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7 new papers this month, from Ausoni-Rognes, DavisD, Gonzalez-Wilson,
Kitchloo-Wilson(2), Neusel, and Neusel-Sezer. 

                  Mark Hovey

New papers appearing on hopf between 8/13/07 and 9/24/07


1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ausoni-Rognes/ausoni-rognes-kku

Title:
Rational algebraic K-theory of topological K-theory.

Authors:
Christian Ausoni and John Rognes.

MSC-class: 19D55; 55N99
arXiv:0708.2160v1 [math.KT]

Christian Ausoni 
Mathematical Institute
University of Bonn

John Rognes
Department of Mathematics
University of Oslo

Abstract:
We show that after rationalization there is a homotopy 
fiber sequence BBU -> K(ku) -> K(Z). We interpret this 
as a correspondence between the virtual 2-vector bundles 
over a space X and their associated anomaly bundles over 
the free loop space LX. We also rationally compute K(KU) 
by using the localization sequence, and K(MU) by a method 
that applies to all connective S-algebras.

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/pcpt

Homotopy type and v1-periodic homotopy groups of p-compact groups

Donald M. Davis

Lehigh University, Bethlehem, PA 18015

Abstract:

We determine the v1-periodic homotopy groups of all irreducible
p-compact groups.  In the most difficult, modular, cases, we follow a
direct path from their associated invariant polynomials to these
homotopy groups.  We show that, with several exceptions, every
irreducible p-compact group is a product of explicit
spherically-resolved spaces which occur also as factors of p-completed
Lie groups.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Wilson/products

The ${BP}$-theory of two-fold  products of projective spaces

Jes\'us Gonz\'alez
Departamento de Matem\'aticas
Centro de Investigaci\'on y de Estudios Avanzados del IPN

W. Stephen Wilson
Department of Mathematics 
Johns Hopkins University 

We compute the BP (co)homology of the product of two
(stunted) projective spaces.  The behavior under maps
(particularly of the Tor term) is studied.  This is used
extensively by Kitchloo and Wilson in their work on
non-immersions.  Additional work with lens spaces is also
included.

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions1

The second real Johnson-Wilson theory and non-immersions of $RP^n$

Nitu Kitchloo
Department of Mathematics  
University of California, 
San Diego (UCSD) 

W. Stephen Wilson
Department of Mathematics 
Johns Hopkins University 

Hu and Kriz construct the real Johnson-Wilson
spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$ periodic, from
the $2(2^n-1)$ periodic spectrum $E(n)$.  $ER(1)$ is just
$KO_{(2)}$ and $E(1)$ is just $KU_{(2)}$.  We compute
$ER(n)^*(RP^\infty)$ and set up a Bockstein spectral sequence to
compute $ER(n)^*(-)$ from $E(n)^*(-)$.  We combine these to
compute $ER(2)^*(RP^{2n})$ and use this to get new
non-immersions for real projective spaces.  Our lowest
dimensional new example is an improvement of 2 for
$RP^{48}$.

5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions2

The second real Johnson-Wilson theory and non-immersions of $RP^n$, Part II.

Nitu Kitchloo
Department of Mathematics  
University of California, 
San Diego (UCSD) 

W. Stephen Wilson
Department of Mathematics 
Johns Hopkins University 

This paper is a continuation of the study begun in the paper
with the same name.  We analyze $ER(2)^{16*+8}(RP^{2n})$ and
compute $ER(2)^*(RP^{16K+1})$ and use these to prove more
non-immersion theorems for $RP^n$ including many in fairly
low dimensions.  In particular, we get 12 new non-immersion
results for $RP^n$ where $n \le 192$, the range included in
Don Davis's tables.  These complement the 10 already found in
part I.

6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmidappendix

title: Degree Bounds and the Regular Representation: Appendix

author: Mara D. Neusel

subjclass[2000]: Primary 13A50
keywords: Invariant
Theory of Finite Permutation Groups, Permutation Representation, Cohen-Macaulay, Gorenstein, Complete Intersection, Hypersurface, Polynomial Algebra, Pseudo-Reflection

abstract:

Let G be a matrix group consisting of permutation matrices. Let F and
K be two different fields. We show that if the polynomial invariants
F[V]^G and K[V]^G are both Cohen-Macaulay, then they are
simultaneously Gorenstein, complete intersections, hypersurfaces,
resp. polynomial. Thus Cohen-Macaulay rings of permutation invariants
are polynomial exactly when G is generated by pseudo-reflections.

7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/psquare

title: the invariants of modular indecomposable representations  of Z_{p^2}

authors: Mara D. Neusel and M\"ufit Sezer

abstract:
We consider the invariant ring for an indecomposable
representation of a cyclic group of order p^2 over a field F of
characteristic p. We describe a set of F-algebra generators of this 
ring of invariants, and thus derive an upper bound for the largest
degree of an element in a minimal generating set for the ring of
invariants. This bound, as a polynomial in p, is of degree two.

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