An Atiyah-Bott formula for the Lefschetz number of a singular foliation

Abstract

This paper presents a formula for the Lefschetz number of a geometric endomorphism in the style of the Atiyah-Bott theorem. The underlying data consist, first, of a compact manifold and a nowhere vanishing smooth real vector field $\mathcal{T}$ that preserves some Riemannian metric, and second, a sequence of first order operators on sections of Hermitian vector bundles with connection whose curvature is annihilated by $\mathcal{T}$ and for which parallel transport along integral curves of $\mathcal{T}$ is unitary. Assuming that the operators of the sequence commute with the various covariant derivatives ${\mathcal{L}}_{\mathcal{T}}={\nabla }_{\mathcal{T}}$ and that their restriction to the spaces of sections annihilated by ${\mathcal{L}}_{\mathcal{T}}$ form a complex, an ellipticity condition gives finite-dimensionality of the resulting equivariant cohomology spaces. The Atiyah-Bott framework, adapted to give a geometric endomorphism only for the complex of ${\mathcal{L}}_{\mathcal{T}}$-parallel sections, together with the finiteness of cohomology allows for the definition of a Lefschetz number. Replacing the condition that the fixed points of the equivariant map $f$ associated with the endomorphism be simple by a condition on wave front sets, which is the underlying condition of Atiyah and Bott, yields that the set of closures of orbits by $\mathcal{T}$ left invariant by $f$ is finite, and then a formula similar to theirs, now relating the Lefschetz number with traces along these orbits.