turkmath.org

A singular (or Hermann) foliation on a smooth manifold M can be seen as a locally finitely generated subsheaf of the sheaf of vector fields on M closed under Lie bracket. We show that if this singular foliation admits a resolution 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. The choices entering the construction of this Lie $\infty$-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie $\infty$-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie $\infty$-algebroid of the singular foliation. For a real analytic or holomorphic singular foliation, it can be chosen, locally, to be a Lie n-algebroid for some finite n. We will show that this universal structure encodes several aspects of the geometry of the leaves of a singular foliation, and that it allows to extend to the singular context some notions (e.g. characteristic classes) until now only defined for regular foliations.