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 =======================