A Transverse Averaging Operator and Cohomology of Quotients by Non-closed Subgroups

TL;DR

This paper introduces a new transverse averaging operator for basic forms in Riemannian foliations, enabling computation of diffeological de Rham cohomology of certain homogeneous spaces via Lie algebra cohomology.

Contribution

It develops a purely infinitesimal averaging operator that preserves basic cohomology classes and applies it to compute cohomology of quotients by non-closed subgroups.

Findings

The operator maps closed basic forms to invariant basic forms within the same cohomology class.

Under certain conditions, the de Rham cohomology of G/H is isomorphic to Lie algebra cohomology.

The isomorphism simplifies when the Lie subalgebra is an ideal, removing the compactness requirement.

Abstract

In this article, we introduce a transverse averaging operator for basic forms on a Riemannian foliation equipped with an isometric transverse Lie algebra action, under the assumption that the leaf closure space is compact. Unlike the classical averaging operator in equivariant geometry, which is defined by integration over a compact Lie group, our operator is built purely from infinitesimal transverse data and does not require any global group action. We show that it sends every closed basic form to an invariant basic form representing the same basic cohomology class. As a main application, we compute the diffeological de Rham cohomology of the homogeneous space $G/H$, where $G$ is a connected Lie group, not necessarily compact, and $H$ is a connected Lie subgroup, not necessarily closed. Let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, respectively. Assuming that $\mathfrak g$ is of compact type and that $G/\overline{H}$ is compact, we prove that \[ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g,\mathfrak h). \] If, in addition, $\mathfrak h$ is an ideal in $\mathfrak g$, then under the weaker assumption that $G/\overline{H}$ is compact, we obtain \[ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g/\mathfrak h), \] without requiring $\mathfrak g$ to be of compact type.