Subject: Re: fixed point theorem
Date: Thu, 23 Jan 2003 01:34:37 -0800
From: Dan Gottlieb
To: Don Davis
CC: gottlieb@math.ucla.edu, "rfb@math.ucla."
Peter McBurney wrote
Subject: Topology list posting
Date: Thu, 16 Jan 2003 07:46:11 +0000
From: Peter McBurney
Hello --
Does anyone know of a generalization of Browder's Fixed Point
Theorem
from R^n to arbitrary topological spaces, or to categories of
same?
*******
Theorem (Browder, 1960): Suppose that S is a non-empty, compact,
convex
subset of R^n, and let
f: [0,1] x S --> S
be a continuous function. Then the set of fixed points
{ (x,s) | s = f(x,s), x \in [0,1] and s \in S }
contains a connected subset A such that the intersection of A with
{0} x
S is non-empty and the intersection of A with {1} x S is
non-empty.
*******
Many thanks,
-- Peter McBurney
University of Liverpool, UK
****************************************************************
In fact Browder's theorem can be generalized to those S which are compact
Euclidean Neighborhood Retracts when f is a homotopy of a map with nonzero
Lefschetz number. In addition, the homotopy f can be generalized to a self
map of
the total space E of a fiber bundle over the identity map of the base B ,
where the
fibre is S and f restricted to a fibre has nonzero Lefschetz number. In this
generalization, B plays the role of I , and the conclusion reads that f has
a
connected set of fixed points which intersects every fibre of the fibre
bundle.
You can find relevant material on this on my webpage
http://www.math.purdue.edu/~gottlieb/Papers/papers.html
Relevant papers are:
11. Vector Fields and Classical Theorems of Topology , Rendiconti del
Seminario Matematico e Fisico, Milano, 60 (1990), pp. 193-203.
- AMS-TeX Source file In dvi format In PostScript format
15. Lectures on Vector Fields and the Unity of Mathematics , Proceedings of
the Topology and Geometry Research Center, TGRC-KOSEF Department of
Mathematics, Kyungpook National University. ed. Jin Suk Pak and Chan-Young,
Vol. 3 (1993), pp. 87-114.
- AMS-TeX Source file In dvi format In PostScript format
9. Vector fields and transfers, , Manuscripta Mathematica , 72 (1991) pp.
111-130. (with J.C. Becker)
- AMS-TeX Source file In dvi format In PostScript format
5. The Index of Discontinuous Vector Fields , New York Journal of
Mathematics , 1 (1994-1995), pp. 130-148 ( with Geetha Samarayanake)
- .tex ( not the official version), .dvi, .ps, .pdf
I don't explicitly prove the theorem I stated above, but section 9 of
papers 11 and 15 give a new proof of what is called Gottlieb's Theorem
in the literature.
The key point in the argument is a version of the theorem
stated above which should make its proof clear. It shows how to replace
the mapping by a vector field, where the fixed point index is the index of
the vector field. Next, the index can be defined for locally defined vector
fields and it is invariant under a generalization of homotopy which I call
otopy, This generalization permits us to consider only the connected
components of the zeros of the otpies , and the invariance of the index
for these otopies proves the required result. This is done carefully in
paper
5. In paper 9, Becker and I show that these otopies give rise to transfers
in fibre bundles, which allows us to generalize the theorem to fibre bundle
maps.
Dan Gottlieb