Mathai-Quillen forms and Lefschetz theory

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.Comment: 44 pages, Late

Traces and Characteristic Classes in Infinite Dimensions

This paper surveys topological results obtained from characteristic classes built from the two types of traces on the algebra of pseudodifferential operators of nonpositive order. The main results are the construction of a universal $\hat A$-polynomial and Chern character that control the $S^1$-index theorem for all circle actions on a fixed vector bundle over a manifold, and $|\pi_1({\rm Diff}(M^5))| = \infty$, for ${\rm Diff}(M^5)$ the diffeomorphism group of circle bundles $M^5$ with large first Chern class over projective algebraic Kaehler surfaces.Comment: Parts of Section 2.3 are not correct. This is discussed in T. McCauley, "S^1-Equivariant Chern-Weil Constructions on Loop Spaces," arXiv:1507.0862