Subject: Re: fixed point theorem
Date: Mon, 20 Jan 2003 10:48:36 -0600 (CST)
From: "Carlos Prieto (113)"
To: Don Davis
As an answer to Peter McBurney's question, there are generalizations to
Brouwer's fixed point theorem as the following.
Theorem. Let X be a contractible compact ENR (or even an ANR) and let f:X
-> X be continuous. Then f has a fixed point.
Pf. Under the hypotheses, the Lefschetz number L(f) is nonzero.
Therefore, by the Lefschetz fixed point theorem, f has a fixed point. qed
There are generalizations of the Lefschetz fixed point theorem to
situations similar to that of Brouwer's theorem of 1960. See, for
instance, [C. Prieto, manuscripta math. 47, 233 - 249 (1984)]. There are
also some results by Dimovsky & Geoghegan [Forum Math. 2 (1990), No. 2
125 -154]. Further results in that direction can be seen in a recent paper
by W. Marzantowicz and myself (preprint available - can be electronically
sent).
Sincerely,
Carlos Prieto
On Mon, 20 Jan 2003, Don Davis 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
>
> ****************************************************************
>
>
--
===================
PROF. CARLOS PRIETO
Instituto de Matemáticas, UNAM
04510 México, DF, MÉXICO
cprieto@math.unam.mx
Tel. (++52-55) 5622-4489,-4520
Fax (++52-55) 5616 0348
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