Smoothness of holonomy covers for singular foliations and essential isotropy

TL;DR

This paper introduces the concept of essential isotropy groups for singular foliations, linking their discreteness to integrability obstructions and their role in the smoothness of holonomy covers and isotropy groups.

Contribution

It establishes the connection between essential isotropy groups and the smoothness of holonomy covers, providing criteria for when these structures are smooth manifolds or Lie groups.

Findings

Discreteness of essential isotropy groups relates to integrability obstruction.

Closeness of the essential isotropy group ensures smoothness of holonomy covers.

Essential isotropy controls the Lie group structure of isotropy groups.

Abstract

Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the {smoothness} of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.