From: Nick Halloway <snowe@rain.org>
Newsgroups: sci.math
Subject: Re: What are the Neatest Fixed Point Theorems/Facts to Teach?
Date: Sat, 2 Nov 1996 08:49:53 -0800
x-no-archive: yes

This brought up the question:  are there some conditions on homology 
groups for a space such that a fixed point theorem holds?  If a 
space is contractible, does a fixed point theorem hold, i.e.
for f continuous: X --> X, does X have a fixed point? 

And, does a fixed point theorem hold for the projective plane?  It
seems like it would, since for S_2 the only functions without fixed
points: S_2 --> S_2 seem to involve inversions.

The Lefschetz fixed-point theorem answers some of these:  it implies
that if X is a path-connected compact polyhedron for which the 
homology groups H_n(X) are finite for n > 0, then every continuous
f: X --> X has a fixed point.  In particular for n even, real
projective space RP^n satisfies a fixed-point theorem.  

What generalizations are there?  If X is a contractible space,
does it satisfy a fixed-point theorem?