The Basic de Rham Complex of a Singular Foliation

Abstract

A singular foliation 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/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 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 F is induced by a linearizable Lie groupoid.