Subject: new Hopf listings
Date: 07 Oct 2002 08:36:40 +0100
From: nsthov01@newton.cam.ac.uk (M.A. Hovey)
To: dmd1@lehigh.edu
The web form for submission to Hopf is a big success! So much so that I
will have to send out these letters more frequently. There are 17 new
papers this time! So I will break this letter up into two parts.
This first part contains 9 new papers this time, from Anton,
Bendersky-DavisD, Blanc-Peschke, 3 from Cisinski, Devinatz,
Ferland-Lewis, and Hovey.
Mark Hovey
New papers appearing on hopf between 09/11/02 and 10/07/02, part 1
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/morava
Title of Paper: On Morava K-theories of an S-arithmetic group
Author: Marian F. Anton
AMS Classification numbers: 55N20,19F27,11F75
Address of Author: Department of Pure Mathematics,
University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
Email address of Author: Marian.Anton@imar.ro
Text of Abstract: We completely describe the Morava K-theories
with respect to the prime p for the etale model of the classifying
space of the general linear group GL(m) over the ring Z[u,1/p]
when p is an odd regular prime and u a primitive p-th root of unity.
For p=3 and m=2 (and conjecturally in the stable range) these
K-theories are the same as those of the classifying space itself.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/SON
v1-periodic homotopy groups of SO(n)
Martin Bendersky and Donald M. Davis
55Q52, 55T15, 57T20
Hunter College, CUNY, NY, NY 10021
Lehigh University, Bethlehem, PA 18015
Abstract
We compute the 2-primary v1-periodic homotopy groups of the special
orthogonal groups SO(n). The method is to calculate
the Bendersky-Thompson spectral sequence, a K*-based unstable homotopy
spectral sequence, of Spin(n). The E2-term is an
Ext group in a category of Adams modules. Most of the differentials
in the spectral sequence are determined by naturality from
those in the spheres.
The resulting groups consist of two main parts. One is summands whose
order depends on the minimal exponent of 2 in several
sums of binomial coefficients times powers. The other is a sum of
roughly [log_2(2n/3)] copies of Z/2.
As the spectral sequence converges to the v1-periodic homotopy groups
of the K-completion of a space, one important part of
the proof is that the natural map from Spin(n) to its K-completion
induces an isomorphism in v1-periodic homotopy groups.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Peschke/BlancPeschke1
Authors: David Blanc and George Peschke
Title: The plus construction, Postnikov towers and universal central
module extensions.
Given a connected space $X$, we consider the effect of Quillen's
plus construction on the homotopy groups of $X$ in terms of its
Postnikov decomposition. Specifically, using universal properties of the
fibration sequence \ $AX\to X\to X^+$, \ we explain the contribution of
\ $\pi_nX$ \ to \ $\pi_nX^+$, \ $\pi_{n+1}X^+$ \ and \ $\pi_nAX$, \
$\pi_{n+1}AX$ \ explicitly in terms of the low dimensional homology of
$\pi_nX$ regarded as a module over $\pi_1X$. \ Key ingredients developed
here for this purpose are universal $\Pi$-central fibrations and a
theory of universal central extensions of modules, analogous to
universal central extensions of perfect groups.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/der2
Images directes cohomologiques dans les categories de modeles
Denis-Charles Cisinski
AMS Classification numbers 55U3, 55U40
Institut de Mathématiques de Jussieu
Université Paris 7
Case 7012
2 place Jussieu
75251 Paris cedex 05 France
E-mail: cisinski@math.jussieu.fr
Abstract
Show that every complete model category M
admits homotopy limits, and more generaly
that every functor between small categories
has a cohomological direct image in M (that is
a homotopy right Kan extension). Furthermore,
we study the local behavor of such constructions.
For this purpose, we introduce Grothendieck's
notion of derivator. Derivators correspond to the
intuition of ``a homotopy complete category''
without speaking about models. Forthcoming
papers will show that this setting is rich
enough to define classical homotopy theory
by a simple universal property.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/propuni
Proprietes universelles et
extensions de Kan derivees
Denis-Charles Cisinski
AMS Classification numbers 55U3, 55U40
Institut de Mathématiques de Jussieu
Université Paris 7
Case 7012
2 place Jussieu
75251 Paris cedex 05 France
E-mail: cisinski@math.jussieu.fr
Abstract
We show that for all small category A,
the derivator associated to the homotopy theory
of presheaves in categories (or in simplicial sets)
on A is the solution of a universal problem (and
a similar statement about the pointed versions
of such derivators is proved). When A is the
final category, this shows that the derivator HOT
associated to the classical homotopy theory is
canonically endowed with a monoidal structure,
and that every derivator admit a canonical
action of HOT. As every model category defines
a derivator, Hovey's homotopy coherence
conjectures are then a consequence of these
constructions.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Cisinski/winfax
Le localisateur fondamental minimal
Denis-Charles Cisinski
AMS Classification numbers 55U40
Institut de Mathématiques de Jussieu
Université Paris 7
Case 7012
2 place Jussieu
75251 Paris cedex 05 France
E-mail: cisinski@math.jussieu.fr
Abstract
Basic localizors were introduced by Grothendieck
in Pursuing Stacks. These are classes of arrows
in the category Cat of small categories
satisfying nice properties of descent (like
Quillen's theorem A).For example, every
cohomology theory defines a basic localizor.
In particular, classical weak equivalences
(i.e. those induced from the simplicial weak equivalences
from th nerve functor) form a basic localizor.
In this paper, we show Grothendieck's conjecture
that Cat's usual weak equivalences
are the smallest basic localizor. This gives in particular
a combinatorial/algebraic way to define classical
homotopy theory.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/LHSspectral
Title: A Lyndon-Hochschild-Serre spectral sequence for certain homotopy
fixed point spectra
Author: Ethan S. Devinatz
AMS Subject Classification: 55N20, 55P43, 55T15
Address: Department of Mathematics, University of Washington, Box 354350,
Seattle, WA 98195
e-mail: devinatz@math.washington.edu
Abstract: Let H and K be closed subgroups of the n th Morava stabilizer
group with H normal in K. We construct a spectral sequence of the
expected form connecting the homotopy of the continuous homotopy H fixed
points of the Landweber exact spectrum E_n with the homotopy of the
continuous K fixed points of E_n. These continuous homotopy fixed point
spectra are the spectra constructed by Devinatz and Hopkins. This
spectral sequence turns out to be an Adams spectral sequence in an
appropriate category of module spectra.
8.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ferland-Lewis/FerlandLewis
Title: The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell
Complexes with Even-Dimensional Cells for $G = \mathbb{Z}/p$
Authors: Kevin K. Ferland and L. Gaunce Lewis, Jr.
AMS Classification numbers: Primary 55M35, 55N91, 57S17; Secondary
14M15 55P91
Addresses: Department of Mathematics, Bloomsburg University,
Bloomsburg, PA 17815
and
Department of Mathematics, Syracuse University, Syracuse NY 13244-1150
email: kferland@bloomu.edu
lglewis@syr.edu
Abstract: It is well known that the homology of a CW-complex with cells
only in even dimensions is free. The equivariant analog of this result
for generalized $G$-cell complexes is, however, not obvious, since
\roG-graded homology cannot be computed using cellular chains. We
consider $G = \mathbb{Z}/p$ and study $G$-cell complexes constructed
using the unit disks of finite dimensional $G$-representations as cells.
Our main result is that, if $X$ is a $G$-complex containing only
even-dimensional representation cells and satisfying certain finiteness
assumptions, then its \roG-graded equivariant ordinary homology
\HoeX{G}{X}{A} is free as a graded module over the homology \HoPt of a
point. This extends a result due to the second author about equivariant
complex projective spaces with linear $\mathbb{Z}/p$-actions. Our new
result applies more generally to equivariant complex Grassmannians with
linear $\mathbb{Z}/p$-actions.
Two aspects of our result are particularly striking. The first is that,
even though the generators of \HoeX{G}{X}{A} are in one-to-one
correspondence with the cells of $X$, the dimension of each generator is
not necessarily the same as the dimension of the corresponding cell.
This shifting of dimensions seems to be a previously unobserved
phenomenon. However, it arises so naturally and ubiquitously in our
context that it seems likely that it will reappear elsewhere in
equivariant homotopy theory. The second unexpected aspect of our result
is that it is not a purely formal consequence of a trivial algebraic
lemma. Instead, we must look at the homology of $X$ with several
different choices of coefficients and apply the Universal Coefficient
Theorem for \roG-graded equivariant ordinary homology.
In order to employ the Universal Coefficient Theorem, we must introduce
the box product of \roG-graded Mackey functors. We must also compute
the $RO(G)$-graded equivariant ordinary homology of a point with an
arbitrary Mackey functor as coefficients. This, and some other, basic
background material on \roG-graded equivariant ordinary homology is
presented in a separate part at the end of the paper.
9.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/comodule
Author: Mark Hovey
Title: Homotopy theory of comodules over a Hopf algebroid
Given a good homology theory E and a topological space X, the E-homology
of X is not just an E_{*}-module but also a comodule over the Hopf
algebroid (E_{*}, E_{*}E). We establish a framework for studying the
homological algebra of comodules over a well-behaved Hopf algebroid (A,
Gamma ). That is, we construct the derived category Stable(Gamma) of
(A, Gamma) as the homotopy category of a Quillen model structure on the
category of unbounded chain complexes of Gamma-comodules. This derived
category is obtained by inverting the homotopy isomorphisms,
NOT the homology isomorphisms. We establish the basic
properties of Stable(Gamma), showing that it is a compactly
generated tensor triangulated category.
---------------------Instructions-----------------------------
To subscribe or unsubscribe to this list, send a message to Don Davis at
dmd1@lehigh.edu with your e-mail address and name.
Please make sure he is using the correct e-mail address for you.
To see past issues of this mailing list, point your WWW browser to
http://math.wesleyan.edu/~mhovey/archive/
If this doesn't work or is missing a few issues, try
http://www.lehigh.edu/~dmd1/algtop.html
which also has the other messages sent to Don's list.
To get the papers listed above, point your WWW client (Netscape or
Internet Explorer) to the URL listed. The general Hopf archive URL is
http://hopf.math.purdue.edu
There is a web form for submitting papers to Hopf on this site as well.
You can also use ftp, explained below.
The largest archive of math preprints is at
http://xxx.lanl.gov
There is an algebraic topology section in this archive. The most useful
way to browse it or submit papers to it is via the front end developed
by Greg Kuperberg:
http://front.math.ucdavis.edu
To get the announcements of new papers in the algebraic topology section
at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe"
(without quotes), and with the body of the message "add AT" (without
quotes).
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/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.
------- End of forwarded message -------