From: Nick Halloway
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?