Lefschetz Theory, Geometric Thom Forms and the Far Point Set

Abstract

The far point set of a self-map of a closed Riemannian manifold $M$ is defined to be the set of points mapped into their cut locus. We prove that the far point set of a map $f$ with Lefschetz number $L(f) \neq \chi(M)$ is infinite unless $M$ is a sphere. There are homology classes supported near $\text{Far}(f)$ which determine $L(f)-\chi(M).$ Using geometric representatives of Thom classes, we obtain a geometric integral formula for the the Lefschetz number, which specializes to the Chern-Gauss-Bonnet formula when $f=\text{Id}.$ We compute this formula explicitly for constant curvature metrics. Finally, we give upper and lower bounds for $L(f)$ in terms of the geometry and topology of $M$ and the differential of $f$.