Uniform Lefschetz fixed-point theory

TL;DR

This paper extends Lefschetz fixed-point theory to noncompact manifolds with bounded geometry using a new uniform bounded cohomology, establishing conditions for fixed points of uniformly continuous maps.

Contribution

It introduces a uniform Lefschetz class for noncompact manifolds and develops a new cohomology theory to analyze fixed points in this setting.

Findings

The uniform Lefschetz class vanishes if and only if the map is homotopic to a fixed-point free map.

Development of uniform bounded cohomology as a tool for fixed-point theory.

Establishment of an obstruction theory based on the new cohomology.

Abstract

We develop the Lefschetz fixed-point theory for noncompact manifolds of bounded geometry and uniformly continuous maps. Specifically, we define the uniform Lefschetz class $\mathscr{L}(f)$ of a uniformly continuous map $f\colon M\to M$ of a uniform simply-connected noncompact complete Riemannian manifold of bounded geometry $M$ satisfying $d(f,1)<\infty$, and prove that $\mathscr{L}(f)=0$ if and only if $f$ is uniformly homotopic to a strongly fixed-point free (without fixed-points on $M$ and at infinity) uniformly continuous map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, which is a variant of bounded cohomology, and develop an obstruction theory formulated in terms of this cohomology.