Lie $\infty$-algebroids and singular foliations

TL;DR

This paper constructs a universal Lie infinity-algebroid structure on resolutions of singular foliations, providing a homotopy-invariant framework that generalizes classical Lie algebroids to singular settings.

Contribution

It introduces the concept of a universal Lie infinity-algebroid for singular foliations, extending the classical theory to resolutions with finite type graded vector bundles.

Findings

Existence of a Lie -algebroid structure on resolutions of singular foliations.

The universal Lie -algebroid is unique up to homotopy.

Provides a homotopy-invariant approach to singular foliations.

Abstract

A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie bracket of vector fields to a Lie $\infty$-algebroid structure on this resolution, that we call a universal Lie $\infty$-algebroid associated to the foliation. The name is justified because it is isomorphic (up to homotopy) to any other Lie $\infty$-algebroid structure built on any other resolution of the given singular foliation.

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