Subject: new Hopf listings
From: Mark Hovey
Date: 05 Jun 2006 08:59:53 -0400
There are 7 new papers this time, from Bartels-Rosenthal,
Chermak-Oliver-Shpectorov, Dwyer-Wilkerson, Jardine (2),
Stacey-Whitehouse, and Wilkerson.
Mark Hovey
New papers appearing on hopf between 5/3/06 and 6/5/06
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Rosenthal/asymptotic
Authors: Arthur Bartels, David Rosenthal
arXiv submission number: math.KT/0605088
Abstract: It is proved that the assembly maps in algebraic K- and
L-theory with respect to the family of finite subgroups is injective for
groups with finite asymptotic dimension that admit a finite model for
the classifying space for proper actions. The result also applies to
certain groups that admit only a finite dimensional model for this
space. In particular, it applies to discrete subgroups of virtually
connected Lie groups.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chermak-Oliver-Shpectorov/fundsol
The simple connectivity of $B\Sol(q)$
by Andrew Chermak, Bob Oliver, and Sergey Shpectorov
Andrew Chermak
Kansas State University
Bob Oliver
LAGA, Institut Galil\'ee
Sergey Shpectorov
University of Birmingham
Abstract: A $p$-local finite group is an algebraic structure which
includes two categories, a fusion system and a linking system, which mimic
the fusion and linking categories of a finite group over one of its Sylow
subgroups. The $p$-completion of the geometric realization of the linking
system is the classifying space of the finite group. In this paper, we
study the geometric realization, \emph{without} completion, of linking
systems of certain exotic 2-local finite groups whose existence was
predicted by Solomon and Benson, and prove that they are all simply
connected.
The file "Co3graph.eps" must be included with the dvi file.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants
POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS
W. G. DWYER AND C. W. WILKERSON
In this note we use elementary methods to prove Steinberg's result
for fields of characteristic 0 or of characteristic prime to the order of
W .
This gives a new proof even in the characteristic zero case.
1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W
a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra
on V # the k-dual of V, and R = S^W the ring of invariants of under the
natural action of W on S. Define P* to be the quotient algebra S
i\tensor_R k. If the characteristic of k is zero or prime to the order
of W and P* satisfies Poincar'e duality, then R is isomorphic to a
polynomial algebra on r generators.
Steinberg [9] has shown that R is polynomial if k is the field
of complex numbers and the quotient algebra P* = S\tensor_R k satisfies
Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4]
to other fields of characteristic zero, and by T.-C. Lin [5] to the case
in
which k is a finite field of characteristic prime to the order of W .
The current proof is independent of previous methods.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/coc-cat3
Title: Cocycle categories
Author: J.F. Jardine
arXive submission number: math.AT/0605198
Abstract: A cocycle category H(X,Y) is defined for objects X and Y in a
model category, and it is shown that the set of morphisms [X,Y] is
isomorphic to the set of path components of H(X,Y) provided the ambient
model category is right proper and satisfies the extra condition that
weak equivalences are closed under finite products. Various applications
of this result are displayed, including the homotopy classification of
torsors, abelian cohomology groups, group extensions and gerbes. The
older classification results have simple new proofs involving
canonically defined cocycles.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/gerbes6
Title: Homotopy classification of gerbes
Author: J.F. Jardine
arXive submission number: math.AT/0605200
Abstract: Gerbes are locally connected presheaves of groupoids. They are
classified up to local weak equivalence by path components in a
2-cocycle category taking values in all sheaves of groups, their
isomorphisms and homotopies. If F is a full presheaf of sheaves of
groups, isomorphisms and homotopies, then [*,BF] is isomorphic to
equivalence classes of gerbes locally equivalent to groups appearing in
F. Giraud's non-abelian cohomology object of equivalence classes of
gerbes with band L is isomorphic to morphisms in the homotopy category
from the point * to the homotopy fibre over L for a map defined on BF
and taking values in the classifying space for the stack completion of
the fundamental groupoid of F.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2
Title: Stable and Unstable Operations in mod p Cohomology Theories
Authors: Andrew Stacey and Sarah Whitehouse
Other useful information: math.AT/0605471
Abstract:
We consider operations between two multiplicative, complex orientable
cohomology theories. Under suitable hypotheses, we construct a map from
unstable to stable operations, left-inverse to the usual map from stable
to unstable operations. The main example is where the target theory is
one of the Morava K-theories in which case our map is closely related to
the Bousfield-Kuhn functor.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/newfred
Loop Spaces and Finiteness
Clarence W. Wilkerson
Purdue University
This expository note began as comments on a shorter note of F.R. Cohen
\cite{cohen}. Cohen's paper is an elegant application of powerful
recent results in unstable homotopy theory to a problem of interest to
analysts.
{\sl {\bf Theorem :} (F.\,R. Cohen,\cite{cohen}) Let $X$ be a simply
connected
finite complex which is not contractible and let $\Omega^j_0X$ be
the component of the constant map in the $j$-th pointed loop space of
$X$. If $j \geq 2$, then the Lusternik-Schnirlman category of
$\Omega^j_0X$ is not finite. }\\
This note includes a rederivation of the above theorem using
H-space methods of W. Browder from the 60's, \cite{Browder-loop},
\cite{Browder-Torsion}. The aim is to reduce the prerequisites for
Cohen's theorem to those available after a second course in algebraic
topology. We end with a discussion of recent work of Lannes-Schwartz on
various notions of finiteness properties and the behavior under looping.
The common theme is extensive use of the action of the Steenrod algebra
on the cohomology of a topological space.
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