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 with 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 1980s.

Appendix

1.1 Morita equivalence for open topological groupoids and Lie groupoids

Definition 2.25, on Morita equivalence of Lie groupoids, is equivalent to several other characterizations, as was proved in [18], (see also [16]). An analogue statement holds also for open topological groupoids, upon replacing submersions with continuous open maps. In this Appendix, we recall these facts and prove some implications that are used in the main body of the paper, the main one being Corollary A.7.

We start recalling the notion of weak equivalence, as given in [20, §1.3], and of bitorsor.

Definition A.1

Let $G\rightrightarrows M$ and $\varGamma \rightrightarrows P$ be two Lie groupoids (respectively, topological groupoids). A morphism $\widehat{\pi }:\varGamma \rightarrow G$ is a weak equivalence if:

• (i) $\varGamma \rightarrow \pi ^{-1} G; \;\;\gamma \mapsto (\mathbf {t}(\gamma ),\widehat{\pi }(\gamma ),\mathbf {s}(\gamma ))$ is an isomorphism,

• (ii) $\mathbf {t}\circ \textit{Pr}_1:G {}_{\mathbf {s}}\!\times _{\pi } P\rightarrow M$ is a surjective submersion (resp. a surjective continuous and open map).

Here $\pi :P\rightarrow M$ denotes the base map covered by $\widehat{\pi }$.

Remark A.2

• (i) Looking at a groupoid as a small category, a weak equivalence is the same thing as a fully faithful and essentially surjective functor.

• (ii) When a map $\pi :P\rightarrow M$ is completely transverse (transverse to the orbits and meeting every orbit) to a Lie groupoid $G\rightrightarrows M$, then the projection $\pi ^{-1} G \rightarrow G$ is a weak equivalence.

• (iii) If G is an open topological groupoid and ${\pi }$ is a continuous, open and surjective map, then condition (ii) in Definition A.1 is automatically satisfied, as can be shown using Lemma 3.35.

Definition A.3

Let $G\rightrightarrows M$ be a Lie groupoid (respectively, a topological open groupoid) and $\pi :P\rightarrow M$ a surjective submersion (resp. a surjective continuous and open map). A G-action over a not necessarily Hausdorff manifoldP is a smooth (resp. continuous) map $\star :G {}_\mathbf {s}\!\times _\pi P \rightarrow P$ such that for all $g,h\in G$ and $p\in P$:

$$\begin{aligned} \pi (g\star p)=\mathbf {t}(g),\quad g\star (h \star p)=(gh)\star p,\quad e_{\pi (p)}\star p=p. \end{aligned}$$

Such a manifold P with a G-action is called a G-module, and $\pi$ is called its moment map. If the G-action is free and proper, then P / G is a manifold and we say that P is a G-principal bundle.

A (G, H)-bimodule for the Lie groupoids $G\rightrightarrows M$ and $H\rightrightarrows N$ is a (not necessarily Hausdorff) manifold P with two actions commuting with each other. A (G, H)-bimodule P that is principal with respect to both actions and such that $G{\backslash } P\cong N$ and $P/H\cong M$ is called a (G, H)-bitorsor.

The following statement can be found in [18, §2.5]

Lemma A.4

Consider a Lie groupoid $\varGamma \rightrightarrows K$, a $\varGamma$-principal bundle S, a $\varGamma$-module Q and a map $f:Q\rightarrow S$ preserving the $\varGamma$ actions. Then, $Q/\varGamma$ is a manifold.

Proposition A.5

Let $G\rightrightarrows M$ and $H\rightrightarrows N$ be Lie groupoids. The following statements are equivalent:

• (i) There exists a Lie groupoid $\varGamma$ and two weak equivalences $\varGamma \rightarrow G$ and $\varGamma \rightarrow H$.

• (ii) There exists a (G, H)-bitorsor P.

• (iii) G and H are Morita equivalent (Definition 2.25).

The proof of this statement can be found in [18] and [16, prop. 2.4]; nevertheless, we review its proof here.

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$: Consider a Lie groupoid $\varGamma \rightrightarrows K$ with weak equivalences $\widehat{\pi }_M:\varGamma \rightarrow G$ and $\widehat{\pi }_N:\varGamma \rightarrow H$. Therefore, $\varGamma =\pi _M^{-1} G\cong \pi _N^{-1} H$. We get that $Q_G:=G {}_{\mathbf {s}}\!\times _{\pi _M} K$ is a $(G,\varGamma )$-bitorsor. Using a similar argument, we get that $Q_H$ a $(\varGamma ,H)$-bitorsor. The (not necessarily Hausdorff) manifold $Q:=(Q_G\times _K Q_H)$ has a diagonal $\varGamma$-action with the canonical map to K as moment map. Applying Lemma A.4 to the map $Q\rightarrow Q_G$, we see that

$$\begin{aligned} P:=(Q_G\times _K Q_H)/\varGamma \end{aligned}$$

is a (not necessarily Hausdorff) manifold. One can check that it is a (G, H)-bitorsor.

$\mathrm{(ii)}\Rightarrow \mathrm{(iii)}$: Consider a (G, H)-bitorsor P, with moment maps $\pi _M:P\rightarrow M$,$\pi _N:P\rightarrow N$. Then,

$$\begin{aligned} \varGamma = G {}_\mathbf {s}\!\times _{\pi _M} P {}_{\pi _N}\!\times _\mathbf {t}H \end{aligned}$$

has a natural structure of Lie groupoid over P with $\mathbf {t}(g,p,h)=gph$, $\mathbf {s}(g,p,h)=p$ and multiplication given canonically by G and H. Then, the maps $\varGamma \rightarrow \pi _M^{-1}G; \;(g,p,h)\mapsto (p,g^{-1},gph)$ and $\varGamma \rightarrow \pi _N^{-1} H;\;(g,p,h)\mapsto (p,h,gph)$ are isomorphisms of Lie groupoids.

This shows that $\pi _M^{-1}G\cong \pi _N^{-1} H$ as Lie groupoids over the not necessarily Hausdorff manifold P. Now take a Hausdorff cover $\{U_i\}_{i\in I}$ of P and let $\tilde{P}:=\sqcup _i U_i$. There is a canonical submersion $\pi :\tilde{P}\rightarrow P$. It is easy tho see that $(\tilde{P},\pi _M\circ \pi , \pi _N\circ \pi )$ is a Morita equivalence.

$\mathrm{(iii)}\Rightarrow \mathrm{(i)}$: Given a Morita equivalence $(P,\pi _M,\pi _N)$ between G and H, call $\varGamma :=\pi _M^{-1}G\cong \pi _N^{-1}H$. The natural projections $\varGamma \rightarrow G$ and $\varGamma \rightarrow H$ are weak equivalences. $\square$

Remark A.6

Proposition A.5 also holds for open topological groupoids, as can be proven using Lemma 3.35. For arbitrary topological groupoids, this is not the case.

Corollary A.7

Let $k\ge 0$. If $G\rightrightarrows M$ and $H\rightrightarrows N$ are source k-connected Morita equivalent Hausdorff Lie groupoids, then there exists a Hausdorff (G, H)-bitorsor P. Moreover, this bitorsor is a Morita equivalence with k-connected fibres (in the sense of Definition 2.25).

Proof

Following the implications $\mathrm{(iii)}\Rightarrow \mathrm{(i)}\Rightarrow \mathrm{(ii)}$ in the proof of Proposition A.5, one sees that the bitorsor P constructed there is Hausdorff. Then, use the implication $\mathrm{(ii)}\Rightarrow \mathrm{(iii)}$ to prove that P is a Morita equivalence.

Note that being P a bitorsor, the fibres of $\pi _M:P\rightarrow P/H{\cong M}$ are diffeomorphic to the source fibres of H, which are k-connected by assumption. A similar argument holds for the fibres of $\pi _N:P\rightarrow P/G\cong N$. $\square$

Remark A.8

The Morita equivalence P in Corollary A.7 is a (global) bisubmersion for the underlying foliations, as we now show. Using the implication “$\mathrm{(ii)}\Rightarrow \mathrm{(iii)}$” in Proposition A.5, we get an isomorphism of Lie groupoids

$$\begin{aligned} \pi _M^{-1}G\cong \pi _N^{-1} H\cong G\ltimes P\rtimes H. \end{aligned}$$

Denote by $\mathcal {F}_M$ and $\mathcal {F}_N$ the foliations underlying G and H. Using Lemmas 1.14(i) and 1.15, we get that the foliations underlying $\pi _M^{-1}G$ and $\pi _N^{-1} H$ are $\pi _M^{-1}\mathcal {F}_M$ and $\pi _N^{-1}\mathcal {F}_N$, respectively. Since P is a (G, H)-bitorsor, the foliation underlying the Lie groupoid $G\ltimes P\rtimes H$ is $\varGamma _\mathrm{c}(\hbox {ker}(d\pi _M))+\varGamma _\mathrm{c}(\hbox {ker}(d\pi _N))$. Hence,

$$\begin{aligned} \pi _M^{-1}\mathcal {F}_M=\pi _N^{-1}\mathcal {F}_N=\varGamma _\mathrm{c}(\hbox {ker}(d\pi _M))+\varGamma _\mathrm{c}(\hbox {ker}(d\pi _N)). \end{aligned}$$