Uniform Lefschetz fixed-point theory

Tsuyoshi Kato; Daisuke Kishimoto; Mitsunobu Tsutaya

Paper

Uniform Lefschetz fixed-point theory

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

of a uniformly continuous map

of a uniform simply-connected noncompact complete Riemannian manifold of bounded geometry

satisfying

, and prove that

if and only if

is uniformly homotopic to a strongly fixed-point free (without fixed-points on

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.