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4 new papers this time, from BrownR, DavisD, Dwyer, and Gottlieb.  

                  Mark Hovey

New papers appearing on hopf between 4/09/03 and 5/13/03

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/noncommut-at

Title: Towards non commutative algebraic topology
Author: Ronald Brown

AMS Classification numbers: 55D15, 55U40, 18D35

Address of Author: Mathematics Division, School of Informatics,
University of Wales, Bangor, Gwynedd LL57 1UT, UK.

Email address of Author: r.brown@bangor.ac.uk

Text of Abstract: These are the transparencies (slightly edited) 
for a seminar at University College, London, on May 7, 2003. They 
give a quick overview of some background and some directions 
taken for algebraic methods for higher dimensional, non 
commutative, local to global problems, including some algebraic 
models of homotopy types. 

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/E7E8

Representation types and 2-primary homotopy groups of certain
compact Lie groups

Donald M. Davis

55Q52, 55T15, 57T20

Department of Mathematics
Lehigh University
Bethlehem, PA 18015
dmd1@lehigh.edu

Abstract:
Bousfield has shown how the 2-primary v1-periodic homotopy
groups of certain compact Lie groups can be obtained from
their representation ring with its decomposition into types
and its exterior power operations. He has formulated a
Technical Condition which must be satisfied in order that
he can prove his description is valid.

We prove that a simply-connected compact simple Lie group
satisfies his Technical Condition if and only if it is not
E6 or Spin(4k+2) with k not a 2-power.  We then use his
description to give an explicit determination of the
2-primary v1-periodic homotopy groups of E7 and E8.
This completes a program, suggested to the author by
Mimura in 1989, of computing the v1-periodic homotopy
groups of all compact simple Lie groups at all primes.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer/local

                     Localization

                     W. G. Dwyer

This is a largely expository paper, which describes the concept of
localization, as it usually comes up in topology, and gives some
examples of it. The examples include local homology and cohomology,
homological localizations of spaces and spectra, and localization with
respect to a map f. For appropriate choices of the map f, this last
gives constructions related to the Goodwillie calculus and to motivic
homotopy theory. There's also a proof that if a localization functor
exists, the higher order categorical invariants associated to
inverting the local equivalences are trivial. 

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Gottlieb/eigbndl

    EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE
                           Daniel Henry Gottlieb

Given a parameterized space of square matrices, the associated set of
eigenvectors forms some kind of a structure over the parameter space.
When is that structure a vector bundle? When is there a vector field of
eigenvectors? We answer those questions in terms of three obstructions,
using a Homotopy Theory approach. We illustrate our obstructions with
five examples. One of those examples gives rise to a 4 by 4 matrix
representation of the Complex Quaternions.  This representation shows
the relationship of the Biquaternions with low dimensional Lie groups
and algebras, Electro-magnetism, and Relativity Theory.  The
eigenstructure of this representation is very interesting, and our
choice of notation produces important mathematical expressions found in
those fields and in Quantum Mechanics. In particular, we show that the
Doppler shift factor is analogous to Berry's Phase.


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