Abstract
We present Hausdorff versions for Lie Integration Theorems 1 and 2 and apply them to study Hausdorff symplectic groupoids arising from Poisson manifolds. To prepare for these results we include a discussion on Lie equivalences and propose an algebraic approach to holonomy. We also include subsidiary results, such as a generalization of the integration of subalgebroids to the nonwide case, and explore in detail the case of foliation groupoids.
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Acknowledgements
We thank H. Bursztyn, R. L. Fernandes and F. Bischoff for comments and suggestions on the first version of the paper. We also thank the anonymous referee for the thorough report which helped us to improve the article considerably. MdH was partially supported by National Council for Scientific and Technological Development – CNPq Grants 303034/20173 and 429879/20180, and by FAPERJ Grant 210434/2019. DL was partially supported by PhD CNPq Grant 140576/20177.
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Communicated by Adrian Constantin.
A Quotients of (nonHausdorff) manifolds
A Quotients of (nonHausdorff) manifolds
We include in this “Appendix” a proof of the Godement criterion for quotients of smooth manifolds, with emphasis in the nonHausdorff case, and a lemma on Lie groupoid actions which is used along the paper. Our treatment is alternative and complementary to the Hausdorff version of [13, §9] and the analytic version in [26, II.3.12].
Our manifolds M are allowed to be nonHausdorff, unless otherwise specified. Our main tool in proving the criterion is the construction of fibered products between transverse smooth maps.
Lemma A.1
Let \(f_1:M_1\rightarrow N\) and \(f_2:M_2\rightarrow N\) be transverse smooth maps. Then the fibered product \(M_1\times _N M_2\subset M_1\times M_2\) is embedded, and for every \((x_1,x_2)\in M_1\times _N M_2\) the following sequence is exact:
Moreover, if \(f_1\) is an embedding, then so does its basechange \(M_1\times _N M_2\rightarrow M_2\).
Proof
Given a local chart \(U\xrightarrow {\varphi } {\mathbb {R}}^n\) of N, let \(V=f_1^{1}(U)\times f_2^{1}(U)\), and consider the function \(F:V\rightarrow {\mathbb {R}}^n\) given by \(F(x_1, x_2)=\varphi (f_1(x_1))\varphi (f_2(x_2))\). Observe that 0 is a regular value of F, for \(dF=d\varphi (df_1\pi _1df_2\pi _2)\) and \(f_1 \pitchfork f_2\). It follows from the constant rank theorem that \(F^{1}(0)=(M_1\times _N M_2)\cap V\subset V\) is embedded with tangent space \(\ker dF=\ker (df_1\pi _1df_2\pi _2)\).
Suppose now that \(f_1\) is an embedding. Working locally, we can assume that \(M_1={\mathbb {R}}^p\), \(N={\mathbb {R}}^{p+q}\) and that \(f_1\) is just the inclusion \({\mathbb {R}}^p\rightarrow {\mathbb {R}}^p\times {\mathbb {R}}^q\), \(x\mapsto (x,0)\). Then \(M_1\times _N M_2\) identifies with the preimage of 0 along \(\pi _2 f_2:M_2\rightarrow {\mathbb {R}}^q\). This composition is a submersion by the transversality hypothesis. The result follows by the constant rank theorem. \(\square \)
Given M a manifold, possibly nonHausdorff, and \(R\subset M\times M\) an equivalence relation, if the quotient M/R admits a manifold structure so that the projection \(\pi :M\rightarrow M/R\) is a submersion, then it easily follows from A.1 that \(R=M\times _{M/R}M\subset M\times M\) is an embedded submanifold and that the projection \(\pi _1:R\rightarrow M\) is a surjective submersion. It turns out that these necessary conditions for R are also sufficient to garantee that M/R is indeed a manifold.
Proposition A.2
(Godement criterion) Let M be a manifold and \(R\subset M\times M\) be an equivalence relation that is an embedded submanifold and makes \(\pi _2:R\rightarrow M\) a submersion. Then M/R inherits a unique canonical smooth structure such that the projection \(\pi :M\rightarrow M/R\) is a submersion. Moreover, M/R is Hausdorff if and only if \(R\subset M\times M\) is closed.
Proof
Step 1: Describing the orbits Given \(x\in M\), write \(O_x\) for its equivalence class or orbit. It follows from the following fibered product diagram and Lemma A.1 that \(O_x\subset M\) is an embedded submanifold with tangent space given by \(T_yO_x\times 0\cong T_{(y,x)}R \cap (T_yM\times 0)\):
Step 2: Building the charts Let \(S_x\subset M\) be a balllike submanifold through x such that \(T_xS_x\oplus T_xO_x=T_xM\). After eventually shrinking \(S_x\) we can assume that (i) \(T_yS_x\oplus T_yO_y=T_y M\) for all \(y\in S_x\), and (ii) \(\pi :S_x\rightarrow M/R\) is injective. We then use \(\alpha _x:S_x\rightarrow M/R\) to define a chart. Note that (i) is equivalent to \(T_yS_x\times 0\cap T_{(y,y)}R=0\), which is an open condition on y. Regarding (ii), the inclusion \(S_x\times S_x\subset M\times M\) is transverse to \(R\subset M\times M\), because of (i) and because \(O_y\times O_y\subset R\) for every \(O_y\). By Lemma A.1\(P= (S_x\times S_x)\cap R\subset S_x\times S_x\) is an embedded submanifold and \(\dim P=\dim S_x\):
The inclusion \(\Delta (S_x)\subset P\) must be an open embedding, then we can find a balllike open \(x\in U\) such that \(U\times U\cap P\subset \Delta (S_x)\) and \(U\rightarrow M/R\) is injective.
Step 3: Compatibility between charts Given two charts \(S_x\rightarrow M/R\) and \(S_y\rightarrow M/R\) with nontrivial intersection, the inclusion \(S_y\times S_x\subset M\times M\) is transverse to R, by an argument analogous to the one in Step 2. Then by Lemma A.1\(P=(S_y\times S_x)\cap R\subset S_x\times S_y\) is an embedded submanifold with \(\dim P=\dim S_x\). The projections \(\psi _1:P\rightarrow S_y\), \(\psi _2:P\rightarrow S_x\), are injective, and the transition between the two charts is given by the composition \(\psi _1\psi _2^{1}\), so it is enough to show that \(\psi _2:P\rightarrow S_x\) is étale, namely a local diffeomorphism. By Lemma A.1\(T_{(y^{\prime },x^{\prime })}P=(T_{y^{\prime }}S_y\times T_{x^{\prime }}S_x)\cap T_{(y^{\prime },x^{\prime })}R\), and by Step 1 \(\ker d\psi _2=T_{(y^{\prime },x^{\prime })}R\cap (T_{y^{\prime }}M\times 0)=T_{y^{\prime }}O_{y^{\prime }}\times 0\). Then \(d_{(y^{\prime },x^{\prime })}\psi _2:T_{(y^{\prime },x^{\prime })}P\rightarrow T_{x^{\prime }}S_x\) is injective, hence an isomorphism, and \(\psi _2\) is étale.
Step 4: \(\pi \) is a smooth submersion By the Step 1 the intersection \((S_x\times M)\cap R\) is an embedded submanifold of \(M\times M\) of dimension \(\dim M\), and \(\pi _2:(S_x\times M)\cap R\rightarrow M\) has injective differential, hence it is locally invertible. If \(\phi :U\rightarrow (S_x\times M)\cap R\) is a local inverse around x, then \(\pi _1\phi =\alpha _x^{1}\pi :U\rightarrow S_x\), and therefore \(\pi :M\rightarrow M/R\) is smooth. It is also a submersion because it admits local sections induced by the inclusions \(S_x\rightarrow M\). This implies that \(\pi \) is an open map, so the topology induced by our charts is indeed the quotient topology, and that the smooth structure on M/R is uniquely determined by that on M.
Step 5: Hausdorffness If M/R is Hausdorff then \(R=(\pi \times \pi )^{1}(\Delta _{M/R})\subset M\times M\) is closed. Conversely, if R is closed, given \((y,x)\notin R\), we can find a basic open \((y,x)\in V\times U\subset (M\times M){\setminus } R\), and since \(\pi :M\rightarrow M/R\) is open, \(\pi (U)\) and \(\pi (V)\) separate \(\pi (x)\) and \(\pi (y)\) in M/R. \(\square \)
All the relations we consider in the paper arise from actions of Lie groupoids. Given \(G\rightrightarrows M\) a Lie groupoid, and given E a possibly nonHausdorff manifold, a (right) action \(E\curvearrowleft G:\rho \) over \(\mu :E\rightarrow M\) is a map \(\rho :E\times _M G\rightarrow E\), \((e,g)\mapsto eg\), defined on the fibered product between \(\mu \) and t, such that \(\mu (eg)=s(g)\), \(\rho _{hg}=\rho _h\rho _g\) and \(\rho _{1_x}=\mathrm{id}_{E_x}\) (see [10] for more details). We say that \(\rho \) is a principal action if the anchor map \((\pi _1,\rho ):E\times _M G\rightarrow E\times E\) is an embedding. Note that the anchor is injective if and only if the action is free, namely if \(eg=e\) implies that \(g=1_{\mu {e}}\), and that it is a topological embedding if and only if the division map \(\delta :R\rightarrow G\), \((e,eg)\mapsto g\) is continuous, where R is the equivalence relation. It follows from Godement criterion A.2 that the orbit space of a principal action is a welldefined possibly nonHausdorff manifold.
Lemma A.3

(a)
If \(K\subset G\rightrightarrows M\) is a wide embedded subgroupoid then right multiplication \(G\times _M K\rightarrow G\), \((g,k)\mapsto gk\), is a principal action.

(b)
If \(E\curvearrowleft G:\rho \) is principal, \(E^{\prime }\curvearrowleft G:\rho ^{\prime }\) is some other action and \(\phi :E^{\prime }\rightarrow E\) is equivariant, namely \(\mu \phi =\mu ^{\prime }\) and \(\phi (e^{\prime }g)=\phi (e^{\prime })g\) for every \(e^{\prime },g\), then \(\rho ^{\prime }\) is also principal.
Proof
Regarding (a), note that the image of the anchor map \((g,k)\mapsto (g,gk)\) is included in the fibered product \(G {_t\times _t} G\), so we can compose it with the division map \(G {_t\times _t} G\rightarrow G\times G\), \((g,h)\mapsto (g,g^{1}h)\), and recover the canonical embedding \(G\times _M K\rightarrow G\times G\).
To prove (b) we will show that the anchor map \((\pi _1,\rho ^{\prime })\) is injective, immersive and a topological embedding. If \(e^{\prime }g=e^{\prime }\) then \(\phi (e^{\prime })g=\phi (e^{\prime })\), and since \(\rho \) is free, g must be a unit, proving that \(\rho ^{\prime }\) is also free, and \((\pi _1,\rho ^{\prime })\) injective. Now let \((w,v)\in T_{(e^{\prime },g)}(E^{\prime }\times _M G)\subset T_{e^{\prime }}E^{\prime }\times T_gG\) such that \(d(\pi _1,\rho ^{\prime })(w,v)=(w,d\rho ^{\prime }(w,v))=0\). Then \(w=0\) and \(0=d\phi d\rho ^{\prime }(0,v)=d\rho (0,v)\). Since \((\pi _1,\rho )\) is immersive, \(v=0\) and \((\pi _1,\rho ^{\prime })\) is also immersive. Finally, calling \(R^{\prime }\) and R the relations defined by \(\rho ^{\prime }\) and \(\rho \), we can write the division map \(\delta ^{\prime }\) as the composition \(R^{\prime }\xrightarrow {\phi \times \phi } R\xrightarrow \delta G\), hence \(\delta ^{\prime }\) is continuous and the anchor is a topological embedding. \(\square \)
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del Hoyo, M., López Garcia, D. On Hausdorff integrations of Lie algebroids. Monatsh Math (2021). https://doi.org/10.1007/s00605021015357
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Keywords
 Lie groupoids
 Poisson manifolds
 Foliations
Mathematics Subject Classification
 22A22
 53D17
 57R30