De Rham Cohomology of Certain Diffeological Quotients

Hector, Mac\'{\i}as-Virg\'os, and Sanmart\'{\i}n-Carb\'on identified the complex of diffeological differential forms on the leaf space of a foliation with the complex of basic forms on the foliated manifold, yielding a canonical isomorphism of cochain complexes. In this short note we prove an equivariant version of their theorem: if a group $H$ acts smoothly on a foliated manifold $(M,\mathcal F)$ by foliation-preserving diffeomorphisms, so that the action descends to the leaf space $M/\mathcal F$, then this canonical identification 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 $H_0$ be 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 $\pi_H:M\to M/H$ identifies the de Rham complex of diffeological forms on $M/H$ with the complex of $H$-invariant basic forms: $$\Omega^\bullet(M/H)\cong \Omega^\bullet(M,\mathcal F)^H .$$ This places the recent result on homogeneous spaces $G/H$ for dense subgroups $H\subset G$ in a broader foliation-theoretic framework, from which it follows as a direct consequence.