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

TL;DR

This work extends the Atiyah–Bott Lefschetz framework to singular foliations generated by a nowhere-vanishing vector field $\mathcal{T}$ on a compact manifold, via an equivariant cohomology built from $\mathcal{L}_{\mathcal{T}}$-parallel sections and a sequence of first-order operators with compatible connections. A wave-front set transversality condition replaces the traditional fixed-point simplicity assumption, ensuring a finite set of orbit closures fixed by the endomorphism and enabling a Lefschetz formula that sums contributions from closed $G$-orbits with traces along orbits and isotropy data. The method blends a de Rham-type model, elliptic Hodge theory on basic forms, averaging over a compact torus, and $L^2$ trace techniques (heat-kernel regularization) to produce an explicit fixed-point formula. The results generalize classical Atiyah–Bott to singular foliations and relate to boundaries of complex $b$-manifolds and CR structures, with the wave-front approach providing a robust nondegeneracy condition in the presence of singularities.

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.

Theorems & Definitions (50)

• Proposition 1.6
• Proposition 1.8
• Lemma 1.18
• Lemma 1.20
• Theorem 1.23
• Proposition 1.25
• Theorem 1.26
• Theorem 1.28
• Example 2.6
• Proposition 3.2