Transversality and Lefschetz numbers for foliation maps

Abstract

Let $${\mathcal{F}}$$ be a smooth foliation on a closed Riemannian manifold M, and let Λ be a transverse invariant measure of $${\mathcal{F}}$$ . Suppose that Λ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the Λ-Lefschetz number of any leaf preserving diffeomorphism $$(M,{\mathcal{F}}) \rightarrow (M,{\mathcal{F}})$$ 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 Λ-Lefschetz number is equal to the analytic Λ-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 $$\mathcal{F}$$ .