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 Email: sahearn@depauw.edu, njk4x@virginia.edu Abstract: Let Map(K,X) denote the space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum of Map(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on the homology of the mapping space. Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_{infty}--term corresponds to the cup product in h^*(Map(K,X)) in the usual way, and the product on the E_1--term is described in terms of group theoretic transfers. We use explicit equivariant S--duality maps to show that, when K is the n sphere, our constructions at the fiber level have descriptions in terms of the Boardman--Vogt little n--cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to the suspension spectrum of its 0th space. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Dwyer-Goerss/moduli The realization space of a \Pi-algebra: a moduli problem in algebraic topology D. Blanc, W. G. Dwyer, and P. G. Goerss A \PI-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R(A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with \pi_*(X)=A, of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R(A), in which the space at the bottom is BAut(A) and the successive fibres are determined by \Pi-algebra cohomology. (This cohomology is the analog for \Pi-algebras of the Hochschild cohomology of an associative ring or the Andre-Quillen cohomology of a commutative ring.) The main technical tool involves working with simplicial resolutions of spaces rather than with spaces themselves. It seems clear that the deformation theory can be applied with little change to study other moduli questions in topology. In the course of working out the details, we find a simple homotopy theoretic way to identify the space that results from taking a functor from finite sets to sets and applying it dimensionwise to a simplicial set. This gives an easy way to reprove and generalize many classical connectivity theorems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/E6 The K-completion of E6 Donald M. Davis 55T15, 55Q52, 57T20 Department of Mathematics Lehigh University Bethlehem, PA 18015 dmd1@lehigh.edu Abstract: We compute the 2-primary v1-periodic homotopy groups of the exceptional Lie group E6. This is done by computing the Bendersky-Thompson spectral sequence of E6. We conjecture that the natural map from E6 to its K-completion induces an isomorphism in v1-periodic homotopy, and discuss issues related to this conjecture. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Green-Hunton-Schuster/cccGHS Title: Chromatic characteristic classes in ordinary group cohomology Authors: David J. Green John R. Hunton Bj"orn Schuster MSC: 20J06 (primary), 16W30 55P47 55R40 (secondary) arXiv: math.AT/0109019 Status: Submitted for publication, Aug. 2001 Abstract: We study a family of subrings, indexed by the natural numbers, of the mod-p cohomology of a finite group G. These subrings are based on a family of v_n-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H^*(BG)), the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings give rise to a "chromatic" (co)filtration, which has both topological and algebraic definitions, of var(H^*(BG)) whose final quotient is the variety var(Ch(G)). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Gdansknotes Title: Braids, trees, and operads Author: Jack Morava AMS classification: 55R810, 14N35, 20F36 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack@math.jhu.edu Abstract: The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties. This talk [at the Gdansk conference on algebraic topology 05-06-01] is an introduction to their study. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Looptangent Title: The tangent bundle of an almost-complex free loopspace Author: Jack Morava AMS classification: 58Dxx; 53C29, 55P91 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack@math.jhu.edu Abstract: The space LV of free loops on a manifold V inherits an action of the circle group \T. When V has an almost-complex structure, the tangent bundle of the free loopspace, pulled back over a certain infinite cyclic cover \tilde LV, has an equivariant decomposition as a completion of \TV \otimes (\oplus \C(k)), where \TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evaluation at the basepoint (and \oplus \C(k) denotes an algebra of Laurent polynomials). On a flat manifold, this analog of Fourier analysis is classical. This construction uses a model for the universal cover of the space of conjugacy classes in the unitary group (also known as a symmetric product of copies of the circle) which may be of independent interest. This paper appears in the proceedings of the Stanford workshop on equivariant homotopy theory, in Homology, Homotopy and Applications, 3 (2001) 407-415. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/PGGravityfinal Title: A rudimentary theory of topological 4D gravity Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack@math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes some relatively accessible examples of such a thing, suggested by the wall-crossing formulas of Donaldson theory. This is the final version of the paper, to appear in Advances in Theoretical and Mathematical Physics. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/TateHeisenberg Title: Tate cohomology of circle actions as a Heisenberg group Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack@math.jhu.edu Abstract: This is a revision of an earlier posting, with a similar name; the paper has been reorganized, and some howlers related to the Segal conjecture have been eliminated: We study the Madsen-Tillman spectrum \CP^\infty_{-1} as a quotient of the Mahowald pro-object \CP^\infty_{-\infty}, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Virasoro Title: An algebraic analog of the Virasoro group Author: Jack Morava AMS classification: 81R10, 55S25 Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: jack@math.jhu.edu Abstract: The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it behaves as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of two-dimensional topological gravity through the homotopy theory of moduli spaces. This is a version of a talk on 23 June 2001 at the Prague Conference on Quantum Groups and Integrable Systems, published in the Czechoslovak J. Physics 51 (2001). ---------------------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/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.