Cohomology of diffeological spaces and foliations

TL;DR

This paper explores the relationship between basic cohomology of foliated manifolds and the De Rham cohomology of their leaf spaces, establishing an isomorphism under certain conditions.

Contribution

It introduces a morphism linking the De Rham cohomology of leaf spaces to basic cohomology, proving it is an isomorphism for Q-foliations.

Findings

Existence of a morphism from H(DF) to Hb(M,F) for any foliation

Isomorphism between these cohomologies for Q-foliations

Enhanced understanding of leaf space cohomology in diffeological terms

Abstract

Let (M,F) be a foliated manifold. We study the relationship between the basic cohomology Hb(M,F) of the foliation and the De Rham cohomology H(DF) of the space of leaves M/F as a quotient diffeological space. We prove that for an arbitrary foliation there is a morphism from H(DF) to Hb(M,F). It is an isomorphism when F is a Q-foliation.