Subject: new Hopf listings
Date: 11 Feb 2002 13:23:43 -0500
From: Mark Hovey
To: dmd1@lehigh.edu
6 papers this by time, by Ando, Bakuradze-Priddy, Bousfield, Kuhn,
Martino-Priddy, and Zhou. Note that the paper by Zhou claims to prove
that V(n) exists for all n and all p >= 5, contradicting Ravenel's proof
that V(3) does not exist at p=5. Zhou claims that the Toda relation
alpha_1 beta_1^p =0 is false, giving some reasons why Toda's proofs are
wrong, and therefore Ravenel's argument does not apply. I am hoping one
of you will clear this up, but in the meantime I should remind you that
papers on the Hopf archive are not edited for correctness or anything
else.
Mark Hovey
New papers appearing on hopf between 01/02/02 and 02/11/02
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ando/ando-aeso
Title:
The sigma orientation for analytic circle-equivariant elliptic cohomology
Author:
Matthew Ando
MSC:
55N34 (Primary); 55N22, 57R91 (Secondary)
Arxiv:
math.AT/0201092
Address:
Department of Mathematics
University of Illinois at Urbana-Champaign
E-mail:
mando@math.uiuc.edu
Abstract:
Let T be the circle group. We construct a canonical Thom isomorphism
in T-equivariant analytic elliptic cohomology, for T-oriented virtual
vector bundles bundles whose Borel-equivariant second Stiefel-Whitney
and second Chern classes vanish. The construction is natural under
pull-back of vector bundles and exponential under Whitney sum. It
extends in the rational case the non-equivariant sigma orientation of
Hopkins, Strickland, and the author. The construction relates the
sigma orientation to the representation theory of loop groups and
Looijenga's weighted projective space, and sheds light even on the
non-equivariant case. Rigidity theorems of Witten-Bott-Taubes
including generalizations by Kefeng Liu follow.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Bakuradze-Priddy/bp3b
TRANSFER AND COMPLEX ORIENTED COHOMOLOGY RINGS
MALKHAZ BAKURADZE AND STEWART PRIDDY
Keywords: transfer, Chern class, classifying space, complex cobor-
dism, Morava K-theory 55N22, 55R12.
1. Introduction
Let p be a prime and let G be a subgroup of the symmetric
group S_p. In this paper we use the transfer to study homotopy orbit spaces
X^p_hG= EG x_G X^p in complex oriented cohomology. We are particularly
interested in computing the ring structure. Thus we are led to consider
the relation between cup products and transfer known as Fröbenius
reciprocity by analogy with representation theory
Tr*(x)y = Tr*(x rho*(y))
(formula (i) of Section 2) where rho : EG x X^p --> X^p_hG is the covering
projection and
Tr* : E*(X^p) ---> E*(X^p_hG)
is the associated transfer homomorphism. It is worth noting that the
multiplicative structure of the cohomology groups we consider is com-
pletely determined by this formula.
In case E = K(s) is Morava K-theory, G is cyclic of order p, and
X is the classifying space of a finite group, Hopkins-Kuhn-Ravenel [11 ]
have studied these cohomology groups as modules over the coefficient
ring. Our paper builds on their approach by extending their notion
of a good group to spaces. For X = CP^infty we determine the algebra
K(s)*(X^p_hS_p) for Morava K-theory; for complex cobordism we compute
the ring MU*(X^p_hS_p) making additional use of the formal group law.
This enables us to make explicit computations of the transfer in both
cases. In an analogous fashion we compute the algebra BP *(X^p_hS_p).
The starting point and original motivation for our work comes from
Quillen's famous formula for Tr*(1), the stable Euler class, for the uni-
versal Z/p covering. As explained in Section 2, our results for CP^infty
provide a universal example which enable us to compute the stable Eu-
ler classes and the transfer in general for many other cases. For example
universal coverings for some nonabelian p-groups, namely those with
cyclic subgroups of index p and those which are semi-direct products
of elementary abelian p-groups with Z/p.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/cosim
Cosimplicial resolutions and homotopy spectral sequences in model categories
A.K. Bousfield
Mathematics Subject Classification. Primary 55U35; Secondary 18G55, 55P60, 55T15.
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Chicago, IL 60607
bous@uic.edu
We develop a general theory of cosimplicial resolutions, homotopy spectral
sequences, and completions for objects in model categories, extending work of
Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a
generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution
model categories, and we are able to construct our homotopy spectral sequences
and completions using very flexible weak resolutions in the spirit of relative
homological algebra. We deduce that our completion functors have triple
structures and preserve certain fiber squares up to homotopy. We also deduce
that the Bendersky-Thompson completions over connective ring spectra are
equivalent to Bousfield-Kan completions over solid rings. The present work
allows us to show, in a subsequent paper, that the Bendersky-Thompson homotopy
spectral sequences over arbitrary ring spectra have well-behaved composition
pairings.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/kuhn-mc
Title: The McCord model for the tensor product of a space and a
commutative ring spectrum.
Author: Nicholas J. Kuhn
AMS classification: Primary 55P43; Secondary 18G55
Author's address: Department of Mathematics, University of Virginia,
Charlottesville, VA 22904
Email: njk4x@virginia.edu
Abstract:
This paper begins by noting that, in a 1969 paper in the Transactions,
M.C.McCord introduced a construction that can be interpreted as a model
for the categorical tensor product of a based space and a topological
abelian group. This can be adapted to Segal's very special
Gamma--spaces, and then to a more modern situation: (K tensor R) where K
is a based space and R is a unital, augmented, commutative, associative
S--algebra.
The model comes with an easy-to-describe filtration. If one lets K =
S^n, and then stabilize with respect to n, one gets a filtered model for
the Topological Andre--Quillen Homology of R. When R = Omega^{infty}
Sigma^{infty} X, one arrives at a filtered model for the connective
cover of a spectrum X, constructed from its 0th space.
Another example comes by letting K be a finite complex, and R the
S--dual of a finite complex Z. Dualizing again, one arrives at
G.Arone's model for the Goodwillie tower of the functor sending Z to the
suspension spectrum of Map(K,Z).
Applying cohomology with field coefficients, one gets various spectral
sequences for deloopings with known E_1--terms. A few nontrivial
examples are given.
In an appendix, we describe the construction for unital, commutative,
associative S--algebras not necessarily augmented.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Martino-Priddy/mobiushopf
Minami-Webb type decompositions for compact Lie groups
John Martino and Stewart Priddy
We extend to compact Lie groups some stable classifying space
decompositions of Minami, following Webb. One notable feature of Webb's
work is the use of a combinatorial Möbius function to encode p-local
information about the cohomology of a finite group. We wish to show
similar phenomena hold for compact Lie groups. However, for a compact
Lie group G one is faced with the problem of an infinite number of
conjugacy classes of p-toral subgroups, that is, extensions of tori by
finite p-groups. These groups are the analogs of p-groups for finite
groups. We circumvent this problem by considering a certain finite
G-complex which allows us to introduce combinatorial methods in the
compact Lie group case. This complex is based on the notion of
p-stubborn subgroups which arose earlier in modular representation
theory of finite groups (where they were called p-radical groups) in
connection with Alperin's conjecture in group cohomology and in the
study of homotopy classes of maps between classifying spaces of compact
Lie groups. We also derive a decomposition based on the corresponding
complex for elementary abelian p-subgroups. Several examples are given
to illustrate the various decompositions.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zzhou
(See the disclaimer at the top of this announcement).
Smith-Toda Spectrum $V(\infty)$ exists for all $p\geqslant 5$}
Zhou Xueguang
AMS classification numbers: 55Q
Address of author:
Department of Mathematics, Nankai University,
Tianjin 300071, People's Republic of China
Email address of author: zhengqb@eyou.co
Abstract
In this paper, we prove that the Smith-Toda spectrum $V(n)$ exists for
all non-negative integers $n$.
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