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

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.