De Rham Cohomology of Certain Diffeological Quotients

TL;DR

This paper extends the understanding of diffeological de Rham cohomology for leaf spaces of foliations under group actions, providing equivariant isomorphisms and explicit computations for quotients by Lie groups.

Contribution

It proves an equivariant version of a diffeological cohomology theorem and computes the cohomology of quotients from smooth Lie group actions, broadening previous results.

Findings

Established an $H$-equivariant isomorphism between diffeological forms and basic forms.

Computed the diffeological de Rham cohomology for quotients by Lie group actions.

Connected recent results on homogeneous spaces to foliation theory.

Abstract

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.