Hausdorff Morita Equivalence of singular foliations

TL;DR

This paper introduces a new notion of equivalence for singular foliations that preserves their transverse geometry and is compatible with the associated holonomy groupoids, unifying previous concepts and providing new invariants.

Contribution

It defines a Hausdorff Morita equivalence for singular foliations, compatible with holonomy groupoids, and unifies existing notions of transverse equivalence for regular foliations.

Findings

The new equivalence preserves transverse geometric structures.

It is compatible with the holonomy groupoid construction.

Several invariants of singular foliations are derived.

Abstract

We introduce a notion of equivalence for singular foliations - understood as suitable families of vector fields - that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's.