Integration of Singular Foliations via Paths

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.

Integration of Singular Subalgebroids

“…In the approach we propose, the integrals of singular subalgebroids are groupoids endowed with a specific diffeology. This was already announced in [27], and diffeology in the context of the holonomy groupoid was later used by other authors [12] [19] [20]. We elaborate on this below, in the rest of this introduction.…”

“…This would enlarge the integration problem beyond the smooth category. This question is closely related to the integration question for Lie-Rinehart algebras endowed with suitable extra structure (what is left of a singular subalgebroid after one forgets the ambient Lie algebroid), which is addressed in a special case in [12].…”

The holonomy of a singular leaf

“…The class of the map φ modulo exp(I T ) is the holonomy of [AZ13,AZ14]. While finishing this article, we learnt that Alfonso Garmendia and Joel Villatoro are going further in this direction (see [GV19]).…”

On the Chern character in Higher Twisted K-theory and spherical T-duality

“…Diffeological spaces were originally introduced by Soriau [52], who was motivated by problems in quantisation. They are vast generalisations of manifolds, and are used to apply tools of differential geometry to spaces that are not well-behaved by traditional standards, for instance orbifolds [31] and, increasingly, leaf spaces of foliations [27,28,39,40]. Here we choose to work in the diffeological category out of convenience -it allows us to use a single language to speak of the singular de Rham theory of both topological spaces and manifolds as we will now describe.…”