Hausdorff Morita equivalence of singular foliations

Abstract

We introduce a notion of equivalence for singular foliations -understood as suitable families of vector fields -that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's. A Appendix 34 A.1 Morita equivalence for open topological groupoids and Lie groupoids . . . . 34 Definition 1.12. Given a Lie groupoid G ⇒ M and a surjective submersion π :is the space of arrows of a Lie groupoid over P . (The source and target maps are the first and third projections and the multiplication is induced by the multiplication in G). This Lie groupoid is called the pullback groupoid of G by π.Definition 1.13. Given a Lie algebroid A over a manifold M with anchor # : A → T M , and a surjective submersion π : P → M , one checks that π −1 A := π * (A) # × dπ T P is the total space of a vector bundle over P . It has a natural Lie algebroid structure, with anchor# := pr 2 : π −1 A → T P being the second projection. The Lie bracket is determined by its restriction to "pullback sections", which is given by the Lie brackets in X(P ) and Γ(A). We call this Lie algebroid the pullback algebroid of A over π.These two definitions are nicely related by the following lemma:Lemma 1.14. Consider a surjective submersion π : P → M .(i) Let G be a Lie groupoid over M , denote by A its Lie algebroid. The Lie algebroid of the Lie groupoid π −1 G is π −1 A.(ii) Let A be an integrable Lie algebroid over M , denote by G the source simply connected Lie groupoid integrating it. If the map π has simply connected fibers, then the source simply connected Lie groupoid integrating π −1 A is π −1 G.Proof. The proof of part (i) can be found in [17, §4.3], so we address only the proof of part (ii). The Lie groupoid π −1 G integrates π −1 A by part (i). Therefore we need to only show that π −1 G is source simply connected. Take p ∈ P . Its source fiber is s −1 (p) = {(q, g, p) : π(p) = s(g) and π(q) = t(g)} P π × t s −1 (π(p)).The canonical submersion s −1 (p) → s −1 (π(p)) has simply connected fibers, since the πfibers are simply connected. Using that s −1 (π(p)) is simply connected and lemma 1.11 we conclude that s −1 (p) is simply connected.

Quotients of singular foliations and Lie 2-group actions

“…We check that Ξ is a continuous open map. This holds because in the following commutative diagram the quotient maps Q and Q M are continuous and open (see [8,Lemma 3.1]), and because Q is surjective.…”

“…In this case F = F big ∶= π −1 F M is the pullback of F M by π. We will need [8,Thm. 3.21], stated as follows:…”

The Holonomy Groupoids of Singularly Foliated Bundles

“…One can alternatively describe a singular foliation of a manifold M as a locally finitely generated submodule of the compactly supported vector fields on M which is closed under Lie brackets, as in [2]. By [31,Remark 1.8] these definitions are equivalent. One of the most important facts regarding singular foliations is the Stefan-Sussmann integration theorem [58,59].…”

“…Holonomy groupoids at this level of generality are topologically pathological, but, as is evident in the recent preprint [60] of Garmendia and Villatoro, are diffeologically quite well-behaved, arising as spaces of classes of leafwise paths identified via their maps on transversal slices. The years since the Androulidakis-Skandalis construction have seen a great deal of further research conducted into singular foliations and their holonomy, see for instance [3,4,5,6,30,31,60].…”

Integration of Singular Foliations via Paths

“…If we are given a Hausdorff Morita equivalence N ← P → M as defined in [GZ19a], we get a pair of morphisms foliated manifolds F M ← F P → F N and a commuting diagram:…”