Re: Generalization of Browder's F.P. Theorem?

[Steven J Vickers <s.j.vickers@cs.bham.ac.uk>]

I'm intrigued by Peter McBurney's question [below]. It looks rather like a
question of the constructive content of Brouwer's fixed point theorem.

Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the
topos of sheaves over [0,1], f is just a continuous endomap of S. If
Brouwer's theorem were constructively true then f would have a fixpoint,
and that would come out as a continuous section of the projection [0,1]xS
-> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x,
g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in
[0,1]} would be as required.

However, the proof of Brouwer that I've seen is not constructive - it goes
by contradiction. So maybe the requirements on A are a way of getting
constructive content in Brouwer's result.

What is known constructively about Brouwer's fixed point theorem?

Steve Vickers.

Peter McBurney wrote:

> 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