Mathai-Quillen forms and Lefschetz theory

Abstract

Mathai-Quillen forms are used to give an integral formula for the Lefschetz number of a smooth map of a closed manifold. Applied to the identity map, this formula reduces to the Chern-Gauss-Bonnet theorem. The formula is computed explicitly for constant curvature metrics. There is in fact a one-parameter family of integral expressions. As the parameter goes to infinity, a topological version of the heat equation proof of the Lefschetz fixed submanifold formula is obtained. As the parameter goes to zero and under a transversality assumption, a lower bound for the number of points mapped into their cut locus is obtained. For diffeomorphisms with Lefschetz number unequal to the Euler characteristic, this number is infinite for most metrics, in particular for metrics of non-positive curvature.