Integration of singular foliations via paths

TL;DR

This paper introduces a new path-based method to construct holonomy and fundamental groupoids of singular foliations, extending classical techniques and clarifying their homotopic nature.

Contribution

It presents a novel quotient-based approach for integrating singular foliations, aligning with the path integration method used for Lie algebroids.

Findings

Provides a clearer homotopic interpretation of groupoids

Establishes functorial properties of the constructions

Extends classical regular foliation methods to singular cases

Abstract

We give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis, our method proceeds by taking a quotient of an infinite dimensional space of paths. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths (per Crainic and Fernandes). In this way, we obtain a characterization of the holonomy and fundamental groupoids of a singular foliation that more clearly reflects the homotopic character of these invariants. As an application of our work, we prove that the constructions of the fundamental and holonomy groupoid of a foliation have functorial properties.