Equivariant basic cohomology of singular Riemannian foliations

TL;DR

This paper develops an equivariant basic cohomology theory for singular Riemannian foliations, demonstrating its invariance, localization properties, and implications for the basic Euler characteristic and cohomology dimensions.

Contribution

It introduces equivariant basic cohomology for singular Riemannian foliations and proves localization results analogous to classical theorems, extending the theory to singular settings.

Findings

Equivariant basic cohomology is invariant under homotopies.

Localization of cohomology to closed leaves is established.

Basic Euler characteristic localizes to the set of closed leaves.

Abstract

We introduce the notion of equivariant basic cohomology for singular Riemannian foliations with transverse infinitesimal actions, and prove some elementary properties such as its invariance under homotopies. For the particular case of singular Killing foliations, there is the natural transverse action of its structural algebra. We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel. As applications, we obtain that the basic Euler characteristic also localizes to this set, and that the dimension of the basic cohomology of the localized foliation is less than or equal to that of the whole foliation, with equality occurring precisely in the equivariantly formal case.