Uniform Lefschetz fixed-point theory
We develop a Lefschetz fixed-point theory for uniformly continuous self-maps of noncompact manifolds of bounded geometry. For a uniform, simply-connected, complete Riemannian manifold $M$ of bounded geometry and a uniformly continuous map $f\colon M\to M$ satisfying $d(f,1)<\infty$, we define a uniform Lefschetz class $\mathscr{L}(f)$ and show that $\mathscr{L}(f)=0$ if and only if $f$ is uniformly homotopic to a map having neither fixed points nor fixed points at infinity; an analogous obstruction class characterizes the existence of a nonvanishing bounded uniformly continuous vector field on $M$. Both classes take values in the $0$-th uniformly finite homology of Block and Weinberger, and hence vanish automatically when $M$ is nonamenable. When $M$ is a Galois covering of a closed manifold, we further establish a Lefschetz-Hopf-type formula for $\mathscr{L}(f)$ and an extension of the Poincaré-Hopf theorem for vector fields. The key new tool is \emph{uniform bounded cohomology}, a variant of Gromov's bounded cohomology for metric spaces satisfying uniform homotopy invariance, excision and Poincaré duality. As a further application, we show that uniform and Lipschitz homotopy theory coincide for uniform simplicial complexes; combined with our obstruction theory, this allows us to obtain Lipschitz analogues of the above results.