Diffeological approaches to leaf spaces of foliations

A singular foliation $\mathcal{F}$ partitions a manifold $M$ into leaves. The leaf space $M/\mathcal{F}$ is often quite singular, but still carries a natural quotient diffeology. We take three approaches to the question: what data in $\mathcal{F}$ is preserved in $M/\mathcal{F}$? First, we give sufficient conditions for the preservation of the basic cohomology. Second, we show that regular Riemannian foliations are determined, up to transverse equivalence, by their leaf spaces. Finally, we define a transverse equivalence of singular foliations (Molino transverse equivalence), and show this induces a diffeomorphism of leaf spaces.