Points fixes des applications compactes dans les espaces ULC

Abstract

A topological space is locally equiconnected if there exists a neighborhood $U$ of the diagonal in $X\times X$ and a continuous map $\lambda:U\times[0,1]\to X$ such that $\lambda(x,y,0)=x$, $\lambda(x,y,1)=y$ et $\lambda(x,x,t)=x$ for $(x,y)\in U$ and $(x,t)\in X\times[0,1]$. This class contains all ANRs, all locally contractible topological groups and the open subsets of convex subsets of linear topological spaces. In a series of papers, we extended the fixed point theory of compact continuous maps, which was well developped for ANRs, to all separeted locally equiconnected spaces. This generalization includes a proof of Schauder's conjecture for compact maps of convex sets. This paper is a survey of that work. The generalization has two steps: the metrizable case, and the passage from the metrizable case to the general case. The metrizable case is, by far, the most difficult. To treat this case, we introduced in [4] the notion of algebraic ANR. Since the proof that metrizable locally equiconnected spaces are algebraic ANRs is rather difficult, we give here a detaled sketch of it in the case of a compact convex subset of a metrizable t.v.s.. The passage from the metrizable case to the general case uses a free functor and representations of compact spaces as inverse limits of some special inverse systems of metrizable compacta.