There are 8 new papers this time, from Andersen-Grodal (the completion of the classification theorem for p-compact groups!), Benson-Chebolu-Christensen-Minac, BrownR-Sivera, Bousfield, Kadzisa-Mimura, Kuhn, Morel, and Snaith. Mark Hovey New papers appearing on hopf between 11/5/06 and 12/7/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Andersen-Grodal/2classification Title: The classification of 2-compact groups Authors: Kasper K. S. Andersen and Jesper Grodal Abstract: We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group. Combined with our earlier work with Moeller and Viruel for p odd, this establishes the full classification of p-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over the p-adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson-Chebolu-Christensen-Minac/GH-pgroup-new Title: Freyd's generating hypothesis for the stable module category of a $p$-group Authors: David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac. Abstract: Freyd's generating hypothesis, interpreted in the stable module category of a finite $p$-group $G$, is the statement that a map between finite-dimensional $kG$-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial $p$-group $G$ if and only if $G$ is either $\mathbb{Z}/2$ or $\mathbb{Z}/3$. We also give various conditions which are equivalent to the generating hypothesis. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/normalisation Title: Normalisation for the fundamental crossed complex of a simplicial set Author(s): Ronald Brown, Rafael Sivera R. Brown University of Wales, Bangor, Dean St., Bangor, Gwynedd LL57 1UT, U.K. R. Sivera, Departamento de Geometria y Topologia, Universitat de Valencia, 46100 Burjassot, Valencia, Spain Abstract: Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes, such as the monoidal closed structure. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/Klocal On the 2-adic K-localizations of H-spaces A.K. Bousfield Department of Mathematics University of Illinois at Chicago Chicago, IL 60607 We determine the 2-adic K-localizations for a large class of H-spaces and related spaces. As in the odd-primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary v1-periodic homotopy groups of our spaces. The present v1-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic K-cohomology algebras (including the 2-adic KO-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic K-cohomology algebras with specified operations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kadzisa-Mimura/mbflsc1 Authors: Hiroyuki Kadzisa, Mamoru Mimura Title: Morse-Bott functions and the Lusternik-Schnirelmann category, I The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used to give an upper bound for Lusternik-Schnirelmann categories of topological spaces. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C^1 and their gradient flows, and to apply the result to some homogeneous spaces to determine their Lusternik-Schnirelmann categories. In particular, the Morse-Bott functions on the Stiefel manifolds considered by Frankel are effectively used for constructing all the cone-decompositions in this paper. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/primitives2 Title: Primitives and central detection numbers in group cohomology Author: Nicholas J. Kuhn Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 abstract: Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we show that d_0(G) is bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity Conjecture holds. In particular, this holds for all groups such that the p--rank of G minus the depth of H^*(G) is at most 2. When we look at examples with p=2, we learn that d_0(G) is at most 7 for all groups with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and d_0(G)=14). Enroute we recover and strengthen theorems of Adem and Karagueuzian on essential cohomology, and Green on depth essential cohomology, and prove theorems about the structure of cohomology primitives associated to central extensions. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morel/A1homotopy A^1-algebraic topology over a field Fabien Morel Mathematisches Institut der Universität München Theresienstr. 39 D-80333 München Text of Abstract: In this work we prove some basic results in the context of A1-homotopy theory of smooth schemes over a field k : the analogue of the Brouwer degree, the Hurewicz theorem, the theory of A1-coverings and its relationship to the fundamental A1-homotopy sheaf, and some fundamental computations involving unramified Milnor-Witt K-theory like the fundamental A1-homotopy sheaves of P^n and SL_n . 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Snaith/UTTArf Title: Upper triangular technology and the Arf-Kervaire invariant Author: Victor Snaith address: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, England Abstract. This paper introduces the upper triangular technology (UTT) into classical homotopy theory. This is a new and easy to use method to calculate the effect of the left unit map in 2-adic connective K-theory; the map which is the basis for operations in bu-theory. By way of application, UTT is used to give a new, very simple proof of a conjecture of Barratt- Jones-Mahowald, which rephrases K-theoretically the existence of framed manifolds of Arf-Kervaire invariant one. ----------------------------