Subject: new Hopf listings
From: Mark Hovey
Date: 04 Mar 2000 10:31:18 -0500
Sorry for the delay; I seem to be getting old and tired.
6 new papers this time.
Mark Hovey
New papers uploaded to hopf between 1/29/00 and 3/4/00.
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Morava/amrrrfls
A renormalized Riemann-Roch formula and the Thom isomorphism for the
free loop space
Authors:
Matthew Ando
mando@math.uiuc.edu
Jack Morava
jack@math.jhu.edu
We show that the fixed-point formula in an equivariant
complex-oriented cohomology theory $E$, applied to the free loop space
of a manifold $X$, may be viewed as a (renormalized) Riemann-Roch
formula for the quotient of the group law of $E$ by a free cyclic
subgroup. If $E$ is $K$-theory, this explains how the elliptic genus
associated to the Tate elliptic curve emerges from Witten's analysis
of the fixed-point formula in $K$-theory. In general this quotient
is not representable, but we show that its torsion subgroup is. In the
case that $E$ is the Borel theory associated to the Lubin-Tate theory
$E_n$, this leads to a description of the functor represented by
$E_n[[q]], analogous to the relationship between the Tate curve and
$K$-theory. For a more general equivariant $E$, we show that the
formal products which arise in this discussion may be naturally
viewed as Thom classes for Thom prospectra as considered by
Cohen-Jones-Segal. These prospectra seem to define interesting models
for the physicists' space of `small' loops on $X$.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Crespo-Saumell/aqfh
Title: Non-simply connected $H$-spaces with finiteness conditions
Authors: Carlos Broto, Juan A. Crespo and Laia Saumell
e-mail addresses: broto@mat.uab.es, chiqui@crm.es, and laia@mat.uab.es
This article is concerned with homotopy properties of $H$-spaces $X$
that are reflected in the module of indecomposables $QH^*(X;\F_p)$. It
is shown that mod $p$ $H$-spaces $X$ of finite type with finite
transcendence degree mod $p$ cohomology and locally finite
$QH^*(X;\F_p)$ are $B\Z/p$-null spaces, Eilenberg-MacLane spaces
$K(\padic,2)$, $K(\Z/p^r,1)$, and extensions of those. If we restrict
attention to $H$-spaces with noetherian mod $p$ cohomology algebra, then
we are left with finite mod $p$ $H$-spaces and Eilenberg-MacLane spaces.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fisher/bous
Title: A Proof of an Exponent Conjecture of Bousfield
Author: Michael J. Fisher
Email: mjf7@lehigh.edu
Abstract: Let p be a fixed odd prime. In this paper we prove an exponent
conjecture of Bousfield, namely that the p-exponent of the spectrum Phi SU(n)
is (n-1) + nu_p((n-1)!) for n >= 2.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Grodal/limsub
Title: Higher limits via subgroup complexes
Author: Jesper Grodal
Email: jg@math.mit.edu
Abstract:
We study the higher derived functors of the inverse limit of a functor
F: D --> Z_{(p)}-mod, where D is one of the standard categories which
arise when studying the homotopy theory of the classifying space of a
finite group G, e.g., the orbit category or the Quillen category of G.
These higher limits are of importance e.g., for the study of maps
between classifying spaces as well as for group cohomology.
We show that these higher limits can be identified with the
G-equivariant Bredon cohomology of the subgroup complex of
p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G)
with values in a G-local coefficient system. We examine when smaller
complexes can be used e.g., taking only p-radical subgroups,
p-centric subgroups, elementary abelian p-subgroups or various
subcollections thereof.
Since the subgroup complexes are finite complexes, and often rather
small, this provides concrete, computable formulas for these higher
limits, generalizing earlier work of especially Jackowski-McClure-
Oliver. It also gives a conceptual explanation of high dimensional
vanishing results previously established in more indirect ways.
As an application we look at the special case where all the higher
limits vanish, as for example is the case for group cohomology. If F
is a functor on the orbit category our formulas for the higher limits
in this case yield five different expressions of F(G) in terms of
values of F on proper subgroups. Two of these are `classical' namely
Webb's exact sequence of Mackey functors and a formula for calculating
stable elements, previously obtained using Alperin's fusion theorem.
Examining this case also leads to improvements of sharpness results
of homology decompositions due to Dwyer and others.
Central to many of the proofs are properties of the Steinberg chain
complex of a finite group G, as well as other concepts from the
emerging Lie theory for arbitrary finite groups of Alperin, Webb, and
others.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhang-Woo/forgetnew1
Title: Phantom maps and Forgetful maps
Authors:
Jianzhong Pan
Institute of Math.,Academia Sinica ,Beijing China and
Department of Mathematics Education , Korea University , Seoul , Korea
email: pjz62@hotmail.com
Moo Ha Woo
Department of Mathematics Education , Korea University , Seoul , Korea
ABSTRACT:
In this note, we attack a question posed ten years ago by Tsukiyama
about the injectivity of the so- called Forgetful map. We show that we
can insert the Forgetful map in an exact sequence and that the problem
can be reduced to the computation of the sequence which turns out
unexpectedly to be related to the phantom map problem and the famous
Halperin conjecture in rational homotopy theory.
Remark:This is an upgraded version of a preprint which has been
on the archive. A problem in Theorem2.8 has been corrected
following a suggestion from K.Iriye.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Karoubi/A_descent_theorem
Max KAROUBI
A descent theorem in topological K-theory
karoubi@math.jussieu.fr
Let A be a Banach algebra and A' its complexification. In this paper we
show that the homotopy fixed point set of K(A'), the topological K-theory
space of A', under complex conjugation is just K(A), the topological
K-theory space of A. This result generalizes the well known fact that BO is
BU^hZ/2. The proof uses in an essential way Atiyah's KR theory and the
Clifford algebra definition of higher K-groups.
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