Subject: new Hopf listings
From: Mark Hovey <hovey@picard.math.wesleyan.edu>
Date: 26 Nov 2000 09:01:35 -0500

Twelve new papers this time.

Mark Hovey

New papers appearing on hopf between 11/8/00 and 11/26/00.

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dorabiala/transfer

Abstract: The goal of this paper is to show that if a smooth fiber bundle
has a compact Lie group as structure group, then the transfer map for the
algebraic K-theory of spaces satisfies analogs of the Mackey Double coset
formula and Feshbach's sum formula. We also prove a "cut and
paste" formula for parametrized Reidemeister torsion.


Wojtek Dorabiala

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/local

Author: Stephen A. Mitchell
Title: The algebraic K-theory spectrum of a 2-adic local field
e-mail: mitchell@math.washington.edu

We explicitly determine the homotopy type of the 2-completed algebraic K-theory
spectrum KF, where F is an arbitrary finite extension of the 2-adic
rational numbers.


3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localhom

Author: Stephen A. Mitchell
Title: The mod 2 homology of the general linear group of a 2-adic local field
e-mail: mitchell@math.washington.edu

Let F be a finite extension of the 2-adic rational numbers. We compute
the mod 2 homology of the general linear group GL(F) as a Hopf algebra over
the Steenrod algebra.

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morton-Strickland/cost-hrk

The Hopf Rings for KO and KU

Dena S. Cowen Morton and  Neil P. Strickland

55N15; 55P43

math.AT/0011125

Department of Mathematics
Xavier University
Cincinnati
OH 45207
USA

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

We compute the mod two homology Hopf rings of the spectra KO and KU.
The spaces in these spectra are the infinite classical groups and
their coset spaces, and their homology was first calculated in the
Cartan seminars, but the Hopf ring structure was first determined in
the second author's unpublished PhD thesis.  The presentation given
here serves as an introduction to the first author's much more
intricate work on the connective spectrum bo.  The Hopf ring viewpoint
turns out to be very convenient for understanding the homological
effect of various maps between classical groups and fibrations of
their connective covers.


5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Saneblidze-Umble/SUpaper

Title: A Diagonal on the Associahedra
Authors: Samson Saneblidze and Ronald Umble
MSC-class: 57T30; 55U10; 55N20; 55N10
xxx.LANL.gov:  math.AT/0011065
Author's Addresses:
        A. Razmadze Mathematical Institute, M. Aleksidze St., 1, 380093 Tbilisi,
 Georgia
        Department of Mathemaitcs, Millersville Univ. of PA, Millersville, PA 17
551
Author's e-mail addresses:
        sane@rmi.acnet.ge
        ron.umble@millersville.edu

ABSTRACT:
An associahedral set is a combinatorial object generated by Stasheff
associahedra K_n and equipped with appropriate face and degeneracy operators.
Associahedral sets are similar in many ways to simplicial or cubical sets. In
this paper we give a formal definition of an associahedral set, discuss some
naturally occurring examples and construct an explicit geometric diagonal
\Delta :C_*(K_n) --> C_*(K_n) \otimes C_*(K_n) on the cellular chains C_*(K_n).
The diagonal \Delta, which is analogous to the Alexander-Whitney diagonal on
the simplices, gives rise to a diagonal on any associahedral set and leads
immediately to an explicit diagonal on the A_\infty operad. As an application
of this, we use the diagonal \Delta to define a tensor product in the A_\infty
category.

6.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-bce

The BP<n> cohomology of elementary Abelian groups

Neil P. Strickland

20J06; 55N20; 14L05

math.AT/0011120

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

In this paper we study E^*BV_k, where E=BP<m,n> is a cohomology theory
with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n]
(if m=0).  We use ideas from the theory of multiple level structures,
developed in earlier work of the author with John Greenlees.  Our
results apply when k is less than or equal to w=n+1-m.  If k<w we find
that E^*BV_k has no v_m-torsion.  When k=w, we show that the
v_m-torsion is annihilated by the ideal I_{n+1}=(v_m,...,v_n), and
that it is a free module on one generator over the ring
F_p[[x_0,...,x_{w-1}]].  We give three very different formulae for
this generator; it is not at all obvious that these give the same
element, and we only have a rather indirect proof of this.

7.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-cag

Chern approximations for generalised group cohomology
Neil P. Strickland

55N20; 55R40; 14L05
math.AT/9906111

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

Let G be a finite group and E is a suitable generalised cohomology
theory.  We define and study a ring C(E,G) that is the best possible
approximation to E^0BG that can be built using knowledge of the
complex representations of G.  We give a description of the
rationalisation of C(E,G) in terms of generalised characters, and we
study some cases in which the map C(E,G) -> E^0BG is an isomorphism.

8.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-csi

Common subbundles and intersections of divisors

Neil P. Strickland
55N20; 14L05; 14M15

math.AT/0011123

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

Let V_0 and V_1 be complex vector bundles over a space X.  We
use the theory of divisors on formal groups to give obstructions in
generalised cohomology that vanish when V_0 and V_1 can be
embedded in a bundle U in such a way that V_0\cap V_1 has
dimension at least k everywhere.  We study various algebraic
universal examples related to this question, and show that they arise
from the generalised cohomology of corresponding topological
universal examples.  This extends and reinterprets earlier work on
degeneracy classes in ordinary cohomology or intersection theory.

9.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-fsfg

Formal schemes and formal groups

Neil P. Strickland
14L05; 55N22
math.AT/0011121

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

We set up a framework for using algebraic geometry to study the
generalised cohomology rings that occur in algebraic topology.  This
idea was probably first introduced by Quillen and it underlies much of
our understanding of complex oriented cohomology theories, exemplified
by the work of Morava.  Most of the results have close and well-known
analogues in the algebro-geometric literature, but with different
definitions or technical assumptions that are often inconvenient for
topological applications.  We merely put everything together in a
systematic and convenient way.

10.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-ghd

Gross-Hopkins duality

Neil P. Strickland
55N20; 55P42; 20E18
math.AT/0011108

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

We give a new and simpler proof of a result of Hopkins and Gross
relating Brown-Comenetz duality to Spanier-Whitehead duality in the
K(n)-local stable homotopy category.


11.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-kld

K(n) local duality for finite groups and groupoids

Neil P. Strickland
55P42; 55P60; 55R40
math.AT/0011109

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

Included postscript file: st-kld.eps

We define an inner product (suitably interpreted) on the K(n)-local
spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid.
This gives an inner product on E^*BG_+ for suitable K(n)-local ring
spectra E.  We relate this to the usual inner product on the
representation ring when n=1, and to the Hopkins-Kuhn-Ravenel
generalised character theory.  We show that LG is a Frobenius algebra
object in the K(n)-local stable category, and we recall the connection
between Frobenius algebras and topological quantum field theories to
help analyse this structure.  In many places we find it convenient to
use groupoids rather than groups, and to assist with this we include a
detailed treatment of the homotopy theory of groupoids.  We also
explain some striking formal similarities between our duality and
Atiyah-Poincare duality for manifolds.

12.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/st-pmm

Products on MU-modules

Neil P. Strickland

55T25

math.AT/0011122

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

Included postscript file: st-pmm.eps

We use the new categories of spectra and MU-modules constructed by
Elmendorf, Kriz, Mandell and May to get improved results about
multiplicative structures on spectra such as P(n) and E(n),
particularly in the case p=2.


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