Subject: new Hopf listings
Date: 25 Nov 2003 13:32:14 -0500
From: Mark Hovey
Reply-To: mhovey@wesleyan.edu
To: dmd1@lehigh.edu
12 new papers this time, from Aouina-Klein, Chalupnik (3), Fausk-Oliver,
Grandis (2), Knudson-Walker, Notbohm-Ray, Sauvageot, Troesch,
and ZhouXueguang.
Mark Hovey
New papers appearing on hopf between 10/24/03 and 11/25/03
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Aouina-Klein/config_stable
Title: On the homotopy invariance of configuration spaces
Author(s): Mokhtar Aouina and John R. Klein
Author's e-mail address: aouina@math.wayne.edu, klein@math.wayne.edu
AMS classification number: Primary 55R80; Secondary 57Q35, 55R70.
Abstract:
For a closed PL manifold M, we consider the configuration space F(M,k)
of ordered k-tuples of distinct points in M. We show that a suitable
iterated suspension of F(M,k) is a homotopy invariant of M. The number
of suspensions we require depends on three parameters: the number of
points k, the dimension of M and the connectivity of M. Our proof uses
a mixture of embedding theory and fiberwise algebraic topology.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/cohsdr
Title of Paper: Schur_De-Rham complex and its cohomology
Author: Marcin Chalupnik
Email: mchal@mimuw.edu.pl
Abstract: We associate to a Young diagram a complex of strict
polynomial functors which we call the Schur-De-Rham complex. Its
cohomology turns out to reflect deep combinatorial properties of a
diagram. We show that if a ground field is of characteristic p,
the Schur-De-Rham complex is acyclic when the p-core of a diagram
is nontrivial. We also compute its cohomology for a diagram with a
trivial p-core and p-quotient consisting of a single diagram.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extpol
Title of Paper: Extensions of strict polynomial functors
Author: Marcin Chalupnik
Email: mchal@mimuw.edu.pl
Abstract: We compute Ext-groups between various strict polynomial
functors important in representation theory (eg. between twisted
Weyl and Schur functors). Our method utilizes: computation of the
Ext-groups between twisted divided and symmetric powers due to
Franjou-Friedlander-Scorichenko-Suslin, resolutions of functors by
divided and symmetric powers, interplay between functors and
representations of the symmetric group.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extws
Title of Paper: Extensions of Weyl and Schur functors
Author: Marcin Chalupnik
Abstract: We use here the Schur-De-Rham complex to extend
calculations of the Ext-groups between twisted Weyl and Schur
functors initiated in the paper ``Extensions of strict polynomial
functors''. The main result is a full calculation of those groups
in the case of a pair of diagrams which can be obtained from
diagrams of the same weights by the operation F described in
``Schur-De-Rham complex and its cohomology''.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Oliver/piperfect
Title: Continuity of p-perfection for compact Lie groups
Authors: Halvard Fausk and Bob Oliver
Author's e-mail address: fausk@math.uio.no and bob@math.univ-paris13.fr
AMS classification number: 55P91
Abstract: Let G be a compact Lie group, and let pi be any prime or
set of primes. We construct a ``pi-perfection map'': a continuous
function from the space of conjugacy classes of all closed
subgroups of G to the space of conjugacy classes of pi-perfect
subgroups with finite index in their normalizer. We use this to
show that the idempotent elements of the Burnside ring of G
localized at pi are in bijective correspondence with the open and
closed subsets of the space of conjugacy classes of pi-perfect
subgroups of G with finite index in their normalizer.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Bsy2
(Note: this paper is only available in pdf format)
Normed combinatorial homology and noncommutative tori
Marco Grandis
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups.
Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
http://www.dima.unige.it/~grandis/
Notes: Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p.
Abstract. Cubical sets have a directed homology, studied in a previous
paper and consisting of preordered abelian groups, with a positive cone
generated by the structural cubes. By this additional information,
cubical sets can provide a sort of "noncommutative topology", agreeing
with some results of noncommutative geometry but lacking the metric
aspects of C*-algebras.
Here, we make such similarity stricter by introducing normed cubical
sets and their normed directed homology, formed of normed preordered
abelian groups. The normed cubical sets associated with "irrational"
rotations have thus the same classification up to isomorphism as the
well-known irrational rotation C*-algebras.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Dht1
(Note: this paper is only available in pdf format)
Directed homotopy theory, I. The fundamental category
Marco Grandis
Key words: homotopy theory, homotopical algebra, directed homotopy, fundamental category.
Dipartimento di Matematica
Universita di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
http://www.dima.unige.it/~grandis/
Notes: to appear in: Cahiers Topologie Geometrie Differentielle Categoriques
Preprint: Dip. Mat. Univ. Genova, Preprint 443 (2001), 26 p.
Revised version: 5 Nov 2001.
Abstract. Directed Algebraic Topology is beginning to emerge from
various applications.
The basic structure we shall use for the foundations of such a theory, a
d-space, is a topological space equipped with a family of directed
paths, closed under some operations. This allows for directed
homotopies, generally non reversible, represented by a cylinder and
cocylinder functors. The existence of 'pastings' (colimits) yields a
geometric realisation of cubical sets as d-spaces, together with
homotopy constructs which will be developed in a sequel. Here, the
fundamental category of a d-space is introduced and a 'Seifert-van
Kampen' theorem proved; its homotopy invariance rests on directed
homotopy of categories. In the process, new shapes appear, for d-spaces
but also for small categories, their elementary algebraic model.
Applications of such tools are briefly considered or suggested, for
objects which model a directed image, or a portion of space-time, or a
concurrent process.
8.
http://hopf.math.purdue.edu/cgi-bin/generate?/Knudson-Walker/hom11-19
Title: Homology of linear groups via cycles in BG x X
Author1: Kevin P. Knudson
Author2: Mark E. Walker
email1: knudson@math.msstate.edu
email2: mwalker@math.unl.edu
Abstract:
Let G be an algebraic group and let X be a smooth integral scheme over a
field k. In this paper we construct homology-type groups H_i(X,G) by
considering cycles in the simplicial scheme BG x X (an idea suggested by
Andrei Suslin). We discuss the basic properties of these groups and
construct a spectral sequence, beginning with the groups
H_i(\Delta^j,G), which converges to the etale cohomology of the
simplicial group BG. These groups are therefore connected with the
study of Friedlander's generalized isomorphism conjecture. We also
compute some examples, focusing in particular on the case X=Spec(k). In
the case where k is the real numbers, there is a connection between the
groups H_i and the Z/2-equivariant cohomology of the classifying space
of the discrete group G(R).
9.
http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm-Ray/djthrational
On Davis-Januszkiewicz Homotopy Types I;
Formality and Rationalisation
by Dietrich Notbohm} and Nigel Ray
For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have
defined a family of homotopy equivalent CW-complexes whose integral
cohomology rings are isomorphic to the Stanley-Reisner algebra of
$K$. Subsequently, Buchstaber and Panov gave an alternative
construction, which they showed to be homotopy equivalent to Davis and
Januszkiewicz's examples. It is therefore natural to investigate the
extent to which the homotopy type of a space $X$ is determined by having
such a cohomology ring. We begin this study here, in the context of
model category theory. In particular, we extend work of Franz by showing
that the singular cochain algebra of $X$ is formal as a differential
graded noncommutative algebra. We then specialise to the rationals, by
proving the corresponding property for Sullivan's {\it commutative\/}
cochain algebra; this confirms that the rationalisation of $X$ is
unique. In a sequel, we will consider the uniqueness of $X$ at each
prime separately, and apply Sullivan's arithmetic square to produce
global results in special families of cases.
10.
http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/simpl-hopf-model
A simplicial model for the Hopf map
Orin R. Sauvageot
Ecole Polytechnique Federale de Lausanne
orin.sauvageot@epfl.ch
We give an explicit simplicial model for the Hopf map S^3 -> S^2. For
this purpose, we construct a model of S^3 as a principal twisted
cartesian product K x_{eta} S^2, where K is a simplicial model for S^1
acting by left multiplication on itself, S^2 is given the simplest
simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We
construct a Kan complex for the simplicial model K of S^1. The
simplicial model for the Hopf map is then the projection K x_{eta} S^2
-> S^2.
11.
http://hopf.math.purdue.edu/cgi-bin/generate?/Troesch/troesch_Resolution_of_symmetric_powers
Title:
A propos d'une question de Friedlander et Suslin I --
Une r'esolution injective des puissances sym'etriques twist'ees
(in French)
Author: Alain Troesch
Address of Author: Institut de Mathematiques de Jussieu, Case 82
4 place Jussieu, F-75252 PARIS CEDEX 05
e-mail address: troesch@math.jussieu.fr
Abstract.
Some years ago, Friedlander and Suslin constructed an explicit
injective resolution of twisted symmetric powers in the category of
strict polynomial functors over a ground field of characteristic
2. The factors in this resolutions are given by direct sums of tensor
products of (non twisted) symmetric powers. The case of a symmetric
power twisted only once is a well-known result: it is some kind of
Koszul complex.
In characteritic p>2, nothing similar was known up to now, even for a
single twist. In this paper, we construct such injective
resolutions. The resolutions we construct are in fact "p-resolutions",
that is, the differential does not vanish when composed twice, but
only when composed p times.
This result should unable us to constuct an injective resolution of
any twisted functor if we know an injective resolution of the
corresponding non twisted functor. This will be the subject of another
paper.
12.
http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhou2
title of the paper: The answer to an email of Mr. Douglas C. Ravenel
author: Zhou Xueguang
Address of author:Department of Mathematics, Nankai University,
Tianjin 300071, People's Republic of China
Email address of author: zhengqb@eyou.com
Abstract: In this paper, we answer the question why V(n) exists for
all non-negative integers n.
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