Fixed-point theories on noncompact manifolds

Abstract

This paper gives three noncompact variants of Lefschetz–Nielsen fixed-point theory parallel to developments that have occurred in surgery theory. Thus, we study when a proper map can be properly homotoped, or boundedly homotoped, or even homotoped in a Cr bounded fashion to a fixed-point free map. As an example, the universal cover of a compact aspherical manifold always has a fixed-point free self-diffeomorphism Cr close to the identity for all r (although this is not the case, in general, for arbitrary infinite covers of such manifolds, or general universal covers).