# The Universal Lie $∞$-Algebroid of a Singular Foliation

### Camille Laurent-Gengoux

Institut Elie Cartan de Lorraine (UMR 7502), Université de Lorraine, 3 rue Augustin Fresnel, 57000 Metz-Technopôle, France### Sylvain Lavau

Université Paris Diderot, Bâtiment Sophie Germain, 8 Place Aurélie Nemours, 75013 Paris, France### Thomas Strobl

CNRS-2924, Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil, and Institut Camille Jordan, Université Claude Bernard, Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

## Abstract

We consider singular foliations $F$ as locally finitely generated $O$-submodules of $O$-derivations closed under the Lie bracket, where $O$ 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 $F$ in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold $M$ if $F$ admits real analytic generators.

We show that every complex of vector bundles $(E_{∙},d)$ over $M$ providing a resolution of a given singular foliation $F$ in the above sense admits the definition of brackets on its sections such that it extends these data into a Lie $∞$-algebroid. This Lie $∞$-algebroid, including the chosen underlying resolution, is unique up to homotopy and, moreover, every other Lie $∞$-algebroid inducing the given $F$ or any of its sub-foliations factors through it in an up-to-homotopy unique manner. We therefore call it the universal Lie $∞$-algebroid of $F$.

It encodes several aspects of the geometry of the leaves of $F$. In particular, it permits us to recover the holonomy groupoid of Androulidakis and Skandalis. Moreover, each leaf carries an isotropy Lie $∞$-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 $F$ 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 $r$.

## Cite this article

Camille Laurent-Gengoux, Sylvain Lavau, Thomas Strobl, The Universal Lie $∞$-Algebroid of a Singular Foliation. Doc. Math. 25 (2020), pp. 1571–1652

DOI 10.4171/DM/782