The Basic de Rham Complex of a Singular Foliation

TL;DR

This paper investigates the relationship between two complexes of differential forms associated with singular foliations, establishing isomorphisms under specific conditions such as regularity, embedded leaf dimension strata, and linearizable Lie groupoid structures.

Contribution

It proves that the pullback by the quotient map induces an isomorphism between the complexes of diffeological and basic forms in key cases of singular foliations.

Findings

Isomorphism of complexes for regular foliations

Isomorphism when leaves of same dimension form an embedded submanifold

Special case for foliations induced by linearizable Lie groupoids

Abstract

A 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.