Subject: new Hopf listings
From: Mark Hovey
Date: 03 May 2006 11:05:07 -0400
There are 7 new papers this time, from Kuhn, Neusel-Sezer (2),
Pengelley-Williams, RadulescuBanu, and SanchezGarcia (2)
Mark Hovey
New papers appearing on hopf between 4/7/06 and 5/3/06
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/autgrp
Title: The nilpotent filtration and the action of automorphisms on the
cohomology of finite $p$--groups
Author: Nicholas J. Kuhn
AMS classification number: 20J06
Abstract: We study H^*(P), the mod p cohomology of a finite p--group P,
viewed as an Out(P)--module. In particular, we study the conjecture,
first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a
primitive idempotent associated to an irreducible Z/p[Out(P)]--module S,
then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is
an upper bound by Quillen's work, and the conjecture can be viewed as
the statement that every irreducible Z/p[Out(P)]--module occurs as a
composition factor in H^*(P) with similar frequency.
In summary, our results are as follows. A strong form of the conjecture
is true when p is odd. The situation is much more complex when p=2, but
is reduced to a question about 2--central groups (groups in which all
elements of order 2 are central), making it easy to verify the
conjecture for many finite 2--groups, including all groups of order 128,
and all groups that can be written as the product of groups of order 64
or less.
Featured is the nilpotent filtration of the category of unstable
A--modules. Also featured are unstable algebras of cohomology
primitives associated to central group extensions.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-I
The Noether Map I
Mara D Neusel and M"ufit Sezer
Abstract:
Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a
finite group G. In this paper we study the image of the associated
Noether map
$
\eta_G^G: F[V(G)]^G \longrightarrow F[V]^G.
$
It turns out that the image of the Noether map characterizes the ring of
invariants in the sense that its integral closure
$\overline{Im(\eta_G^G)} =F[V]^G$. This is true without any
restrictions on the group, representation, or ground field. Moreover,
we show that the extension $Im (\eta_G^G) \subseteq F[V]^G$ is a finite
$p$-root extension. Furthermore, we show that the Noether map is
surjective, i.e., its image integrally closed, if $V=F^n$ is a
projective $FG$-module. We apply these results and obtain upper bounds
on the degrees of a minimal generating set of $F[V]^G$ and the
Cohen-Macaulay defect of $F[V]^G$. We illustrate our results with
several examples.
Note that this paper together with noether-map-II contain stronger results
than the authors' previous paper Neusel-Sezer/noether.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-II
The Noether Map II
Mara D Neusel and M"ufit Sezer
Abstract:
Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a
finite group G. In this paper we proceed with the study of the image of
the associated Noether map
\[
\eta_G^G: F[V(G)]^G \longrightarrow F[V]^G.
\]
In [Noether Map I] it has been shown that the Noether map is surjective
if $V$ is a projective $FG$-module. This paper deals with the
converse. The converse is in general not true: we illustrate this with
an example. However, for $p$-groups (where $p$ is the characteristic of
the ground field $F$) as well as for permutation representations of any
group the surjectivity of the Noether map implies the projectivity of
$V$.
Note that this paper together with noether-map-I contain stronger results
than the authors' previous paper Neusel-Sezer/noether.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/peng-will-oddkam
The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations,
and invariant theory
David J. Pengelley and Frank Williams
New Mexico State University
Las Cruces, NM 88003
Primary 16W22; Secondary 16W30, 16W50, 55S10, 55S12, 55S99, 57T05.
We describe bialgebras of lower-indexed algebraic Steenrod operations
over the field with p elements, p an odd prime. These go beyond the
operations that can act nontrivially in topology, and their duals are
closely related to algebras of polynomial invariants under subgroups of
the general linear groups that contain the unipotent upper triangular
groups. There are significant differences between these algebras and
the analogous one for p=2 , in particular in the nature and consequences
of the defining Adem relations.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/RadulescuBanu/cofib-cat
Title: Cofibrations in Homotopy Theory
Author: Andrei Radulescu-Banu
Author's mailing address: 86 Cedar St, Lexington, MA 02421
Abstract:
We define Anderson-Brown-Cisisnski (ABC) cofibration categories, and
construct homotopy colimits of diagrams of objects in ABC cofibraction
categories. Homotopy colimits for Quillen model categories are obtained
as a particular case. We attach to each ABC cofibration category a right
derivator. A dual theory is developed for homotopy limits in ABC
fibration categories and for left derivators. These constructions
provide a natural framework for 'doing homotopy theory' in ABC
(co)fibration categories.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/bredon
Title:
Bredon homology and equivariant K-homology of SL(3,Z)
Author:
Ruben Sanchez-Garcia
Author's address:
Department of Pure Maths, Hicks Building
University of Sheffield
Sheffield S3 7RH, United Kingdom
Included ps or eps files:
SouleFundamentalDomainLabelled.eps
TruncatedCube.eps
AMS classification number:
19L47, 55N91 (Primary); 19K99, 46L80 (Secondary)
Other useful information:
arXiv:math.KT/0601587
Abstract:
We obtain the equivariant K-homology of the classifying space
\underline{E}SL(3,Z) from the computation of its Bredon homology with
respect to finite subgroups and coefficients in the representation
ring. We also obtain the corresponding results for GL(3,Z). Our
calculations give therefore the topological side of the Baum-Connes
conjecture for these groups.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/coxeter
Title:
Equivariant K-homology for some Coxeter groups
Author:
Ruben Sanchez-Garcia
Author's address:
Department of Pure Maths, Hicks Building
University of Sheffield
Sheffield S3 7RH, United Kingdom
Included eps files:
hexagon3.eps
hexagon4.eps
interval.eps
tessellation0.eps
trianglesd2.eps
AMS classification number:
19L47, 55N91 (Primary); 19K99, 46L80 (Secondary)
Other useful information:
arXiv:math.KT/0604402
Abstract:
We obtain the equivariant K-homology of the classifying space
\underline{E}W for W a right-angled or, more generally, an even Coxeter
group. The key result is a formula for the relative Bredon homology of
\underline{E}W in terms of Coxeter cells. Our calculations amount to the
K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly
map.
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