Points fixes des applications compactes dans les espaces ULC

TL;DR

This paper surveys the extension of fixed point theory for compact maps from ANRs to all locally equiconnected spaces, including a detailed proof for the metrizable case and a method for the general case.

Contribution

It extends fixed point theory to locally equiconnected spaces, proving Schauder's conjecture for convex sets and introducing algebraic ANRs for the metrizable case.

Findings

Extended fixed point theory to locally equiconnected spaces.

Proved Schauder's conjecture for compact maps of convex sets.

Developed methods for passing from metrizable to general spaces.

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.