The Universal Lie $\infty$-Algebroid of a Singular Foliation

Abstract

We consider singular foliations \mathcal{F} as locally finitely generated \mathscr{O} -submodules of \mathscr{O} -derivations closed under the Lie bracket, where \mathscr{O} is the ring of smooth, holomorphic, or real analytic functions on a correspondingly chosen manifold. We first collect and/or prove several results about the existence of resolutions of such an \mathcal{F} in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold M if \mathcal{F} admits real analytic generators. We show that every complex of vector bundles (E_\bullet,\mathrm{d}) over M providing a resolution of a given singular foliation \mathcal{F} in the above sense admits the definition of brackets on its sections such that it extends these data into a Lie \infty -algebroid. This Lie \infty -algebroid, including the chosen underlying resolution, is unique up to homotopy and, moreover, every other Lie \infty -algebroid inducing the given \mathcal{F} or any of its sub-foliations factors through it in an up-to-homotopy unique manner. We therefore call it the universal Lie \infty -algebroid of \mathcal{F} . It encodes several aspects of the geometry of the leaves of \mathcal{F} . In particular, it permits us to recover the holonomy groupoid of Androulidakis and Skandalis. Moreover, each leaf carries an isotropy Lie \infty -algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation \mathcal{F} generated by r vector fields for which we show by these techniques that, even locally, it cannot result from a Lie algebroid of the minimal rank r .