De Rham cohomology of diffeological spaces and foliations

TL;DR

This paper establishes a canonical isomorphism between the base-like cohomology of a foliated manifold and the De Rham cohomology of its leaf space viewed as a diffeological space, linking foliation theory with diffeology.

Contribution

It proves the isomorphism between the base-like cohomology of foliations and the De Rham cohomology of the leaf space as a diffeological space, providing a new perspective.

Findings

Base-like cohomology is isomorphic to the De Rham cohomology of the leaf space.

The leaf space can be effectively studied using diffeological methods.

The result bridges foliation theory and diffeological space analysis.

Abstract

Let $(M,\mathcal{F})$ be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms $\Omega^*_b(M,\mathcal{F})$ 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 $H^*_b(M,\mathcal{F})$ and $H^*(M/\mathcal{F})$ are isomorphic.