Transversality and Lefschetz numbers for foliation maps

TL;DR

This paper develops a topological approach to defining the Lefschetz number for foliation maps with invariant measures, extending transversality results and exploring connections to analytic Lefschetz numbers and trace formulas.

Contribution

It introduces a topological definition of the $ ext{Lambda}$-Lefschetz number for foliation maps, extending smooth approximation and transversality to foliations, and investigates its relation to analytic Lefschetz numbers.

Findings

Defined topological $ ext{Lambda}$-Lefschetz number for foliation maps.

Extended transversality results to the context of foliations.

Connected the topological and analytic Lefschetz numbers, proposing a Lefschetz trace formula.

Abstract

Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $\Lambda$ be a transverse invariant measure of $F$. Suppose that $\Lambda$ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the $\Lambda$-Lefschetz number of any leaf preserving diffeomorphism $(M,F)\to(M,F)$ is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological $\Lambda$-Lefschetz number is equal to the analytic $\Lambda$-Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to $F$.