Groupoids and singular foliations

TL;DR

This thesis develops a new notion of equivalence for singular foliations that preserves their transverse geometry and explores the structures behind their quotients, linking them to holonomy groupoids and Lie groupoids.

Contribution

It introduces a compatible equivalence for singular foliations and analyzes their quotient structures in relation to holonomy groupoids and Lie groupoids.

Findings

Defined a new equivalence notion for singular foliations

Connected quotient structures of foliations with holonomy groupoids

Provided foundational insights into singular foliations and their relations to Lie groupoids

Abstract

This doctoral thesis has two objectives. The first objective is to introduce a notion of equivalence for singular foliations that preserves their transverse geometry and is compatible with the notions of Morita equivalence of the holonomy groupoids and the transverse equivalence for regular foliations that appeared in the 1980's. The second one is to describe the structures behind quotients of singular foliations and to connect these results with their associated holonomy groupoids. It also wants to give an introduction to the notion of singular foliations as given by Androulidakis and Skandalis in arXiv:math/0612370, as well as to their relation with Lie groupoids and Lie algebroids.