--------------------------------------- 10 new papers this time, from Anderson-Grodal-Moller-Viruel, Anjos-Granja, Bauer-McCarthy, Behrens-Pemmaraju, Budney-Conant-Scannell-Sinha, Donadze-Inassaridze-Porter, Dorabiala-Johnson, Kitchloo-Laures-Wilson, Salvatore, and Sinha. Mark Hovey New papers appearing on hopf between 3/01/03 and 4/09/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anderson-Grodal-Moller-Viruel/classificationpodd Title: The classification of p-compact groups for p odd Authors: Kasper K. S. Andersen, Jesper Grodal, Jesper M. M{\o}ller, Antonio Viruel Subj-class: AT Algebraic Topology (GR Group theory; RT Representation Theory) MSC-class: 55R35 (Primary) 55P35, 57T10, 20G20 (Secondary) Comments: 87 pages \\ A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our method however leads to a largely self-contained proof of the entire classification theorem. \\ 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Anjos-Granja/homotopy.decomp.symplect Title: Homotopy decomposition of a group of symplectomorphisms of S^2\times S^2 Authors: Silvia Anjos and Gustavo Granja AMS Classification numbers: 57S05, 57R17, 55R35 Address of Authors: Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais 1049-001 Lisboa Portugal Email address of Authors: sanjos@math.ist.utl.pt ggranja@math.ist.utl.pt Abstract: We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bauer-McCarthy/mcbauer3 Kristine Bauer Department of Mathematics Johns Hopkins University 3400 N. Charles St. Baltimore, MD 21218 USA kbbauer@math.jhu.edu Randy McCarthy Department of Mathematics University of Illinois 1409 W. Green St. Urbana, IL 61801 USA randy@math.uiuc.edu On vanishing Tate cohomology and decompositions in Goodwillie calculus Mathematical Subject Classification: 55P65 (55P45, 13D03) Our main result is that if F is a functor from a pointed category C to spectra, the Goodwillie tower of F evaluated at X splits rationally when X is a co-H-object of C. We show that the layers of F(X) in this case are easy to identify. The splitting of the Goodwillie tower gives a decomposition of F(X) into a product of its layers. We use this to recover the rational decompositions of Hochschild and higher Hochschild homology by Pirashvili, Loday,and Gerstenhaber-Schack. Finally, we extend the main theorem to include dual calculus to recover the Poincar\'e-Birkhoff-Witt theorem, and improve the theorem in the special case in which the comultiplication map is cocommutative. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Pemmaraju/v2 On the existence of the self map v_2^9 on the Smith-Toda complex V(1) at the prime 3 Mark Behrens Department of Mathematics University of Chicago Chicago, IL 60637, U.S.A. mbehrens@math.uchicago.edu Satya Pemmaraju Fixed Income Derivatives UBS Warburg Stamford, CT 06901, U.S.A. Satya.Pemmaraju@ubsw.com AMS Classification: 55Q51; 55Q45, 55T15 math.AT/0303223 submitted to proceedings of the Northwestern University conference on algebraic topology, March 2002 Included EPS files: assE2.eps bss.eps eo_2V1.eps eo_2V1ASS.eps extP.eps splitting.eps Note: there is one chart created using the landscape package in LaTeX. On some dvi viewers, this chart does not display properly, but is viewable when converted to Postscript. Abstract Let V(1) be the Smith-Toda complex at the prime 3. We prove that there exists a map v_2^9: \Sigma^{144}V(1) \to V(1) that is a K(2) equivalence. This map is used to construct various v_2-periodic infinite families in the 3-primary stable homotopy groups of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Budney-Conant-Scannell-Sinha/selflink Title: New perspectives on self-linking Author: Ryan Budney, James Conant, Kevin P. Scannell, Dev P. Sinha AMS Class: 57M27; 55R80; 57R40; 57M25; 55P99 LANL ID: math.AT/0303034 Addresses: Departments of Mathematics, Rochester University, Cornell University, St. Louis University, University of Oregon Email: rybu@math.rochester.edu, jconant@polygon.math.cornell.edu, scannell@slu.edu, dps@math.uoregon.edu Abstract: We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. Note: The .dvi version is missing many (fun) figures - we strongly recommend downloading the .pdf file. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Donadze-Inassaridze-Porter/Hopfder N-fold Cech derived functors and generalized Hopf type formulas by Guram Donadze, Nick Inassaridze, and Timothy Porter, In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k. The paper ends with an application to algebraic K-theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Dorabiala-Johnson/torsion The product theorem for parametrized topological Reidemeister torsion Wojtek Dorabiala Mark W. Johnson Primary: 19D10; Secondary: 18F25, 19Exx, 55R70 Department of Mathematics Penn State Altoona Altoona, PA 16601-3760 wud2@psu.edu mwj3@psu.edu The goal of this article is to prove the product formula for parametrized topological Reidemeister torsion. The theorem states that the product of the parametrized Euler characteristic of one fibration with the parametrized Reidemeister torsion class of another fibration yields the parametrized Reidemeister torsion class of the product fibration. In the process of establishing the theorem, several new products must be defined involving (derivative theories of) parametrized $\Aof$-theory and a detailed description of the coassembly map for parametrized $\Aof$-theory is included. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Laures-Wilson/klw2 Splittings of bicommutative Hopf algebras Nitu Kitchloo, Gerd Laures and W. Stephen Wilson Department of Mathematics Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA Mathematisches Institut der Universit\"at Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany wsw@math.jhu.edu, nitu@math.jhu.edu, gerd@laures.de Abstract: We use the theory of Dieudonne modules to show that certain types of short exact sequences of Hopf algebras split. Several examples occur naturally with Morava K-theory. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Salvatore/config Title: Configuration spaces on the sphere and higher loop spaces Author: Paolo Salvatore AMS classification numbers: 55P48, 55R80, 55S12 xxx number: math.AT/0303290 Address: Dipartimento di Matematica, Universita` di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: salvator@mat.uniroma2.it Abstract: We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/semifree Title: Bordism of semi-free $S^1$-actions. Author: Dev P. Sinha AMS Class: 57R85 (primary); 55R40 (secondary). LANL ID: math.AT/0303100 Addresses: Department of Mathematics, University of Oregon, Eugene OR Email: dps@math.uoregon.edu Abstract: We calculate the geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. To calculate the geometric theory, we prove a case of the geometric realization conjecture, which in general would determine the geometric theory in terms of the homotopical. The determination of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry. ---------------