De Rham cohomology of diffeological spaces and foliations

Abstract

Let $(M,\mathcal{F})$ be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms ${\Omega }_b^{\ast }(M,\mathcal{F})$ of the foliation and the "De Rham complex" of the space of leaves $M/\mathcal{F}$ when considered as a "diffeological" quotient. Consequently, the two corresponding cohomology groups $H_b^{\ast }(M,\mathcal{F})$ and $H^{\ast }(M/\mathcal{F})$ are isomorphic.

13 pages