Subject: new Hopf listings
From: Mark Hovey
Date: 07 Dec 2006 14:15:28 -0500
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 C1 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
A1-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.
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