[Paper Review] The Universal Lie $\infty$-Algebroid of a Singular Foliation

Camille Laurent-Gengoux, Sylvain Lavau+1 more|arXiv (Cornell University)|Jun 2, 2018

Spinal Hematomas and Complications5 citations

TL;DR

This paper introduces the universal Lie ∞-algebroid of a singular foliation, a unique (up to homotopy) higher algebroid structure that encodes the geometry of singular leaves and holonomy. It proves existence of geometric resolutions for singular foliations on compact manifolds and shows that this universal structure universally factors all other Lie ∞-algebroids inducing the same foliation, recovering the holonomy groupoid and defining isotropy Lie ∞-algebras on each leaf with higher brackets as new invariants.

ABSTRACT

We consider singular foliations ${\cal F}$ 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 ${\cal F}$ in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold $M$ if ${\cal F}$ admits real analytic generators. We show that every complex of vector bundles $(E_\bullet,\dd)$ over $M$ providing a resolution of a given singular foliation ${\cal F}$ 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 ${\cal F}$ 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 ${\cal F}$. It encodes several aspects of the geometry of the leaves of ${\cal F}$. 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 ${\cal F}$ 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 $r$.

Motivation & Objective

• To establish a universal Lie ∞-algebroid structure for singular foliations, generalizing Lie algebroids to singular settings.
• To prove the existence of geometric resolutions (vector bundle complexes) of singular foliations on compact manifolds with real analytic generators.
• To show that every such resolution can be uniquely extended to a Lie ∞-algebroid structure up to homotopy.
• To demonstrate that this universal structure universally factors all other Lie ∞-algebroids inducing the same foliation or its sub-foliations.
• To recover the holonomy groupoid of Androulidakis and Skandalis and define isotropy Lie ∞-algebras on each leaf as higher invariants.

Proposed method

• Define singular foliations as locally finitely generated O-submodules of O-derivations closed under Lie bracket, where O is smooth, holomorphic, or real analytic functions.
• Construct geometric resolutions of such foliations as complexes of vector bundles (E•, d) over a compact manifold M, with length at most dim M + 1.
• Equip each resolution with brackets on sections to extend it into a Lie ∞-algebroid, using arity-deformation and cohomological techniques.
• Use homotopy theory of Lie ∞-algebroids to prove universality: any Lie ∞-algebroid inducing F or a sub-foliation factors through the universal one up to homotopy.
• Define E-paths and homotopies of E-paths in the universal Lie ∞-algebroid to construct its fundamental groupoid.
• Show that the fundamental groupoid of the universal Lie ∞-algebroid is the universal cover of the holonomy groupoid of Androulidakis and Skandalis.

Experimental results

Research questions

• RQ1 Does every singular foliation on a compact smooth manifold admit a geometric resolution of length at most dim M + 1?
• RQ2 Can every geometric resolution of a singular foliation be uniquely extended to a Lie ∞-algebroid structure?
• RQ3 Is there a unique (up to homotopy) Lie ∞-algebroid that universally factors all other Lie ∞-algebroids inducing the same singular foliation?
• RQ4 Can the holonomy groupoid of a singular foliation be recovered from the fundamental groupoid of its universal Lie ∞-algebroid?
• RQ5 What higher invariants arise from the isotropy Lie ∞-algebra structure on each leaf of a singular foliation?

Key findings

• On a compact smooth manifold, every singular foliation admitting real analytic local generators admits a geometric resolution of length at most dim M + 1.
• Every geometric resolution (E•, d) of a singular foliation F admits a unique extension to a Lie ∞-algebroid structure up to homotopy.
• The universal Lie ∞-algebroid of F is unique up to homotopy and universally factors all other Lie ∞-algebroids inducing F or any of its sub-foliations.
• The fundamental groupoid of the universal Lie ∞-algebroid is isomorphic to the universal cover of the holonomy groupoid of Androulidakis and Skandalis.
• Each leaf of the foliation carries a canonical isotropy Lie ∞-algebra structure, unique up to isomorphism, extending the minimal isotropy Lie algebra by higher brackets.
• The paper constructs a singular foliation generated by r vector fields that cannot locally arise from a Lie algebroid of minimal rank r, demonstrating the necessity of higher algebroid structures.