2026/05/03 by Yi Lin
Mathematics · #math.DG #math.SG
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 H0 denote its identity component, and let \mathcal F be the foliation by H0-orbits. If H is second countable, or, in the non-second-countable case, if the induced component-group action on M/H0 satisfies a natural subduction condition, then pullback by the quotient map πH:M\longrightarrow M/H induces a canonical isomorphism of cochain complexes Ω^\bullet(M/H)≅Ω^\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⊂ G into a broader foliation-theoretic framework, from which it follows as a direct consequence.