The holonomy of a singular leaf

TL;DR

This paper defines a new notion of holonomy for singular leaves in foliations using sequences of group morphisms, connecting it to existing holonomy groupoids and Lie $ $-algebroid structures.

Contribution

It introduces a novel concept of holonomy for singular leaves, linking it to universal Lie $ $-algebroids and existing holonomy groupoids in foliation theory.

Findings

Defines holonomy of a singular leaf via group morphisms.

Establishes a long exact sequence relating holonomy to Lie $ $-algebroids.

Connects the new holonomy concept to existing groupoid constructions.

Abstract

We introduce the holonomy of a singular leaf $L$ of a singular foliation as a sequence of group morphisms from $\pi_n(L)$ to the $\pi_{n-1}$ of the universal Lie $\infty$-algebroid of the transverse foliation of $L$. We include these morphisms in a long exact sequence, thus relating them to the holonomy groupoid of Androulidakis and Skandalis and to a similar construction by Brahic and Zhu for Lie algebroids.