The holonomy Lie $\infty$-groupoid of a singular foliation I
Ruben Louis, Camille Laurent-Gengoux
TL;DR
This work constructs a finite-dimensional holonomy para-Lie $\infty$-groupoid $K_\bullet$ that integrates the universal Lie $\infty$-algebroid of a singular foliation $\mathcal{F}$, under the mild hypothesis that $\mathcal{F}$ admits a geometric resolution. The integration is achieved via a recursive tower of bi-submersions, yielding a Kan-like simplicial structure whose full tangent complex recovers a geometric resolution of $\mathcal{F}$ and whose 1-truncation matches the classical Androulidakis-Skandalis holonomy groupoid. Unlike infinite-dimensional Sullivan–Getzler approaches, this method provides a finite-dimensional global integration, with dimensions explicitly controlled by the ranks in the geometric resolution. The construction offers a new pathway to integrating higher Lie algebroids associated to singular foliations and suggests robustness to generalizations beyond geometric resolutions. Together, these results connect higher groupoid techniques with concrete holonomy structures in singular foliation theory, providing both conceptual and computational advantages.
Abstract
We construct a finite-dimensional higher Lie groupoid integrating a singular foliation $\mathcal F$, under the mild assumption that the latter admits a geometric resolution. More precisely, a recursive use of bi-submersions, a tool coming from non-commutative geometry and invented by Androulidakis and Skandalis, allows us to integrate any universal Lie $\infty$-algebroid of a singular foliation to a Kan simplicial manifold, where all components are made of non-connected manifolds which are all the same finite dimension that can be chosen to be equal to the ranks of a given geometric resolution. Its 1-truncation is the Androulidakis-Skandalis holonomy groupoid.