Subject: new Hopf listings
From: Mark Hovey
Date: 16 Jun 2000 09:31:36 -0400
7 new papers this time. Sometimes there is a considerable delay between
the time the author puts a paper on Hopf and the time it is announced.
This delay is sometimes at my end, and sometimes at Clarence's end. I
believe the delay on Clarence's end is longer when the author e-mails
him the paper, as Clarence then has to do more work. I believe this is the
reason that some of the papers announced this time were actually
submitted sooner than some of the papers announced last time.
Mark Hovey
New papers appearing on hopf between 6/4/00 and 6/16/00.
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenD-CohenF-Xicotencatl/CCX
Title: Lie algebras associated to fiber-type arrangements
Authors: Daniel C. Cohen, Frederick R. Cohen, Miguel Xicotencatl
math.AT/0005091
Addresses of Authors
D. Cohen, Department of Mathematics, Louisiana State University,
Baton Rouge, LA 70803
F. Cohen, Department of Mathematics, University of Rochester,
Rochester, NY 14627
M. Xicotencatl, Depto. de Mathematicas, Cinvestav del IPN, Mexico City
Max-Plank-Institut fur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany
Email address of Authors
cohen@math.lsu.edu
cohf@math.rochester.edu
xico@@mpim-bonn.mpg.de
Abstract:
Given a hyperplane arrangement in a complex vector space of dimension n,
there is a natural associated arrangement of codimension k subspaces in a
complex vector space of dimension k*n. Topological invariants of the
complement of this subspace arrangement are related to those of the
complement of the original hyperplane arrangement. In particular, if the
hyperplane arrangement is fiber-type, then, apart from grading, the Lie
algebra obtained from the descending central series for the fundamental
group of the complement of the hyperplane arrangement is isomorphic to the
Lie algebra of primitive elements in the homology of the loop space for the
complement of the associated subspace arrangement. Furthermore, this last
Lie algebra is given by the homotopy groups modulo torsion of the loop
space of the complement of the subspace arrangement. Looping further
yields an associated Poisson algebra, and generalizations of the
"universal infinitesimal Poisson braid relations."
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fausk-Lewis-May/FLMApril20
The Picard Group of Equivariant Stable Homotopy Theory
by H. Fausk, L.G. Lewis, Jr, and J.P. May
The University of Chicago (Fausk and May)
Syracuse University (Lewis)
fausk@math.uchicago.edu,
lglewis@mailbox.syr.edu,
may@math.uchicago.edu
April 20, 2000
Let G be a compact Lie group. We describe the Picard group Pic(HoGS) of
invertible objects in the stable homotopy category of G-spectra in terms
of a suitable class of homotopy representations of G. Combining this
with results of tom Dieck and Petrie, which we reprove, we deduce an
exact sequence that gives an essentially algebraic description of
Pic(HoGS) in terms of the Picard group of the Burnside ring of G. The
deduction is based on an embedding of the Picard group of the
endomorphism ring of the unit object of any stable homotopy category C
in the Picard group of C.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model
Spectra and symmetric spectra in general model categories
by Mark Hovey
Wesleyan University
hovey@member.ams.org
June, 2000
(This is an updated version; following an idea of Voevodsky, we
strengthen our proof that stable homotopy isomorphisms agree with stable
equivalences of ordinary spectra so that it applies to one version of
motivic homotopy theory. )
The basic idea is to automate the passage
from unstable to stable homotopy theory, so that it applies in
particular to the A^1 category of Voevodsky. So if we start with a model
category C and a left Quillen endofunctor G of C, we want to make a new
model category, the stabilization of C, where G becomes a Quillen
equivalence. The simplest way to do this is with ordinary spectra.
Thanks to Hirschhorn's localization technology, we can construct the
stable model structure on ordinary spectra with almost no hypotheses on
C and G. A new feature of this revision is that we show that, under
strong smallness hypotheses on G and C, the stable equivalences coincide
with the appropriate generalization of stable homotopy isomorphisms.
If C has a tensor product, and G is given by tensoring with a cofibrant
object K, then we also can construct symmetric spectra. The
localization techniques apply here as well, so we get a stable model
structure of symmetric spectra without having to assume anything like the
Freudenthal suspension theorem. In particular, this is a new
construction of the stable model structure on simplicial symmetric
spectra. Symmetric spectra form a monoidal model category, unlike
ordinary spectra, but we are unable to prove that the monoid axiom holds
in general.
Also new to this revision is a much more careful comparison between
symmetric spectra and ordinary spectra when both are defined. Symmetric
spectra and ordinary spectra are not always Quillen equivalent; we need
the cyclic permutation map on K tensor K tensor K to be homotopic to the
identity. Under some additional technical hypotheses (which again are
satisfied in the A^1 category), we construct a zigzag of Quillen
equivalences between symmetric spectra and ordinary spectra.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May/MMM
Equivariant orthogonal spectra and S-modules
by M.A. Mandell and J.P. May
The University of Chicago
mandell@math.uchicago.edu
may@math.uchicago.edu
April 20, 2000
The last few years have seen a revolution in our understanding
of the foundations of stable homotopy theory. Many symmetric monoidal
model categories of spectra whose homotopy categories are equivalent
to the stable homotopy category are now known, whereas no such categories
were known before 1993. The most well-known examples are the category of
S-modules and the category of symmetric spectra. We focus on the category
of orthogonal spectra, which enjoys some of the best features of S-modules
and symmetric spectra and which is particularly well-suited to equivariant
generalization. We first complete the nonequivariant theory by comparing
orthogonal spectra to S-modules. We then develop the equivariant theory.
For a compact Lie group G, we construct a symmetric monoidal model category
of orthogonal G-spectra whose homotopy category is equivalent to the
classical stable homotopy category of G-spectra. We also complete the
theory of S_G-modules and compare the categories of orthogonal G-spectra
and S_G-modules. A key feature is the analysis of change of universe, change
of group, fixed point, and orbit functors in these two highly structured
categories for the study of equivariant stable homotopy theory.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/PicApril20
Picard groups, Grothendieck rings, and Burnside rings
of categories
J.P. May
The University of Chicago
may@math.uchicago.edu
For Saunders Mac Lane, on his 90th birthday
April 20, 2000
We discuss the Picard group, the Grothendieck ring, and the Burnside
ring of a symmetric monoidal category, and we consider examples from
algebra, homological algebra, topology, and algebraic geometry.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May-Neumann/MNApril20
On the cohomology of generalized homogeneous spaces
by J.P. May and F. Neumann
The University of Chicago
Georg-August-Universit\"at, G\"ottingen, Germany
may@math.uchicago.edu
neumann@cfgauss.uni-math.gwdg.de
April 20, 2000
We observe that work of Gugenheim and May on the cohomology of classical
homogeneous spaces G/H of Lie groups applies verbatim to the calculation
of the cohomology of generalized homogeneous spaces G/H, where G is a
finite loop space or a p-compact group and H is a ``subgroup'' in the
homotopical sense.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Santos/equivariant-D-T
A note on the equivariant Dold-Thom theorem
by Pedro F. dos Santos
Addresses of Author:
Department of Mathematics,
Texas A&M University,
College Station TX-77840
Department of Mathematics,
Instituto Superior Tecnico,
1049 Lisboa, Portugal
Email: pedfs@math.ist.utl.pt
In this note we prove a version of the classical Dold-Thom theorem for
the RO(G)-graded equivariant homology functors H^G_*(-;RM), where G is a
finite group, M is a discrete Z[G]-module, and RM is the Mackey functor
associated to M. In the case where M=Z with the trivial G-action, our
result says that, for a G-CW-complex X, and for a finite dimensional
G-representation V, there is a natural isomorphism
[S^V,Z_0(X)]_G \cong H^G_V(X;RM);
where Z_0(X) denotes the free abelian group on X.
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You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and
login as ftp. Then cd to pub. Files are organized by author name, so
papers by me are in pub/Hovey. If you want to download a file using ftp,
you must type
binary
before you type
get .
To put a paper of yours on the archive, cd to /pub/incoming. Transfer
the dvi file using binary, by first typing
binary
then
put
You should also transfer an abstract as well. Clarence has explicit
instructions for the form of this abstract: see
http://hopf.math.purdue.edu/pub/new-html/submissions.html
In particular, your abstract is meant to be read by humans, so should be
as readable as possible. I reserve the right to edit unreadable
abstracts. You should then e-mail Clarence at wilker@math.purdue.edu
telling him what you have uploaded.
I am solely responsible for these messages---don't send complaints
about them to Clarence. Thanks to Clarence for creating and maintaining
the archive.