Subject: Re: fixed point theorem Date: Thu, 23 Jan 2003 01:34:37 -0800 From: Dan Gottlieb To: Don Davis CC: gottlieb@math.ucla.edu, "rfb@math.ucla." Peter McBurney 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 **************************************************************** In fact Browder's theorem can be generalized to those S which are compact Euclidean Neighborhood Retracts when f is a homotopy of a map with nonzero Lefschetz number. In addition, the homotopy f can be generalized to a self map of the total space E of a fiber bundle over the identity map of the base B , where the fibre is S and f restricted to a fibre has nonzero Lefschetz number. In this generalization, B plays the role of I , and the conclusion reads that f has a connected set of fixed points which intersects every fibre of the fibre bundle. You can find relevant material on this on my webpage http://www.math.purdue.edu/~gottlieb/Papers/papers.html Relevant papers are: 11. Vector Fields and Classical Theorems of Topology , Rendiconti del Seminario Matematico e Fisico, Milano, 60 (1990), pp. 193-203. - AMS-TeX Source file In dvi format In PostScript format 15. Lectures on Vector Fields and the Unity of Mathematics , Proceedings of the Topology and Geometry Research Center, TGRC-KOSEF Department of Mathematics, Kyungpook National University. ed. Jin Suk Pak and Chan-Young, Vol. 3 (1993), pp. 87-114. - AMS-TeX Source file In dvi format In PostScript format 9. Vector fields and transfers, , Manuscripta Mathematica , 72 (1991) pp. 111-130. (with J.C. Becker) - AMS-TeX Source file In dvi format In PostScript format 5. The Index of Discontinuous Vector Fields , New York Journal of Mathematics , 1 (1994-1995), pp. 130-148 ( with Geetha Samarayanake) - .tex ( not the official version), .dvi, .ps, .pdf I don't explicitly prove the theorem I stated above, but section 9 of papers 11 and 15 give a new proof of what is called Gottlieb's Theorem in the literature. The key point in the argument is a version of the theorem stated above which should make its proof clear. It shows how to replace the mapping by a vector field, where the fixed point index is the index of the vector field. Next, the index can be defined for locally defined vector fields and it is invariant under a generalization of homotopy which I call otopy, This generalization permits us to consider only the connected components of the zeros of the otpies , and the invariance of the index for these otopies proves the required result. This is done carefully in paper 5. In paper 9, Becker and I show that these otopies give rise to transfers in fibre bundles, which allows us to generalize the theorem to fibre bundle maps. Dan Gottlieb