Quotients of singular foliations and Lie 2-group actions

Abstract

Every singular foliation has an associated topological groupoid, called holonomy groupoid [1]. In this note we exhibit some functorial properties of this assignment: if a foliated manifold (M, F M ) is the quotient of a foliated manifold (P, F P ) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid. Contents

Integration of Singular Foliations via Paths

“…According to Example 6.15, the condition in Theorem 6.21 that (F, f ) is a morphism of foliated manifolds is equivalent to the claim that F M is contained in f −1 (F N ). This result is an improvement on theorems that appear in preprints [Zam18] and [GZ19b]. However, by decomposing the problem into the cases f = Id M and F M = f −1 (F N ), one can mostly recover Theorem 6.21 as a corollary of these papers.…”