Blow-up groupoid of singular foliations

TL;DR

This paper presents a novel blow-up construction for singular foliations and their holonomy groupoids, providing a desingularisation that maintains some smooth structure, advancing the understanding of singular foliation geometry.

Contribution

It introduces a new blow-up method for singular foliations and their holonomy groupoids, resulting in a desingularised, locally compact, locally Hausdorff groupoid with retained smooth features.

Findings

Constructs a desingularised groupoid retaining smooth structure

Produces a locally compact, locally Hausdorff groupoid

Provides a new framework for analyzing singular foliations

Abstract

We introduce a blow-up construction of a smooth manifold along the singular leaves of an arbitrary singular foliation in the sense of Stefan and Sussmann, as well as a blow-up construction of the holonomy groupoid defined by Androulidakis and Skandalis. Our construction gives a locally compact locally Hausdorff groupoid, which can be regarded as a desingularisation of the singular foliation. We show that it retains some smooth structure.