Homotopies and the universal fixed point property

TL;DR

This paper explores the universal fixed point property in topological spaces, revealing constraints on continuous fixed point selections and analyzing implications across various mathematical structures.

Contribution

It introduces the universal fixed point property and investigates its implications in topology, Banach spaces, manifolds, CW complexes, and combinatorics.

Findings

Restrictions on continuous fixed point selections

Application to convex subspaces of Banach spaces

Insights into topology of manifolds and CW complexes

Abstract

A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.