Holonomy transformations for singular foliations

TL;DR

This paper extends the concept of holonomy from regular to singular foliations by introducing holonomy transformations linked to the holonomy groupoid, aiding the linearization problem analysis.

Contribution

It introduces holonomy transformations for singular foliations, connecting linearization issues with groupoid compactness and proper Lie groupoids.

Findings

Holonomy transformations are attached to elements of the holonomy groupoid.

The assignment of holonomy transformations is injective.

Link between linearization problem and isotropy group compactness.

Abstract

In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation. This assignment is injective. Holonomy transformations allow us to link the linearization problem with the compactness of the isotropy group of the holonomy groupoid, as well as with the linearization problem for proper Lie groupoids.