Abstract
Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold $(M,\mathcal{F}\_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}\_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.
Highlights
The space of leaves of a foliation is typically not smooth, and might fail to be Hausdorff
When F contains the infinitesimal generators of the G-action, the induced morphism Ξ is the quotient map of a Lie group action in the category of topological groupoids
We summarize as follows Ex. 4.7, Prop. 4.9, Prop. 4.11 and Prop. 4.15: Proposition. i) The open surjective morphism Ξ∶ H(F) → H(FM ) generally fails to be a fibration of topological groupoids
Summary
The space of leaves of a foliation is typically not smooth, and might fail to be Hausdorff. When F contains the infinitesimal generators of the G-action, the induced morphism Ξ is the quotient map of a Lie group action in the category of topological groupoids. This is useful because when Ξ is a fibration of Lie groupoids [9], it allows to describe the holonomy groupoid H(FM ) without knowing FM.
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