The Basic de Rham Complex of a Singular Foliation

Abstract

AbstractA singular foliation $\mathcal {F}$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal {F}$ and that of the basic differential forms on $M$. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when $\mathcal {F}$ is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of $M$, and, as a special case of the latter, when $\mathcal {F}$ is induced by a linearizable Lie groupoid or is a singular Riemannian foliation.