De Rham Cohomology of Certain Diffeological Quotients

Hector, Mac\'ıas-Virgós, and Sanmart\'ın-Carbón identified the complex of diffeological differential forms on the leaf space of a foliation with the complex of basic differential forms on the foliated manifold, yielding a canonical isomorphism of cochain complexes. In this paper, we prove an equivariant version of their theorem. More precisely, let a group $H$ act smoothly on a foliated manifold $(M,\mathcal F)$ by foliation-preserving diffeomorphisms, so that the action descends to the leaf space $M/\mathcal F$. We show that the canonical identification between diffeological differential forms on $M/\mathcal F$ and basic differential forms on $(M,\mathcal F)$ is $H$-equivariant.
As an application, we compute the diffeological de Rham cohomology of quotients $M/H$ arising from smooth, locally free actions of Lie groups that are not necessarily connected or second countable. More precisely, let $H$ be a Lie group, not necessarily second countable, acting smoothly and locally freely on a second countable manifold $M$. Let $H_0$ denote its identity component, and let $\mathcal F$ be the foliation by $H_0$-orbits. If $H$ is second countable, or, in the non-second-countable case, if the induced component-group action on $M/H_0$ satisfies a natural subduction condition, then pullback by the quotient map $ \pi_H:M\longrightarrow M/H$ induces a canonical isomorphism of cochain complexes \[ \Omega^\bullet(M/H)\cong\Omega^\bullet(M,\mathcal F)^H. \] This places the recent computation of the diffeological de Rham cohomology of homogeneous spaces $G/H$ for dense Lie subgroups $H\subset G$ into a broader foliation-theoretic framework, from which it follows as a direct consequence.