Subject: new Hopf listings--January 2001
From: Mark Hovey
Date: 05 Feb 2001 09:52:19 -0500
These are the January papers, of which there are 13.
Mark Hovey
New papers appearing on hopf between 1/1/01 and 2/3/01
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker/regquotients
On the homology of regular quotients
Andrew Baker
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland.
a.baker@maths.gla.ac.uk
We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is
generated by a (finite or infinite) regular sequence. This generalizes
the Koszul complex for the case $s=1$. We easily deduce that for $s>1$,
the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction
$R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker-Lazarev/Rmod-ASS
On the Adams Spectral Sequence for $R$-modules
Andrew Baker \& Andrej Lazarev
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland.
a.baker@maths.gla.ac.uk
Department of Mathematics, University of Bristol, Bristol BS8 1TW, England.
A.Lazarev@bris.ac.uk
We consider the Adams Spectral Sequence for $R$-modules based on
commutative localized regular quotient ring spectra of a commutative
$S$-algebra $R$ in the sense of Elmendorf, Kriz, Mandell, May and
Strickland. The formulation of this spectral sequence is similar to the
classical case, and we reduce to algebra involving the cohomology of
certain `brave new Hopf algebroids' $E^R_*E$. In order to work out the
details we resurrect Adams' original approach to Universal Coefficient
Spectral Sequences for modules over an $R$ ring spectrum.
We show that the Adams Spectral Sequence for $S_R$ based on
$E=R/I[X^{-1}]$ converges to the homotopy of the $E$-nilpotent
completion which has homotopy
\[
\pi_*\hat{\mathrm{L}}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}.
\]
We also show that $\hat{\mathrm{L}}^R_ES_R$ is equivalent to
$\L^R_ES_R$, the Bousfield localization of $S_R$ with respect to
$E$-theory. This seems surprising since the spectral sequence collapses
at $\E_2$, but $\E_r$ does not have a vanishing line because of the
presence of polynomial generators of positive cohomological degree, thus
only one of Bousfield's two standard convergence criteria applies here
even though we have this equivalence. The details involve a construction
of the internal $I$-adic tower \[ R/I\la R/I^2\la\cdots\la R/I^s\la
R/I^{s+1}\la\cdots \] whose homotopy limit is $\hat{\mathrm{L}}^R_ES_R$.
Finally, we describe some examples for the case $R=MU$.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Greenlees/kubg
The Connective K-theory of Finite Groups
Robert Bruner and John Greenlees
MSC2000: Primary 19L41, 19L47, 19L64, 55N15.
Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30.
Department of Mathematics, School of Mathematics and Statistics,
Wayne State University, Hicks Building,
Detroit MI 48202-3489, Sheffield S3 7RH,
USA. UK.
rrb@math.wayne.edu, j.greenlees@sheffield.ac.uk
Included graphics files:
AdamsA4.eps
AdamsBip.eps
AdamsC2.eps
AdamsC4.eps
AdamsC5.eps
AdamsD8.eps
AdamsQ8.eps
AdamsSl23.eps
AdamsV2.eps
AdamsX.eps
ExtIE.eps
Extku.eps
Extl.eps
L.eps
Qrank4.eps
Qrank4lc.eps
T3rank6.eps
T3rank6lc.eps
Xku.eps
rank8.eps
string.eps
tku2.eps
Abstract:
This paper is devoted to the connective K homology and cohomology of
finite groups G. We attempt to give a systematic account from several
points of view.
In Chapter 1, following Quillen, we use the methods of algebraic
geometry to study the ring ku^*(BG) where ku denotes connective complex
K-theory. We describe the variety in terms of the category of abelian
p-subgroups of G for primes p dividing the group order. The variety is
obtained by splicing that of periodic complex K-theory and that of
integral ordinary homology, the interest lying in the way these parts
fit together. The main technical obstacle is that the Kunneth spectral
sequence does not collapse, so we have to show that it collapses up to
isomorphism of varieties.
In Chapter 2 we give several families of new complete and explicit
calculations of the ring ku^*(BG).
In Chapter 3 we consider the associated homology ku_*(BG), as a module
over ku^*(BG) by using the local cohomology spectral sequence. This
gives new specific calculations, but also illuminating structural
information, including remarkable duality properties.
Finally, in Chapter 4 we make a particular study of elementary abelian
groups V. Despite the group-theoretic simplicity of V, the detailed
calculation of ku^*(BV) and ku_*(BV) exposes a very intricate
structure, and gives a striking illustration of our methods. Unlike
earlier work, our description is natural for the action of GL(V).
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/JohnsonM/shfloop
Loop Spaces as Sheaves: A Sheaf-Theoretic View of Loop Spaces
Mark W. Johnson
\address {Department of Mathematics\\
University of Notre Dame\\
Notre Dame, IN 46556}
\email{johnson.295@nd.edu}
The context of enriched sheaf theory introduced in \cite{thesis}
provides a convenient viewpoint for models of the stable homotopy
category as well as categories of finite loop spaces. Also, the
languages of algebraic geometry and algebraic topology have been
interacting quite heavily in recent years, primarily due to the work of
Voevodsky and that of Hopkins. Thus, the language of Grothendieck
topologies is becoming a necessary tool for the algebraic topologist.
The current document is intended to give a somewhat relaxed introduction
to this language of sheaves in a topological context, using familiar
examples such as $n$-fold loop spaces and pointed $G$-spaces. This
language also includes the diagram categories of spectra from
\cite{MMSS} as well as spectra in the sense of \cite{Lewis}, which will
be discussed in some detail.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision
Title: Excision for simplicial sheaves on the Stein site
and Gromov's Oka Principle
Author: Finnur Larusson
AMS classification numbers: Primary: 32Q28; secondary: 18F10,
18F20, 18G30, 18G55, 32E10, 32H02, 55U35
arXiv:math.CV/0101103
Department of Mathematics
University of Western Ontario
London, Ontario N6A 5B7
Canada
larusson@uwo.ca
ABSTRACT: A complex manifold $X$ satisfies the Oka-Grauert property
if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak
equivalence for every Stein manifold $S$, where the spaces of
holomorphic and continuous maps from $S$ to $X$ are given the
compact-open topology. Gromov's Oka principle states that if $X$ has a
spray, then it has the Oka-Grauert property. The purpose of this paper
is to investigate the Oka-Grauert property using homotopical algebra.
We embed the category of complex manifolds into the model category of
simplicial sheaves on the site of Stein manifolds. Our main result is
that the Oka-Grauert property is equivalent to $X$ representing a finite
homotopy sheaf on the Stein site. This expresses the Oka-Grauert
property in purely holomorphic terms, without reference to continuous
maps.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert
This is another revised version of the proof of the Hilbert-Smith
conjecture.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Looptan
Title: The equivariant tangent bundle of a free smooth loopspace
Author: Jack Morava
AMS classification: 58Dxx; 53C29, 55P91
Address: The Johns Hopkins Uniperversity
e-mail: jack@math.jhu.edu
ABSTRACT: The space of free loops on a manifold X inherits an action of
the circle group \T. A Riemannian metric on X defines an equivariant
isomorphism of the complexified tangent bundle of the loopspace with
\bT X \otimes (\oplus \C(n)), where \C(n) is the standard one-dimensional
representation of \T, and \bT X \otimes \C is an equivariant bundle on the
loopspace, nonequivariantly isomorphic to the pullback of the complexified
tangent bundle of X along evaluation at the basepoint. On a flat manifold,
this analogue of Fourier analysis is quite familiar.
[Perhaps this is all nonsense; if so, please let me know.]
8.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity5
Title: Pretty Good Gravity
Author: Jack Morava
AMS Classification: 19Dxx, 57Rxx, 83Cxx
(not yet on xxx, but will be soon)
Address: Dept. of Mathematics, the Johns Hopkins Uniperversity
e-mail address: jack@math.jhu.edu
Abstract: A theory of topological gravity is a homotopy-theoretic
representation of the Segal-Tillmann topologification of a two-category
with cobordisms as morphisms. This note describes a relatively accessible
example of such a thing, suggested by the wall-crossing formulas of
Donaldson theory.
[This is a writeup of a talk at the RIMS Symposium on algebraic geometry
and integrable systems related to string theory, June 12-16, 2000.]
9.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Tate2MU
Title: Duality of Tate cohomology of framed circle actions
Author: Jack Morava
AMS classification: 19Dxx, 57Rxx, 83Cxx
Address: The Johns Hopkins University
Baltimore 21218 Maryland
e-mail:
Abstract:
The complex Mahowald pro-spectrum \CP^{\infty}_{-\infty} is not, as might
seem at first sight, Spanier-Whitehead self-dual; rather, its S-dual is its
own double suspension. This assertion makes better sense as a claim about
the Tate cohomology spectrum t_{\T}S^0 defined by circle actions on framed
manifolds. A subtle twist in some duality properties of infinite-dimensional
projective space results, which has consequences [via work of Madsen and
Tillmann] for the Virasoro symmetries [discovered by Witten and Kontsevich]
of the stable cohomology of the Riemann moduli space.
10.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moreno/moreno
Author: Guillermo Moreno
Title: Alternative elements in the Cayley--Dickson algebras
We describe the alternative elements in the Cayley-Dickson algebras
for n>3. Also we ``measure'' the failure of these algebras of being a
normed algebra in terms of the alternative elements.
11.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk-Schwede-Shipley/simplicia
l
Title: Simplicial structures on model categories and functors
Authors: Charles Rezk, Stefan Schwede, Brooke Shipley
To appear in American Journal of Mathematics
Institute for Advanced Study
School of Mathematics
Olden Lane
Princeton, NJ 08540, USA
rezk@ias.edu
Fakultat fur Mathematik
Universitat Bielefeld
33615 Bielefeld, Germany
schwede@mathematik.uni-bielefeld.de
Department of Mathematics
Purdue University
West Lafayette, IN 47907, USA
bshipley@math.purdue.edu
We produce a highly structured way of associating a simplicial category
to a model category which improves on work of Dwyer and Kan and answers
a question of Hovey. We show that model categories satisfying a certain
axiom are Quillen equivalent to simplicial model categories. A simplicial
model category provides higher order structure such as composable mapping
spaces and homotopy colimits. We also show that certain homotopy
invariant functors can be replaced by weakly equivalent simplicial,
or `continuous', functors. This is used to show that if a simplicial
model category structure exists on a model category then it is unique
up to simplicial Quillen equivalence.
12.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/ks-hgr
The homotopy groups $\pi_*(L_nT(m)\wedge V(n-2))$
Katsumi Shimomura
Department of Mathematics,
Faculty of Science,
Kochi University,
Kochi, 780-8520
Japan
katsumi@math.kochi-u.ac.jp
Let $V_{T(m)}(n)$ denote the spectrum such that
$BP_*(V_{T(m)}(n))=BP_*/I_{n+1}[t_1,\dots, t_m]$ for the ideal
$I_{n+1}=(p,v_1,\dots, v_{n})$. In the title, we write $T(m)\wedge
V(n-2)$ as $V_{T(m)}(n-2)$. Ravenel determined the structure of the
Adams-Novikov $E_2$-term for the homotopy groups
$\pi_*(L_nV_{T(m)}(n-1))$ for $n\le m+2$ and $n3$.
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