On Hausdorff integrations of Lie algebroids

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 non-wide case, and explore in detail the case of foliation groupoids.

A Quotients of (non-Hausdorff) manifolds

We include in this “Appendix” a proof of the Godement criterion for quotients of smooth manifolds, with emphasis in the non-Hausdorff 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 non-Hausdorff, 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:

$$\begin{aligned} 0\rightarrow T_{(x_1,x_2)}(M_1\times _N M_2)\rightarrow T_{x_1}M_1\times T_{x_2}M_2 \xrightarrow {df_1\pi _1-df_2\pi _2} T_xN\rightarrow 0. \end{aligned}$$

Moreover, if $f_1$ is an embedding, then so does its base-change $M_1\times _N M_2\rightarrow M_2$.

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 $dF=d\varphi (df_1\pi _1-df_2\pi _2)$ 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 $\ker dF=\ker (df_1\pi _1-df_2\pi _2)$.

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 non-Hausdorff, 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 ball-like 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 ball-like 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 non-Hausdorff 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 well-defined possibly non-Hausdorff 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. $\square$