j. differential geometry 58 (2001) 467-500

HOLONOMY GROUPOIDS OF SINGULAR FOLIATIONS

CLAIRE DEBORD

Abstract

We give a new construction of Lie groupoids which is particularly well adapted to the generalization of holonomy groupoids to singular foliations. Given a family of local Lie groupoids on open sets of a smooth manifold M, satisfying some hypothesis, we construct a Lie groupoid which contains the whole family. This construction involves a new way of considering (local) Morita equivalences, not only as equivalence relations but also as generalized isomorphisms. In particular we prove that almost injective Lie algebroids are integrable.

Introduction

To any regular foliation is associated its holonomy groupoid. This groupoid was introduced in the topological context by C. Ehresmann [14], and later the differentiable case was done independently by J. Pradines and H.E. Winkelnkemper [25, 32]. It is the smallest Lie groupoid whose orbits are the leaves of the foliation [23], and every regular foliation is given by the action of its holonomy groupoid on its space of units.

The holonomy Lie groupoid of a foliation has been the starting point of several studies. A. Connes used it to define the Von Neumann algebra and the C^{∗}-algebra of a foliation [8]; A. Haefliger pointed out that, considered up to a suitable equivalence, the holonomy groupoid defines the transverse structure of the foliation [16]; it plays a crucial role in the index theory for foliations which was developed by A. Connes and G. Skandalis [10] as well as for the definition of the transverse fundamental class of a foliation [9]. See [7] for a complete bibliography on this subject.

Received July 2, 2001.

On the other hand symplectic geometry and control theory have focused attention on foliations generated by families of vector fields. These foliations were introduced independently by P. Stefan and H.J. Sussman and are usually called Stefan foliations [28, 29]. By definition, a Stefan foliation F on a manifold M is the partition of M by the integral manifolds of a completely integrable differentiable distribution on M. In such foliations the dimension of the leaves can change. Our study concerns the Stefan foliations for which the union of leaves of maximal dimension is a dense open subset of the underlying manifold, these foliations are called almost regular.

There are plenty of examples of Stefan foliations: orbits of a Lie group action, faces of a manifold with corners, symplectic foliation of a Poisson manifold, etc. Thus it is a natural question to extend the notion of holonomy Lie groupoid to such foliations, it is the first step toward getting transverse invariants, to compute a C^{∗}-algebra or to do index theory. This paper is devoted to the study of this question. Precisely the question we answer is: given an almost regular foliation how can one find a Lie groupoid which induces the foliation on its space of units and which is the smallest in some way?

The first idea is to define holonomy for Stefan foliations. This has already been done under some hypothesis by M. Baeur and P. Dazord [2, 12]. But this process involves loss of information. For example the holonomy group of any singular leaf reduced to a single point will always vanish and so we will miss information to compute a Lie groupoid. In [25], J. Pradines constructs the holonomy groupoid from the local transverse isomorphisms, that is the diffeomorphisms between small transversals to the foliation. This method also fails in the singular case because we lose local triviality of the foliation as well as the notion of small transversals.

We are looking at Lie groupoids which are “the smallest in some way”. In other word we do not want unnecessary isotropy. This leads us to consider a special case of Lie groupoids namely the quasi-graphoids. A quasi-graphoid has the property that the foliation it induces on its units space is almost regular. Furthermore, restricted to the regular part, it is isomorphic to the usual holonomy groupoid of the corresponding regular foliation. So these groupoids are good candidates to be holonomy groupoids of almost regular foliations.

On the other hand, we remark that the problem of finding a Lie groupoid associated to a singular foliation F on a manifold M has two aspects:

The local problem which is to find for each point x of M a (local) Lie groupoid associated to the foliation induced by F on a neighborhood of x. This problem is obvious for the regular foliations. Since the equivalence relation to be on the same leaf is locally regular in this case, one can just take the graph of this regular equivalence relation over each distinguished open set. When the (singular) foliation is defined by an almost injective Lie algebroid over M, that is a Lie algebroid whose anchor is injective when restricted to a dense open subset of M, this problem reduced to the local integration of the algebroid. This has already been done in detail by the author in [13].

The global problem which consists in finding a Lie groupoid on M made from these (local) groupoids. Even in the regular case, this aspect is more difficult to understand because holonomy is involved.

However, the study of the regular case gives rise to the following remark. Suppose that (M, F) is a regular foliation and take a cover of M by distinguished open sets. All the open sets of this cover are equipped with a simple foliation and these foliations fit together to give (M, F). In the same way, to each of these open sets, the graph of the corresponding regular equivalence relation is associated and these Lie groupoids fit together to generate the holonomy groupoid of (M, F), denoted Hol(M, F). The first step was to understand how these groupoids fit together. The conclusion of this study is the following:

If γ is an element of Hol(M, F) whose source is x and range is y, let U_{0} (resp. U_{1}) be a distinguished neighborhood of x (resp. y), T_{0} (resp. T_{1}) be a small transversal passing through x (resp. y) and G_{0} (resp. G_{1}) the holonomy groupoid of the simple foliation (U_{0}, F|_{U_{0}}) (resp. (U_{1}, F|_{U_{1}})). Then the data of γ is equivalent to:

- The holonomy class of a path c tangent to the foliation, starting at x and ending at y (usual construction [32]).

- The germ at x of a diffeomorphism from T_{0} to T_{1} (J. Pradines’ construction [25]).

- The germ of a Morita equivalence between G_{0} and G_{1} (defined here in Section 3).

The transverse isomorphism mentioned above will just be the holonomy isomorphism associated to the path c. Morita equivalence between two groupoids was defined by M. Hilsum and G. Skandalis [17] to be generalized isomorphisms between the orbit spaces of the groupoids. Here, by construction, the orbit space of G_{i} is naturally isomorphic to T_{i}, i = 0, 1.

This new approach will be very useful, because even if the notion of holonomy classes of paths tangent to the foliation and transverse isomorphisms can’t be defined for almost regular foliations, the notion of Morita equivalence remains.

Finally the construction of a Lie holonomy groupoid of an almost regular foliation can be decomposed in two steps. The first step is a construction of a generalized atlas of the foliation. This “atlas” will be made up of local Lie groupoids. The existence of such an atlas will replace the local triviality of the foliation. The second step is the construction of a Lie groupoid associated to such an atlas using (local) Morita equivalences.

This paper is organized as follows:

In Section 1 we recall briefly the definition of Stefan foliations.

In Section 2 we study quasi-graphoids and we define the general atlases of a foliation.

In Section 3 we first extend the notion of Morita equivalence to local groupoids. Next we show that given a generalized atlas, the set of local Morita equivalence between elements of this atlas behaves almost like a pseudo-group of local diffeomorphisms. We finish by defining the germ of a Morita equivalence, and we construct the groupoids of the germs of the elements of the pseudo-group.

In the last section we apply the obtained results to almost regular foliations, and we give several examples.

I want to address special thanks to Georges Skandalis for his relevant suggestions.

1. Singular foliations

A generalized distribution D = ∪_{x∈M}D_{x} ⊂ T M on a smooth manifold M is smooth when for each (x, v) ∈ D_{x}, there exists a smooth local tangent vector field X on M such that:

x ∈ Dom(X), X(x) = (x, v) and ∀y ∈ Dom(X), X(y) ∈ D_{y}.

It is integrable when for all x ∈ M, there exists an integral manifold passing through x, that is, an immersed submanifold F_{x} of M such that x belongs to F_{x} and T_{y}F_{y} = D_{y} for all y ∈ F_{x}.

The maximal connected integral manifolds of an integrable differentiable distribution D on the manifold M define a partition F of M. Such a partition is what we call a Stefan foliation on the manifold M [28]. The elements of the partition are called the leaves. Contrary to the regular case the dimension of the leaves may change. A leaf which is not of dimension equal to max_{x∈M} (dim D_{x}) is singular and it is regular otherwise. Because of the rank lower semi-continuity the dimension of the leaves increases around a singular leaf.

When the union M_{0} of the leaves of maximal dimension is a dense open subset of M we say that the Stefan foliation (M, F) is almost regular. In this case, the foliation F restricted to M_{0} is a regular foliation denoted by (M_{0}, F_{0}) and called the maximal regular subfoliation of (M, F).

Examples.

1. A regular foliation is a Stefan foliation.

2. The partition of a manifold M into the orbits of a differentiable action of a Lie group on M is a Stefan foliation.

3. The integral manifolds of an involutive family of tangent vector fields on a manifold M which is locally of finite type are the leaves of a Stefan foliation [29].

4. Let N be a manifold equipped with two regular foliations F_{1} and F_{2} where F_{1} is a subfoliation of F_{2} (that is every leaf of F_{1} lies on a leaf of F_{2}). Then F_{1} × {0} and F_{2} × {t} for t ≠ 0 defines an almost regular foliation on M × ℝ.

5. We recall that an quartering on a manifold M is the data of a finite family {V_{i}, i ∈ I} of codimension 1 submanifolds of M such that for all J ⊂ I, the family of the inclusions of the V_{j}, j ∈ J is transverse [19]. The faces of a quartering on M are the leaves of an almost regular foliation.

6. Recall that a Lie algebroid A = (p : A → T M, [ , ]_{A}) on a smooth manifold M is a vector bundle A → M equipped with a bracket [ , ]_{A} : Γ(A) × Γ(A) → Γ(A) on the module of sections of A together with a morphism of fiber bundles p : A → T M from A to the tangent bundle T M of M called the anchor, such that: i) The bracket [ , ]_{A} is ℝ-bilinear, antisymmetric and satisfies to the

Jacobi identity.

ii) [X, fY]_{A} = f[X, Y]_{A} + p(X)(f)Y for all X, Y ∈ Γ(A) and f a smooth function of M.

iii) p([X, Y]_{A}) = [p(X), p(Y)] for all X, Y ∈ Γ(A).

The distribution defined by p(A) on M is involutive and locally of finite type, so it is integrable [29]. The corresponding Stefan foliation on M is said to be defined by the Lie algebroid A.

Suppose now that G ⇉_{r}^{s} M is a Lie groupoid on the manifold M

having s as source map and r as range map.

Recall that if x belongs to M the orbit of G passing through x is the set

{y ∈ M | ∃γ ∈ G such that s(γ) = x and r(γ) = y} = r(s^{−1}(x)).

We denote by AG the Lie algebroid of G and by F_{G} the corresponding foliation on M. The leaves of the Stefan foliation F_{G} are the connected components of the orbits of G. Thus the action of the Lie groupoid G on its space of units M defines a Stefan foliation.

2. Quasi-graphoids

These groupoids have been introduced independently by J. Renault [27] who called them essentially principal groupoids and by B. Bigonnet [3] under the name of quasi-graphoids. We will see that quasi-graphoids have the good properties to be holonomy groupoids of almost regular foliations. We recall their definition, and we complete B. Bigonnet’s work by the study of their properties which leads to a geometric justification of this choice.

2.1 Definition and properties

Recall that a manifold S equipped with two submersions a and b onto a manifold B is called a graph over B. If (a′, b′) : S′ → B × B is another graph over B, a morphism of graphs from S to S′ is a smooth map φ : S → S′ such that a′ ∘ φ = a and b′ ∘ f = b [26].

Definition-Proposition 1 ([3]). Let G ⇉_{r}^{s} G^{(0)} be a Lie groupoid

over G^{(0)}, s its domain map, r its range map and u : G^{(0)} ⟶ G the units inclusion. The two following assertions are equivalent:

1. If ν : V ⟶ G is a local section of s then

r ∘ ν = 1_{V} if and only if ν = u|_{V} .

2. If N is a manifold, f and g are two smooth maps from N to G such that:

i) s ∘ f = s ∘ g and r ∘ f = r ∘ g.

ii) One of the maps s ∘ f and r ∘ f is a submersion.

Then f = g.

A quasi-graphoid is a Lie groupoid which satisfies these equivalent properties.

In other words G is a quasi-graphoid when for any graph S over G^{(0)} there exists at most one morphism of graphs from S to G.

Examples.

1. The holonomy groupoid of a regular foliation is a quasi-graphoid.

2. If H × M → M is a differentiable action of a Lie group H on a manifold M such that there is a saturated dense open subset of M over which the action is free, the groupoid of the action is a quasi-graphoid.

One can remark that the algebraic structure of a quasi-graphoid is fixed by the source and range maps. More precisely if G ⇉_{r}^{s} G^{(0)} is a

quasi-graphoid then:

– The inverse map is the unique smooth map i : G → G such that s ∘ i = r and r ∘ i = s.

– The product is the unique smooth map

p : G^{(2)} = {(γ, η) ∈ G × G | s(γ) = r(η)} → G

such that s ∘ p = s ∘ pr_{2} and r ∘ p = r ∘ pr_{1} where pr_{1} (resp. pr_{2}) is the projection onto the first (resp. second) factor.

In the same way, let G⇉B and H⇉B be two Lie groupoids having the same units and φ : G → H a morphism of graphs. If H is a quasigraphoid then a morphism of graphs from G to H has to be a morphism of Lie groupoids.

An other important property of quasi-graphoids is that a local morphism of graphs to a quasi-graphoid is always extendible.

Proposition 1. Let G ⇉_{r_{G}}^{s_{G}} B and H ⇉_{r_{H}}^{s_{H}} B be two Lie groupoids

having the same units space and φ : V → G a smooth map from a neighborhood of B in H to G which satisfies s_{H}∘φ = s_{G} and r_{H}∘φ = r_{G}.

If H is s-connected and G is a quasi-graphoid then there exists a unique morphism of Lie groupoid φ̃ : H → G such that the restriction of φ̃ to V is equal to φ.

Proof. We have already noticed that if φ extends to a smooth morphism of graphs φ̃ then φ̃ will be a morphism of Lie groupoid. The definition of quasi-graphoids implies that such a morphism must be unique. So it remains to show that φ extends to a smooth morphism of graphs.

Recall that a local section ν of s_{H} is said to be admissible when r_{H} ∘ ν is a local diffeomorphism of the space of units. When ν_{1} and ν_{2} are two local admissible sections such that r_{H}(ν_{2}(dom(ν_{2}))) ⊂ dom(ν_{1}) we denote by ν_{1}·ν_{2} the local admissible section which sends x ∈ dom(ν_{2}) onto the product ν_{1}(r_{H}(ν_{2}(x))) · ν_{2}(x).

If γ is an element of H then, because H is s-connected, one can find a neighborhood O of s_{H}(γ) in B and local admissible sections ν_{1}, …, ν_{n} such that the image of each ν_{i} is a subset of V, the map ν_{1} · ν_{2} ⋯ ν_{n} is defined on O and ν_{1}·ν_{2} ⋯ ν_{n}(s_{H}(γ)) = γ. Thus for each ν_{i}, the map φ∘ν_{i} is an admissible local section of G and the product φ(ν_{1})·φ(ν_{2}) ⋯ φ(ν_{n}) is defined.

Let U be a neighborhood of s_{H}(γ) in H such that U ⊂ V and U ∩ B = O. Then the following map

\begin{aligned}h : & U & ⟶ & H \\ & η & ↦ & ν_{1} · ν_{2} ⋯ ν_{n}(r_{H}(η)) · η \end{aligned}

gives rise to a diffeomorphism from U onto a neighborhood W of γ in H. We define the morphism of graphs

\begin{aligned}φ̃_{W} : & W & ⟶ & G \\ & h(η) & ↦ & φ(ν_{1}) · φ(ν_{2}) ⋯ φ(ν_{n})(r_{H}(η)) · η \end{aligned} .

By repeating this process we obtain an open covering {W_{i}}_{i∈I} of H and for each i ∈ I, a morphism of graphs φ̃_{i} : W_{i} → G. Because G is a quasi-graphoid, φ̃_{i} and φ̃_{j} must coincide on W_{i} ∩ W_{j}. Thus the map φ̃ : H → G defined by φ̃ = φ̃_{i} on W_{i} is a morphism of graphs. By uniqueness the restriction of φ̃ to V is equal to φ. q.e.d.

Notice that if φ is a diffeomorphism onto its image and if G and H are two s-connected quasi-graphoids then φ̃ is an isomorphism of Lie groupoids.

Corollary 1. Two s-connected quasi-graphoids having the same space of units are isomorphic if and only if their Lie algebroids are isomorphic.

Proof. Suppose that G and H are two s-connected quasi-graphoids having the same space of units B such that their Lie algebroids AG and AH are isomorphic. Then using the exponential map [18, Prop. 4.12, p. 136], one can easily construct a covering {W_{i}}_{i∈I} of B in G and for all i ∈ I a morphism of graphs φ_{i} : W_{i} → H which integrates the isomorphism from AG onto AH. Because H is a quasi-graphoid φ_{i} and φ_{j} must coincide on W_{i} ∩ W_{j}. Thus we obtain in this way a local isomorphism of graphs from G onto H. Using Proposition 1 we conclude that G and H are isomorphic. q.e.d.

The next proposition shows that quasi-graphoids are closely related to almost regular foliations.

Proposition 2. Let G ⇉_{r}^{s} G^{(0)} be a quasi-graphoid, (G^{(0)}, F_{G})

the singular foliation induced by G on G^{(0)} and AG = (p : AG → T G^{(0)}, [ , ]) the Lie algebroid of G.

The following assertions hold:

1. The anchor p is almost injective, that is the set

G_{0}^{(0)} = {x ∈ G^{(0)} | p_{x} : AG_{x} → T_{x}G^{(0)} is infective }

is a dense open subset of G^{(0)}.

2. The foliation (G^{(0)}, F_{G}) is almost regular.

3. The restriction of the s-connected component G^{c} of G to the regular part G_{0}^{(0)} is the holonomy groupoid of the maximal regular subfoliation of F_{G} denoted by Hol(G_{0}^{(0)}, F_{G}|_{G_{0}^{(0)}}).

Proof. 1. The rank lower semi-continuity implies that G_{0}^{(0)} is an open subset of G^{(0)}.

Let U be an open set of G^{(0)}, the rank lower semi-continuity implies that there is a non empty open subset V of U over which the rank of p is maximal. We can consider the fiber bundle ker(p|_{V}) over V. We have to check that ker(p|_{V}) = V × {0}.

Let X be a local section of ker(p|_{V}). According to [18] (prop. 4.1 p 126), for all x_{0} ∈ V, there is an open neighborhood V_{x_{0}} ⊂ V of x_{0}, an ε > 0 and a (unique) smooth family of local sections Exp(tX) of s defined on V_{x_{0}} for |t| < ε such that {r ∘ Exp(tX) : V_{x_{0}} → M} is the one parameter group of local transformations associated to the local vector field p(X) over V_{x_{0}} and

\frac{d}{dt}Exp(tX)|_{0} = X.

Since p(X) = 0 we get that r ∘ Exp(tX) = 1_{V_{x_{0}}} for all t such that |t| < ε. Then, G being a quasi-graphoid, we obtain that for all such t, Exp(tX) = u|_{V_{x_{0}}} , where u is the unit map of G. So X = 0.

2. An immediate consequence of p being almost injective is that the image by p of the restriction of AG to G_{0}^{(0)} is an involutive distribution on G_{0}^{(0)}. So it induces a regular foliation. In other words the restriction of F_{G} to G_{0}^{(0)} is regular.

3. The Lie algebroid of the restriction of G^{c} to G_{0}^{(0)} is equal to the restriction of AG to G_{0}^{(0)} and so it is isomorphic to the tangent space of the regular foliation (G_{0}^{(0)}, F_{G}|_{G_{0}^{(0)}}). Furthermore the tangent space of

the regular foliation (G_{0}^{(0)}, F_{G}|_{G_{0}^{(0)}}) is the Lie algebroid of the holonomy

groupoid of (G_{0}^{(0)}, F_{G}|_{G_{0}^{(0)}}). Finally, the restriction of G^{c} to G_{0}^{(0)} and

Hol(G_{0}^{(0)}, F_{G}|_{G_{0}^{(0)}}) are two s-connected quasi-graphoids having isomorphic Lie algebroids, so by the previous corollary they are isomorphic as well. q.e.d.

2.2 Local quasi-graphoids

We are going to present here the local version of quasi-graphoids. First, we need to recall the notion of local Lie groupoid which is due to Van Est [15].

A local Lie groupoid is given by:

– Two smooth manifolds L and L^{(0)} and an embedding u : L^{(0)} → L. The manifold L^{(0)} must be Hausdorff, it is called the set of units. We usually identify L^{(0)} with its image by u in L.

– Two surjective submersions: L ⇉_{r}^{s} L^{(0)} called the range and source

map, they must satisfy s ∘ u = r ∘ u = id.

– A smooth involution

\begin{matrix}L & ⟶ & L \\ γ & ↦ & γ^{−1} \end{matrix}

called the inverse map. It satisfies s(γ^{−1}) = r(γ) for γ ∈ L.

– An open subset D^{2}L of L^{(2)} = {(γ_{1}, γ_{2}) ∈ L × L | s(γ_{1}) = r(γ_{2})} called the set of composable pairs and a smooth local product:

\begin{matrix}D^{2}L & ⟶ & L \\ (γ_{1}, γ_{2}) & ↦ & γ_{1} · γ_{2} \end{matrix} .

The following properties must be fulfilled:

s(γ_{1} · γ_{2}) = s(γ_{2}) and r(γ_{1} · γ_{2}) = r(γ_{1}) when the product γ_{1} · γ_{2} is defined.

For all γ ∈ L the products r(γ) · γ, γ · s(γ), γ · γ^{−1} and γ^{−1} · γ are defined and respectively equal to γ, γ, r(γ) and s(γ).

If the product γ_{1} · γ_{2} is defined then so is the product γ_{2}^{−1} · γ_{1}^{−1} and (γ_{1} · γ_{2})^{−1} = γ_{2}^{−1} · γ_{1}^{−1}.

If the products γ_{1} · γ_{2}, γ_{2} · γ_{3} and (γ_{1} · γ_{2}) · γ_{3} are defined then so is the product γ_{1} · (γ_{2} · γ_{3}) and (γ_{1} · γ_{2}) · γ_{3} = γ_{1} · (γ_{2} · γ_{3}).

The only difference between groupoids and local groupoids is that in the second case the condition s(γ_{1}) = r(γ_{2}) is necessary for the product γ_{1}.γ_{2} to exist but not sufficient. The product is defined as soon as γ_{1} and γ_{2} are “small enough”, that is “close enough” from units.

Notations Let L ⇉_{r}^{s} L^{(0)} be a (local) Lie groupoid and denote by

p : D^{2}L → L the product of L.

– If O is a subset of L^{(0)} we denote by

L_{O} = s^{−1}(O), L^{O} = r^{−1}(O) and L|_{O} = L_{O}^{O} = s^{−1}(O) ∩ r^{−1}(O).

– If V is an open subset of L such that V ∩ L^{(0)} = L_{V}^{(0)} ≠ ∅, we denote by V^{+} the set {γ ∈ V | γ^{−1} ∈ V }.

We define L(V) = V^{+} ∩s^{−1}(L_{V}^{(0)})∩r^{−1}(L_{V}^{(0)}) and the set of composable pairs of L(V) by D^{2}L(V) = p^{−1}(L(V)) ∩ L(V) × L(V). Then we

define the restriction of L to V to be the local Lie groupoid L(V) ⇉_{r}^{s} L_{V}^{(0)}.

So to any open subset V of L which encounter L^{(0)} on L_{V}^{(0)} we thereby associate an open sub local Lie groupoid of L admitting L_{V}^{(0)} as space of units.

In particular if O is an open subset of L^{(0)}, L|_{O} ⇉_{r}^{s} O is a local Lie

groupoid also called the restriction of L to O when there is no risk of confusion.

As for Lie groupoid there is a natural notion of graphs morphism and local graphs morphism between two local Lie groupoids having the same space of units.

In particular two (local) Lie groupoids L ⇉_{r}^{s} L^{(0)} and L′ ⇉_{r′}^{s′} L^{′(0)} such

that L^{(0)} ∩ L^{′(0)} ≠ ∅ are locally isomorphic as graphs if there is a neighborhood V of L^{(0)} ∩ L^{′(0)} in L, a neighborhood V′ of L^{(0)} ∩ L^{′(0)} in L′ and a diffeomorphism φ : V → V′ such that s′ ∘ φ = s and r′ ∘ φ = r. This also means that there is a restriction of L containing L^{(0)} ∩ L^{′(0)} and a restriction of L′ containing L^{(0)} ∩ L^{′(0)} which are two isomorphic graphs.

Definition 1. A local quasi-graphoid is a local Lie groupoid L ⇉ L^{(0)}, having the property that for all manifolds S equipped with two submersions a and b onto L^{(0)}, there exists at most one morphism of graphs from S to L.

Remark 1.

1. Let L ⇉_{r}^{s} L^{(0)} be a local quasi-graphoid. If D is an open subset of L^{(2)}

there exists at most one smooth map p : D → L such that s ∘ p = s ∘ pr_{2} and r ∘p = r ∘pr_{1}. So there exists a maximal open subset D_{max}^{2}L of L^{(2)} which is the domain of a smooth map p which satisfies s ∘ p = s ∘ pr_{2} and r ∘ p = r ∘ pr_{1}. We call D_{max}^{2}L the maximal set of composable pairs.

We shall always consider local quasi-graphoids equipped with the maximal set of composable pairs, and thus we shall not have to specify what is this set as soon as we know that a local product is defined on a neighborhood of {(γ, γ^{−1}) | γ ∈ L} in L^{(2)}.

2. As for quasi-graphoids a (local) morphism of graphs from a local Lie groupoid to a local quasi-graphoid is a (local) morphism of groupoid.

3. Local quasi-graphoids are stable under restriction.

4. If L ⇉_{r}^{s} L^{(0)} is a local Lie groupoid and x belongs to L^{(0)} we define

the orbit of L passing through x to be the set

{y ∈ L^{(0)} | ∃γ_{1}, …, γ_{n} ∈ L ; s(γ_{1}) = x, r(γ_{1}) = s(γ_{2}),

…, r(γ_{n−1}) = s(γ_{n}), r(γ_{n}) = y}.

As for quasi-graphoids the Lie algebroid of a local quasi-graphoid is almost injective, and its orbits are the leaves of an almost regular foliation.

In the local context, there is a proposition similar to the Definition-Proposition 1:

Proposition 3. Let L⇉L^{(0)} be a local Lie groupoid. The following assertions are equivalent:

1. L is locally isomorphic as graph to a local Lie groupoid L′⇉L^{(0)} having the property that the only local smooth section of both the source s′ and the range r′ is the inclusion of units.

2. L is locally isomorphic as graph to a local quasi-graphoid.

Proof. A local quasi-graphoid has the property that the only local section of both the source and the range map is the unit map. So 2 implies 1.

Suppose that there is an isomorphism of graphs from V′ onto V where V′ (resp. V) is an open subset of L′ (resp. L) containing L^{(0)}.

Because D^{2}L contains {(γ, γ^{−1}) | γ ∈ L}, one can find an open neighborhood W of L^{(0)} in L such that:

– W = W^{−1} where W^{−1} := {γ^{−1} | γ ∈ W }.

– W × W ∩ L^{(2)} ⊂ D^{2}L.

– The image of the local product of L restricted to W × W ∩ L^{(2)} is contained in V.

W inherits from L a structure of local Lie groupoid. Moreover W and L are obviously locally isomorphic as graphs.

Let S be a manifold equipped with two submersions a, b onto L^{(0)} and f_{1}, f_{2} be two morphisms of graphs from S to W.

If σ : U → S is a local section of a we define

\begin{aligned}ν : & U & → & W \\ & x & ↦ & (f_{1}(σ(x)))^{−1} · f_{2}(σ(x)) \end{aligned}

Then ν is a section of both the source s and the range r. Thus ν is the unit map and so f_{1} ∘ σ = f_{2} ∘ σ. By repeating this process for any local section of a we show that f_{1} = f_{2}. Thus W is a local quasi-graphoid.

q.e.d.

2.3 Generalized Atlas

These “atlases” which are made up with local Lie groupoids are defined to replace in the case of singular foliation the atlases by distinguished charts of regular foliations. With such an atlas, we shall be able to do so without local triviality of the foliation and then compute a Lie groupoid associated to it.

Definition 2. Let M be a smooth manifold. A generalized atlas

on M is a set U = {L_{i} ⇉_{r_{i}}^{s_{i}} O_{i}}_{i∈I}, where:

– {O_{i}}_{i∈I} is a covering of M by open subsets.

– For all i ∈ I, L_{i} ⇉_{r_{i}}^{s_{i}} O_{i} is a local quasi-graphoid over O_{i}.

The following gluing condition must be fulfilled:

For all i, j ∈ I, there is a local graphs isomorphism from L_{i} onto L_{j}, that is an open subset H_{i}^{j} (resp.H_{j}^{i}) of L_{i} (resp. L_{j}) which contains O_{i} ∩ O_{j} and an isomorphism of graphs φ_{ji} : H_{i}^{j} ⟶ H_{j}^{i}.

Remark 2.

1. Because we are dealing with local quasi-graphoids the graphs isomorphisms φ_{ji} are unique. So there exists a maximal open subset of L_{i} containing O_{i} ∩ O_{j} which is the domain of a morphism of graphs to L_{j}. We will always suppose that the domain H_{i}^{j} of φ_{ji} is maximal.

2. Because the L_{i} are local quasi-graphoids the following equations are fulfilled when i, j, k belong to I:

i) φ_{ki} = φ_{kj} ∘ φ_{ji} when restricted to H_{i}^{j} ∩ H_{i}^{k} ∩ φ_{ji}^{−1}(H_{j}^{k}).

ii) φ_{ji}^{−1} = φ_{ij}.

We will say that two generalized atlases U and V on the manifold M are equivalent when U ∪ V is again a generalized atlas on M. The generalized atlas U on M is stable under restriction if for any element L ⇉ O of U and any open subset V of L which encounter O the restriction of L to V is an element of U. Any generalized atlas is equivalent to a generalized atlas which is stable under restriction.

If L ⇉ L^{(0)} is a local quasi-graphoid and {O_{i}}_{i∈I} an open covering of L^{(0)} then {L|_{O_{i}}}_{i∈I} defines a generalized atlas on L^{(0)}. Two different coverings of L^{(0)} will give rise to two equivalent atlases by this process. Conversely, let U = {L_{i} ⇉_{r_{i}}^{s_{i}} O_{i}}_{i∈I} be a generalized atlas on a manifold

M. Remark 2 above implies that the relation ∼ defined on ⨆_{i∈I} L_{i} by:

(γ, i) ∼ (η, j) ⇔ \{ \begin{matrix}(γ, i) ∈ H_{i}^{j} and (η, j) ∈ H_{j}^{i} \\ φ_{ji}(γ) = η , \end{matrix}

is a regular equivalence relation. Let us denote by L the quotient manifold of ⨆_{i∈I} L_{i} by this relation. There is a unique structure of local quasi-graphoid on L with units space M such that for all i ∈ I, L_{i} is a sub local Lie groupoid of G. Moreover, the atlases U and {L} are equivalent.

In conclusion, generalized atlases and local quasi-graphoids are two equivalent notions. In particular, a generalized atlas on a manifold M defines naturally an almost regular foliation on M.

Given an almost regular foliation F on a manifold M, we will say that a generalized atlas on M is a a generalized atlas for F when it defines F.

Examples

1. Regular foliation: Let F be a regular foliation on a manifold M and {O_{i}}_{i∈I} a covering of M by distinguished charts. The foliation F_{i} induced by F on O_{i} is such that its space of leaves O_{i}/F_{i} is a manifold. In other words the equivalence relation on O_{i} defined by being on the same leaf of F_{i} is regular. Thus the graph of this equivalence relation, G_{i} = {(x, y) ∈ O_{i} × O_{i} | x and y are on the same leaf of F_{i}} ⇉ O_{i} is a quasi-graphoid.

The family {G_{i}}_{i∈I} is a generalized atlas for F.

2. Local almost free action of a Lie group: Let (M, F) be a singular foliation defined by a symmetric local action of a Lie group H on M [22]. We suppose that there is a dense open subset M_{0} of M on which the action is free. We define M ⋊_{loc}H to be the local Lie groupoid

M ⋊_{loc} H := D ⇉_{r_{⋊}}^{s_{⋊}} M,

where:

– D ⊂ H × M is the domain of the local action.

– The inclusion of units is given by x ↦ (1_{H}, x).

– If (h, x) belongs to D then its source is s_{⋊}(h, x) = x and its range is r_{⋊}(h, x) = h · x.

– The local product is defined on

D^{2}(M ⋊_{loc} H) = {((h, x), (g, y)) ∈ D × D | x = g · y , (hg, y) ∈ D} by

(h, g · y) · (g, y) = (hg, y).

One can easily check that M ⋊_{loc} H is a local quasi-graphoid which defines F, thus {M ⋊_{loc} H} is a generalized atlas for F.

3. Codimension 1 submanifold: Let M be a manifold and N be a submanifold of codimension 1 of M. We define F_{N} to be the foliation on M whose leaves are N and the connected components of M \ N.

Let O be an open subset of M such that O ∩ N ≠ ∅ and f_{O} : O → ℝ be a smooth map such that df_{O}_{x} ≠ 0 for all x in O ∩ N and O ∩ N = f_{O}^{−1}(0). The following map is regular:

\begin{aligned}Φ : & O × O × ℝ_{∗}^{+} & ⟶ & ℝ \\ & (z, y, λ) & ↦ & λf_{O}(z) − f_{O}(y) \end{aligned} .

Thus Φ^{−1}(0) is a submanifold of O × O × ℝ_{∗}^{+} and we define:

GO = Φ^{−1}(0) = {(z, y, λ) ∈ O × O × ℝ_{∗}^{+} | λf_{O}(z) = f_{O}(y)} ⇉_{r_{O}}^{s_{O}} O

where:

– The inclusion of units is the map x ↦ (x, x, 1).

– The source and range maps are given by s_{O}(z, y, λ) = y and r_{O}(z, y, λ) = z.

– The product is defined on GO^{(2)} by (x, y, λ) · (y, t, μ) = (x, t, λμ).

One can find a family {O_{i}}_{i∈I} of open subsets of M such that:

– For all i ∈ I, O_{i} ∩ N ≠ ∅ and O_{i} is equipped with a map f_{O_{i}} : O_{i} → ℝ as above.

– N ⊂ ∪_{i∈I}O_{i}.

Let (M \ N) ⊠ (M \ N) be the subset of (M \ N)×(M \ N) made of pairs of points which are on the same connected component of M \ N. Equipped with the usual pair groupoid structure (M \ N) ⊠ (M \ N) becomes a quasi-graphoid over M \ N.

The set {(M \ N) ⊠ (M \ N)} ∪ {G_{O_{i}}, i ∈ I} is a generalized atlas for the foliation F_{N}.

4. Let N be a manifold equipped with two regular foliations F_{1} and F_{2} where F_{1} is a subfoliation of F_{2}. Let F be the foliation on M = N × ℝ defined by F_{1} × {0} and F_{2} × {t} for t ≠ 0. We construct a generalized atlas for (M, F) in the following way.

Let U be an open subset of N and Γ = {X_{1}, …, X_{k+q}} a family of vector fields defined on U such that:

– Γ is a basis over U of local sections of T F_{2}, the tangent bundle of F_{2}.

– Γ′ = {X_{1}, …, X_{k}} is a basis over U of local sections of T F_{1}.

– [X_{i}, X_{j}] = 0 for all i, j.

Thus we have a free local action φ_{U} : D_{U} ⊂ U × ℝ^{k} × ℝ^{q} → U of ℝ^{k} × ℝ^{q} on U such that for (x, t, ξ) in D_{U} the points x and φ_{U}(x, t, ξ) belong to the same leaf of F_{2} and the points x and φ_{U}(x, t, 0) belong to the same leaf of F_{1}. Such a local action is said to be compatible with the foliations F_{1} and F_{2}. Let ε_{U} > 0 be a real number such that for all λ ∈ ℝ, |λ| < ε_{U} we have that (y, t, λξ) belongs to D_{U} when (y, t, ξ) is in D_{U}.

Then we define the local quasi-graphoid

LU = D_{U}×] − ε_{U}, ε_{U}[⇉_{r_{U}}^{s_{U}} U×] − ε_{U}, ε_{U}[

as follows:

– The inclusion of units is the map (y, λ) ↦ (y, 0, 0, λ).

– The source and range maps are defined by s_{U}(y, t, q, λ) = (y, λ) and r_{U}(y, t, ξ, λ) = (φ(y, t, λξ), λ).

– The inverse is defined by (y, t, ξ, λ)^{−1} = (φ(y, t, λξ), −t, −ξ, λ).

– The product is given by

(φ(y, t_{1}, λξ_{1}), t_{2}, ξ_{2}, λ) · (y, t_{1}, ξ_{1}, λ) = (y, t_{1} + t_{2}, ξ_{1} + ξ_{2}, λ)

whenever it makes sense.

Let {(U_{i}, φ_{i})}_{i∈I} be such that {U_{i}}_{i∈I} is an open covering of N and φ_{i} is a local action of ℝ^{k} × ℝ^{q} on U_{i} compatible with the foliations F_{1} and F_{2}.

Let G = H_{2}×ℝ_{∗} ⇉ N ×ℝ_{∗} be the product groupoid of the holonomy

groupoid H_{2} of F_{2} and the trivial groupoid ℝ_{∗} ⇉^{id}_{id} ℝ_{∗}.

One can prove that the set {G} ∪ {LU_{i}}_{i∈I} is a generalized atlas for the foliation F.

5. Almost injective Lie algebroid: A Lie algebroid A = (p : A → T M, [ , ]) over a manifold M is said to be almost injective when its anchor p is injective in restriction to a dense open subset of M. In other words the anchor p induces an injective morphism from the set of smooth local sections of A onto the set of smooth local tangent vector fields over M. Such a Lie algebroid defines on M an almost regular foliation F_{A}. We have shown in [13] how such an algebroid integrates into a local quasi-graphoid. Here is a brief review of this construction.

Let O be an open subset of M, {Y_{1}, …, Y_{k}} a local basis of sections of A defined on O and X_{i} = p(Y_{i}) the corresponding tangent vector field on M. For simplicity, we suppose that the tangent vector fields X_{i} are complete, and we denote by φ_{i}^{t} the flow of X_{i}.

According to [13], there exist an open subset LO of O × ℝ^{k} and a local quasi-graphoid structure on LO

LO ⇉_{r}^{s} O

such that:

–The unit map is

\begin{aligned}u : & O & ⟶ & LO ⊂ O × ℝ^{k} \\ & x & ↦ & (x, 0) \end{aligned} .

–The source and range maps are defined by:

\begin{aligned}s : & LO & ⟶ & O \\ & (x, t_{1}, …, t_{k}) & ↦ & x \end{aligned}

\begin{aligned}r : & LO & ⟶ & O \\ & (x, t_{1}, …, t_{k}) & ↦ & φ_{1}^{t_{1}} ∘ … ∘ φ_{k}^{t_{k}}(x) \end{aligned} .

With the help of a local trivialisation of the bundle A we construct in this way a generalized atlas U_{A} of F_{A}. Moreover up to local equivalence of generalized atlases, the generalized atlas U_{A} only depends on A.

Of course any of the previous examples are particular cases of this example.

3. Pseudo-group of local Morita isomorphisms and associated groupoids

In the first part of this section we extend the notion of Morita equivalence between Lie groupoids to local Lie groupoids. The following definitions are nearly the same as those encountered in [10, 16, 17]. Actions of local groupoids cannot be defined globally so the usual definitions do not make sense in this case, and we have to take some care when extending these definitions to the local context.

In the second part, we show that the set of local Morita equivalences between the elements of a generalized atlas behaves as a pseudo-group of local diffeomorphisms. We finish this section by the construction of a Lie groupoid associated to such a pseudo-group.

3.1 Local Morita isomorphisms

Definition 3. Let Z be a manifold, a : Z → L^{(0)} be a smooth map and L ⇉_{r}^{s} L^{(0)} be a local Lie groupoid:

[figure: L^{(0)}.; Z; L; r; s; a]

Let D_{Ψ} be an open subset of the fiber product

Z ×_{a,r} L = {(z, γ) ∈ Z × L ; a(z) = r(γ)}.

A right local action of L on Z with domain D_{Ψ} is a surjective smooth map Ψ : D_{Ψ} ⟶ Z denoted by Ψ(z, γ) = z ∗ γ, which satisfies the following properties:

1. For all (z, γ) ∈ D_{Ψ} we have the equality a(z ∗ γ) = s(γ).

2. If (z, γ_{1}) belongs to D_{Ψ}, and if the product γ_{1}.γ_{2} is defined, then if one of the following expression (z ∗ γ_{1}) ∗ γ_{2} or z ∗ (γ_{1} · γ_{2}) is defined, so is the other one and they are equal.

3. For all z ∈ Z, (z, a(z)) belongs to D_{Ψ} and z ∗ a(z) = z.

We will say that the local action is free when z ∗ γ = z implies that γ belongs to L^{(0)}.

We will say that the local action is locally proper if all z ∈ Z has an open neighborhood V_{z} such that the map Υ : D_{Ψ} ⟶ Z × Z defined by Υ(z, γ) = (z, z ∗ γ) is proper when restricted to D_{Ψ} ∩ Υ^{−1}(V_{z} × V_{z}).

We define left local actions in the same way.

Remark. A free local action of the local Lie groupoid L ⇉ L^{(0)} on the manifold Z induces a regular foliation on Z denoted by F_{L,Z}. The leaf of F_{L,Z} passing trough a point z_{0} of Z is the orbit of z_{0} for the local action. In other words it is the set

{z ∈ Z | ∃ γ_{1}, …, γ_{n} ∈ L such that z = (⋯ ((z_{0} ∗ γ_{1}) ∗ γ_{2}) ⋯) ∗ γ_{n}}.

The tangent space of this foliation can be defined in the following way. Let Ψ : D_{Ψ} ⟶ Z be the local action and pr_{1} : D_{Ψ} ⟶ Z the projection onto the first factor. We define F = ker(T pr_{1}) ⊂ T D_{Ψ} to be the kernel of the differential of pr_{1}. One can check that the image T Ψ(F) of F by the differential of Ψ is an involutive subbundle of T Z. Thus T Ψ(F) is the tangent bundle of a regular foliation F_{L,Z} on Z.

Let L_{0} ⇉ O_{0} and L_{1} ⇉ O_{1} be two local Lie groupoids. A local generalized isomorphism f from L_{0} to L_{1} is defined by its graph:

[figure: O_{1}; O_{0} .; L_{1}; Z_{f}; L_{0}; r_{1}; s_{1}; r_{0}; s_{0}; b_{f}; a_{f}]

The graph is a smooth manifold Z_{f} equipped with two surjective submersions (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} in addition with a left local, locally proper, free action of L_{1} and a right local, locally proper, free action of L_{0}. Moreover the following properties must be fulfilled:

1. For all γ_{0} ∈ L_{0}, z ∈ Z_{f} and γ_{1} ∈ L_{1} such that γ_{1} ∗ z and z ∗ γ_{0} are defined we have that a_{f}(γ_{1} ∗ z) = a_{f}(z), b_{f}(z ∗ γ_{0}) = b_{f}(z), and if one of the following expressions (γ_{1} ∗ z) ∗ γ_{0} or γ_{1} ∗ (z ∗ γ_{0}) is defined, so is the other one and they are equal.

2. For all z ∈ Z_{f}, there is an open neighborhood V_{z} of z in Z_{f} such that the action of L_{0} (resp. L_{1}) is transitive on the fibers of b_{f} (resp. a_{f}) restricted to V_{z}. In this case V_{z}/L_{0} and V_{z}/L_{1} make sense and a_{f} (resp. b_{f}) induces a diffeomorphism from V_{z}/L_{1} to a_{f}(V_{z}) (resp. from V_{z}/L_{0} to b_{f}(V_{z})).

Two such graphs (b_{f}, a_{f}) : Z_{f} → O_{1}×O_{0} and (b_{g}, a_{g}) : Z_{g} → O_{1}×O_{0} are equivalent if there exists an isomorphism of graphs from Z_{f} onto Z_{g} which intertwines the actions of L_{0} and L_{1}.

Definition 4. A local Morita isomorphism f from L_{0} onto L_{1} is an equivalence class of local generalized isomorphisms.

It will be denoted by f : L_{1} ↶ L_{0} and each of its representatives will be called a graph for f.

Examples.

1. Let T and N be two manifolds and consider the trivial Lie groupoids T ⇉ T and N ⇉ N having the identity map as source and range maps. A diffeomorphism f : T → N induces a local Morita isomorphism from T onto N for which (pr_{1}, pr_{2}) : Z_{f} = {(f(x), x) | x ∈ T } → N × T is a graph. In this trivial case any local Morita isomorphism is of this type.

2. Let L ⇉ L^{(0)} be a local Lie groupoid of source s and range r and T a closed embedded submanifold of L^{(0)} which encounters all the orbits. The manifold L_{T}^{T} = s^{−1}(T) ∩ r^{−1}(T) inherits from L a structure of local Lie groupoid over T. Then (r, s) : L_{T} = s^{−1}(T) → L^{(0)} × T is a graph of a local Morita isomorphism from L_{T}^{T} onto L, the action being right and left multiplication.

3. Let (M, F) be a regular foliation. Take two distinguished open sets U_{i} ≃ P_{i} × T_{i} where P_{i} is a plaque and T_{i} a transversal, i = 0, 1. Let G_{i} ⇉ U_{i} be the graph of the regular equivalence relation induced by F on U_{i}.

Combining the two previous examples one can show that there is a bijection between the set of Morita equivalences between G_{0} and G_{1} and the diffeomorphisms from T_{0} onto T_{1}.

We will be especially interested in local Morita isomorphisms between local quasi-graphoids. In this situation we have the following lemma:

Lemma 1. Let L_{0} ⇉ O_{0} and L_{1} ⇉ O_{1} be two local Lie groupoids, f : L_{1} ↶ L_{0} a local Morita isomorphism and (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} a graph for f. Let z be a point of Z_{f}.

If L_{0} or L_{1} is a local quasi-graphoid there is an open neighborhood V_{z} of z in Z_{f} such that for any graph (b_{S}, a_{S}) : S → O_{1} × O_{0} there is at most one morphism of graphs from S onto V_{z}.

Proof. Let V_{z} be a neighborhood of z such that b_{f} induces a diffeomorphism between V_{z}/L_{0} and b_{f}(V_{z}). Let (b_{S}, a_{S}) : S → O_{1} × O_{0} be a graph and suppose that there exist two smooth maps Φ_{0}, Φ_{1} : S → V_{z} such that a_{f} ∘ Φ_{0} = a_{S} = a_{f} ∘ Φ_{1} and b_{f} ∘ Φ_{0} = b_{S} = b_{f} ∘ Φ_{1}.

Then we consider the two following differentiable maps:

Ψ : S → L_{0} , x ↦ γ such that Φ_{0}(x) = Φ_{1}(x) ∗ γ,
Ψ′ : S → L_{0} , x ↦ a_{S}(x),

which satisfy s_{0} ∘ Ψ = s_{0} ∘ Ψ′ = a_{S} and r_{0} ∘ Ψ = r_{0} ∘ Ψ′ = a_{S}.

If L_{0} is a local quasi-graphoid there is at most one morphism of graphs from (a_{s}, a_{s}) : S → O_{0} × O_{0} to (r_{0}, s_{0}) : L_{0} → O_{0} × O_{0}. Thus Ψ = Ψ′ and so Φ_{0} = Φ_{1}. q.e.d.

Let (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} and (b_{g}, a_{g}) : Z_{g} → O_{1} × O_{0} be the graphs of two local Morita isomorphisms f and g between two local quasi-graphoids L_{0} and L_{1}. The previous lemma implies that if there is an isomorphism of graphs from Z_{f} onto Z_{g} then it must intertwine the actions. So in this case, the local Morita isomorphisms f and g are equal.

This lemma will play a crucial role in the following for the definition of a smooth structure on the set of germs of local Morita isomorphisms.

3.2 The local pseudo-group structure

Local Morita isomorphisms are going to replace transverse isomorphisms in the construction of the holonomy groupoid. So we have to define identity, composition, inversion and restriction for them.

In the following L_{0}⇉O_{0} and L_{1}⇉O_{1} are two local Lie groupoids. We suppose that we have a local Morita isomorphism f : L_{1} ↶ L_{0} and we take a graph (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} for f.

Identity: Let s_{0} and r_{0} be respectively the source and range maps of L_{0}. We denote by Id_{L_{0}} : L_{0} ↶ L_{0} the local Morita isomorphism represented by the graph Z_{Id_{L_{0}}} = L_{0}, a_{Id_{L}} = s_{0} and b_{Id_{L}} = r_{0}, the two actions being right and left multiplication. This local Morita isomorphism is called the identity.

Inversion: Let Z_{f^{−1}} = Z_{f}, a_{f^{−1}} = b_{f} and b_{f^{−1}} = a_{f}. We consider the right local action of L_{1} on Z_{f^{−1}} (resp. the left local action of L_{0} on Z_{f^{−1}}) which is defined by Ψ_{1}(z, γ_{1}) = γ_{1}^{−1} ∗z (resp. Ψ_{0}(γ_{0}, z) = z ∗γ_{0}^{−1}). Equipped with these actions, (b_{f^{−1}}, a_{f^{−1}}) : Z_{f^{−1}} → O_{0} × O_{1} is a graph of the local Morita isomorphism f^{−1} : L_{0} ↶ L_{1} called the inverse of f.

Restriction: Let H_{0} (resp. H_{1}) be an open sub local groupoid of L_{0} (resp. L_{1}) and V be an open subset of Z_{f} such that a_{f}(V) is the set of units of H_{0} and b_{f}(V) is the set of units of H_{1}. The restriction of f to H_{0}, H_{1} and V is the local Morita isomorphism from H_{0} onto H_{1} which admits as graph the restriction (b_{f}, a_{f})| : V → b_{f}(V) × a_{f}(V) with the local action induced by L_{0} and L_{1}. We denote it by f|_{H_{1},V,H_{0}} : H_{1} ↶ H_{0}.

Local composition: Let g : L_{2} ↶ L_{1} be another local Morita isomorphism admitting the following graph (b_{g}, a_{g}) : Z_{g} → O_{2} × O_{1}, where O_{2} is the set of units of L_{2}:

[figure: O_{2}; O_{1}; O_{0} .; L_{2}; Z_{g}; L_{1}; Z_{f}; L_{0}; r_{1}; s_{1}; r_{0}; s_{0}; r_{2}; s_{2}; b_{g}; a_{g}; b_{f}; a_{f}]

We consider the fiber product

Z_{g} ×_{O_{1}} Z_{f} = {(z_{1}, z_{2}) ∈ Z_{g} × Z_{f} | a_{g}(z_{1}) = b_{f}(z_{2})}.

We define a right local free action of L_{1} on Z_{g} ×_{O_{1}} Z_{f} in the following way: Ψ(z_{1}, z_{2}, γ) = (z_{1} ∗ γ, γ^{−1} ∗ z_{2}) when (z_{1}, z_{2}) belongs to Z_{g} ×_{O_{1}} Z_{f} and γ ∈ L_{1} is such that z_{1} ∗ γ and γ^{−1} ∗ z_{2} exist. This action defines a regular foliation F_{L_{1}} on Z_{g} ×_{O_{1}} Z_{f}.

The local Morita isomorphisms f and g are called composable if the equivalence relation induced by the foliation F_{L_{1}} on Z_{g} ×_{O_{1}} Z_{f} is regular, that is the space of leaves of F_{L_{1}} is a manifold. In this case we denote this space of leaves by Z_{g∘f} := Z_{g} ×_{O_{1}} Z_{f}/F_{L_{1}}. The maps a_{f} ∘ pr_{2} : Z_{g} ×_{O_{1}} Z_{f} → O_{0} and b_{g} ∘ pr_{1} : Z_{g} ×_{O_{1}} Z_{f} → O_{2} induce on the quotient space the maps a_{g∘f} : Z_{g∘f} → O_{0} and b_{g∘f} : Z_{g∘f} → O_{2}. Finally with the obvious local actions of L_{0} on the right and L_{2} on the left, (b_{g∘f}, a_{g∘f}) : Z_{g∘f} → O_{2} ×O_{0} is the graph of a local Morita isomorphism called the composition of g and f and denoted by g ∘ f : L_{2} ↶ L_{0}.

In the general case the foliation F_{L_{1}} is regular and so it is locally trivial. Thus, by restricting f and g, one can make a composition of g and f which depends of the restrictions. Contrary to the case of local diffeomorphisms there is not a better restriction for the composition of local Morita isomorphism. Therefore we talk about local composition.

One must remark that for any point (z_{1}, z_{2}) of Z_{g} ×_{O_{1}} Z_{f}, one can find an open sub local groupoid H_{1} of L_{1}, an open neighborhood V_{1} of z_{1} and an open neighborhood V_{2} of z_{2} such that the restrictions f|_{H_{1},V_{1},L_{0}} and g|_{L_{2},V_{2},H_{1}} are composable.

The previous operations are really an inverse and a composition when you look at them locally. More precisely one can check the following proposition:

Proposition 4. Let f : L_{1} ↶ L_{0} be a local Morita isomorphism from L_{0} ⇉ O_{0} onto L_{1} ⇉ O_{1} for which (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} is a graph. Let z be a point in Z_{f}. The following assertions are fulfilled:

1. There exists an open sub local groupoid H_{0} of L_{0} with O_{0} as space of units and an open sub local groupoid H_{1} of L_{1} with O_{1} as space of units such that:

Id_{H_{1}} ∘ f|_{H_{1},Z_{f} ,H_{0}} = f|_{H_{1},Z_{f} ,H_{0}} ∘ Id_{H_{0}} = f|_{H_{1},Z_{f} ,H_{0}}.

2. There exist an open neighborhood V_{z} of z in Z_{f}, an open sub local groupoid H′_{0} of L_{0} with a_{f}(V_{z}) as space of units and an open sub local groupoid H′_{1} of L_{1} with b_{f}(V_{z}) as space of units such that:

f|_{H′_{1},V_{z},H′_{0}} ∘ (f|_{H′_{1},V_{z},H′_{0}})^{−1} = Id_{H′_{1}},
(f|_{H′_{1},V_{z},H′_{0}})^{−1} ∘ f|_{H′_{1},V_{z},H′_{0}} = Id_{H′_{0}}.

Definition 5. We will say that we have a local pseudo-group of local Morita isomorphisms on the smooth manifold M when we have a generalized atlas U on M stable under restriction together with a set IU of local Morita isomorphisms between elements of U such that:

– “Identity” is in IU, that is for all L ⇉ O in U, Id_{L} : L ↶ L is in IU.

– IU is stable under inversion, local composition and restriction.

3.3 Associated groupoid

A generalized atlas U on a smooth manifold M or equivalently a local quasi-graphoid over M induces a generalized atlas Ũ stable under restriction on M. Then we can consider the local pseudo-group of local Morita isomorphisms between local quasi-graphoids which constitute Ũ. We are going to associate to such a local pseudo-group the Lie groupoid of its germs as it is usually done for a pseudo-group of local diffeomorphisms. To do this we start by defining the germ of a local Morita isomorphism between two local quasi-graphoids.

Let L_{0} ⇉ O_{0} and L_{1} ⇉ O_{1} be two local quasi-graphoids. We suppose that f : L_{1} ↶ L_{0} and g : L_{1} ↶ L_{0} are two local Morita isomorphisms and we take a graph (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} for f and a graph (b_{g}, a_{g}) : Z_{g} → O_{1} × O_{0} for g. We consider an element γ_{f} of Z_{f} and γ_{g} of Z_{g}.

We will say that the germ of f in γ_{f} is equal to the germ of g in γ_{g} and we will denote it by [f]_{γ_{f}} = [g]_{γ_{g}} if there exists an open neighborhood V_{γ_{f}} of γ_{f} in Z_{f}, an open neighborhood V_{γ_{g}} of γ_{g} in Z_{g} and an isomorphism of graphs ϕ : V_{γ_{f}} ⟶ V_{γ_{g}} which sends γ_{f} onto γ_{g}.

One can show that the equality [f]_{γ_{f}} = [g]_{γ_{g}} implies that there exists an open sub local quasi-graphoid H_{0} of L_{0}, an open sub local quasi-graphoid H_{1} of L_{1}, an open neighborhood V_{γ_{f}} of γ_{f} in Z_{f} and an open neighborhood V_{γ_{g}} of γ_{g} in Z_{g} such that f|_{H_{1},V_{γ_{f}} ,H_{0}} = g|_{H_{1},V_{γg} ,H_{0}}.

In the general case it may happen that f : L_{1} ↶ L_{0} and g : L′_{1} ↶ L′_{0} are local Morita isomorphisms between different local quasi-graphoids. When L_{i} ⇉ O_{i} and L′_{i} ⇉ O′_{i} are locally isomorphic we can consider L̃_{i} ⇉ O_{i} ∩ O′_{i} the maximal open sub quasi-graphoid of both L_{i} and L′_{i}, i = 0, 1. Then we can look at the restrictions f|_{L̃_{1},Z_{f} ,L̃_{0}} and g|_{L̃_{1},Z_{f} ,L̃_{0}} and use the previous definition of germ. The gluing condition of generalized atlases ensures that we are always in the case just mentioned when we are dealing with a local pseudo-group of local Morita isomorphisms.

From now on U = {L_{i} ⇉_{r_{i}}^{s_{i}} O_{i}}_{i∈I} is a generalized atlas stable under restriction on a smooth manifold M and IU is the pseudo-group of local Morita isomorphisms between elements of U.

Proposition 5. The set of germs of elements of IU, denoted by GIU, is naturally endowed with a smooth manifold structure. For this structure M is an embedded submanifold of GIU.

Proof. Let [f]_{γ} ∈ GIU, f : L_{1} ↶ L_{0} be one of its representatives which admits Z_{f} as graph and γ ∈ Z_{f}. Lemma 1 ensures the local injectivity of the map

\begin{matrix}Z_{f} & ⟶ & GIU \\ η & ↦ & [f]_{η} \end{matrix} .

This allows us to provide GIU with a smooth structure.

The following map is an embedding of the manifold M in the manifold GIU:

\begin{matrix}M & ⟶ & GIU \\ x & ↦ & [Id_{L}]_{x} \end{matrix}

where L is a map over x which means that L is an element of U and x a unit of L. q.e.d.

Now we can define the Lie groupoid associated to the generalized atlas U. We deduce the following theorem:

Theorem 1. With the following structural maps GIU is a Lie groupoid over M:

– Source and range:

\begin{aligned}s : & GIU & ⟶ & M \\ & [f : L_{1} ↶ L_{0}]_{z} & ↦ & a_{f}(z) \end{aligned}

\begin{aligned}r : & GIU & ⟶ & M \\ & [f : L_{1} ↶ L_{0}]_{z} & ↦ & b_{f}(z) \end{aligned}

where (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} is a graph for f and z belongs to Z_{f}.

– Inverse:

[f : L_{1} ↶ L_{0}]_{z}^{−1} = [f^{−1} : L_{0} ↶ L_{1}]_{z}.

– Product: Let [f : L_{1} ↶ L_{0}]_{z} and [g : L_{2} ↶ L_{1}]_{t} be two elements of GIU. Let’s take a graph (b_{f}, a_{f}) : Z_{f} → O_{1} × O_{0} for f and a graph (b_{g}, a_{g}) : Z_{g} → O_{2} × O_{1} for g such that z belongs to Z_{f} and t belongs to Z_{g}. We suppose that a_{g}(t) = b_{f}(z). Then

[g : L_{2} ↶ L_{1}]_{t} · [f : L_{1} ↶ L_{0}]_{z} = [g ∘ f : L_{2} ↶ L_{0}]_{(t,z)}.

– Units:

\begin{matrix}M & ⟶ & GIU \\ x & ↦ & [Id_{L}]_{x} \end{matrix}

where L is a map over x.

Moreover GIU ⇉_{r}^{s} M is a quasi-graphoid.

Proof. The only point which is not obvious is that GIU is a quasigraphoid. To see this let O be an open subset of M and ν : O → GIU a smooth local section of both s and r. By restricting O if necessary we can suppose that there exists an element L ⇉_{r_{0}}^{s_{0}} O of U and a local Morita

isomorphism f : L ↶ L for which (b_{f}, a_{f}) : L → O × O is a graph and such that the image of ν is a subset of {[f]_{γ} | γ ∈ Z_{f}}. Then there is a unique smooth map ν̃ : O → Z_{f} such that:

i) For all x ∈ O we have that ν(x) = [f]_{ν̃(x)}.

ii) a_{f} ∘ ν̃ = b_{f} ∘ ν̃ = 1_{O}.

The map γ ↦ γ ∗ ν̃(s_{0}(γ)) induces a local isomorphism of graphs from L to Z_{f}. So there is a neighborhood W of O in L such that

[Id_{L}]_{γ} = [f]_{φ(γ)} for all γ in W.

In particular if x belongs to O we get [Id_{L}]_{x} = [f]_{φ(x)} = [f]_{ν̃(x)} = ν(x). So ν is the inclusion of units. q.e.d.

Remark.

1. If L ⇉ M is a local quasi-graphoid let U_{L} be the corresponding generalized atlas stable under restriction of M, that is U_{L} contains L and all its restriction. The biggest quasi-graphoid over M which admits L as a sub local Lie groupoid is GIU_{L}. If moreover L is s-connected, the smallest quasi-graphoid over M which admits L as a sub local Lie groupoid is the s-connected component of GIU_{L}.

2. If we take V to be the set of all local quasi-graphoids over open subsets of M and IV the pseudo-group of local Morita isomorphisms between elements of V, an analogous construction gives rise to a Lie groupoid GIV ⇉ GV which is isomorphic to the universal convector of J. Pradines and B. Bigonnet [3, 26, 4].

Recall that an almost injective Lie algebroid is a Lie algebroid for which the anchor is injective when restricted to a dense open subset of the base space. The local integration of almost injective Lie algebroid [13] ensures that given an almost injective Lie algebroid on a smooth manifold M there exists a local quasi-graphoid L over M which integrates A. One can consider the maximal generalized atlas corresponding to L and then uses the previous theorem to compute a Lie groupoid G on M. Of course the Lie algebroid of G is equal to the Lie algebroid of L that is A. Finally we obtain:

Theorem 2. Every almost injective Lie algebroid is integrable.

In [24], J. Pradines asserted that, as for Lie algebras, every Lie algebroid integrates into a Lie groupoid. In fact this assertion, named Lie’s third theorem for Lie algebroid is false. This was pointed out by a counter example given by P. Molino and R. Almeida in [1]. Since that time lots of work has been done around this problem. For example K. Mackenzie has found obstructions to integrability of transitive Lie algebroids [18], fundamental examples have been studied by A. Weinstein especially in relation with Poisson manifolds [30, 31] and more recently V. Nistor has studied integrability of Lie algebroids over stratified manifolds [21]. A part of this subject is discussed in the book of A. Cannas da Silva and A. Weinstein [6].

During the writing of this paper, a very interesting preprint of M. Crainic and R.L. Fernandes appeared [11]. They give a necessary and sufficient condition for the integrability of Lie algebroids. In particular, they recover the previous theorem.

4. Holonomy groupoid of an almost regular foliation

In this section, we apply the results obtained previously to almost regular foliations. We end with several examples.

The study of quasi-graphoids enables us to propose a reasonable definition of holonomy groupoid for an almost regular foliation:

Definition 6. Let (M, F) be an almost regular foliation. A holonomy groupoid of (M, F) is an s-connected quasi-graphoid G having M as space of units and such that the orbits of G are the leaves of F.

Proposition 2 ensures that this definition coincides with the usual one in the case of a regular foliation.

If F is an almost regular foliation on a smooth manifold M such that the regular leaves are of dimension k then an obvious necessary condition for the existence of an holonomy groupoid for F is that F can be defined by an almost injective Lie algebroid A, that is A is of dimension k. The previous results ensure that this condition is also sufficient. Finally we have the following theorem:

Theorem 3. Let (M, F) be an almost regular foliation and k denote the dimension of the regular leaves. There exists an holonomy groupoid for (M, F) if and only if one of the following equivalent assertions is fulfilled:

1. F can be defined by a quasi-graphoid with space of units M.

2. There exists a generalized atlas for F or equivalently F can be defined by a local quasi-graphoid over M.

3. F can be defined by a Lie algebroid of dimension k over M.

One must remark that for an almost regular foliation (M, F) there exist as many holonomy groupoids as there are maximal generalized atlases or as there are almost regular Lie algebroids over M which define F. Nevertheless some foliations admit a better holonomy groupoid. More precisely we will say that G is the universal holonomy groupoid of (M, F) if for any other holonomy groupoid H of (M, F) there is a morphism of Lie groupoids from H to G. One can check that this is equivalent to the fact that for any almost regular Lie algebroid A which defines F there is a morphism of Lie algebroids from A to AG, the Lie algebroid of G. In such a situation we say that the Lie algebroid AG is extremal for F. A particular case of extremal algebroid for a foliation F is when the anchor of A induces an isomorphism between the module of local smooth sections of A and the module of local vector fields tangent to the foliation F.

In conclusion if (M, F) is an almost regular foliation it satisfies one of the three following properties:

i) The foliation (M, F) does not possess any holonomy groupoid, or equivalently F cannot be defined by an almost injective Lie algebroid. ii) The foliation (M, F) possesses a holonomy groupoid but not a

universal one: that is F can be defined by an almost injective Lie algebroid, but there is no extremal Lie algebroid for F.

iii) The foliation (M, F) possesses an universal holonomy groupoid, or in other words F is defined by an extremal Lie algebroid.

We finish this paper by giving examples of each of these cases.

Examples.

1. Regular foliation: If F is a regular foliation on the manifold M then the tangent bundle T F of F equipped with the inclusion as anchor is the unique almost regular Lie algebroid which defines F. In this case the foliation admits a unique holonomy groupoid: Hol(M, F). The uniqueness of the holonomy groupoid is a characteristic property of a regular foliation.

2. Almost free action of a Lie group: Let (M, F_{H}) be a singular foliation defined by an almost free action of a connected Lie group H on M. We denote by Φ : H × M → M the action and by H the Lie algebra of H.

Let p : M × H → T M be the map defines by p : (x, v) ↦ T Φ(x, v) where T Φ(x, v) denotes the evaluation at (x, 0, 1_{H}, v) ∈ T_{x}M ×T_{1_{H}} H of the differential of Φ. The set A_{H} = M × H is naturally endowed with a structure of Lie algebroid over M whose anchor is p. This algebroid is almost injective and defines F_{H}.

The crossed product groupoid M ⋊ H ⇉ M integrates A and is a holonomy groupoid for F_{H}. This groupoid is not universal in general.

3. Concentric spheres: Let F be the foliation of ℝ^{3} by concentric spheres and O be the singular leaf. Because the 2 sphere is not parallelizable there exists no Lie algebroid of dimension 2 which defines F. Thus such a foliation has no holonomy groupoid.

More generally a foliation F on a manifold M which is locally the foliation by concentric spheres of dimension different from 1, 3 and 7 has no holonomy groupoid [5].

4. Submanifold: Let M be a manifold of dimension m and N a submanifold of M of dimension n < m. We define F_{N} to be the foliation on M whose leaves are N and the connected components of M \ N.

Codimension 1 submanifold: We have already seen in section 2 how to construct a generalized atlas for F_{N}. The holonomy groupoid we obtain using this atlas is a particular case of groupoids related to the notion of explosion of manifolds which where constructed for the study of manifolds with corners [19, 20, 31].

In the simple case where M = N × ℝ we define

\begin{aligned}p : & A_{N} := T M ≃ T N × T ℝ & → & T M ≃ T N × T ℝ \\ & (x, v; t, λ) & ↦ & (x, v; t, t.λ). \end{aligned}

There is a unique structure of Lie algebroid on A_{N} whose anchor is p. This Lie algebroid is an almost injective Lie algebroid which defines F_{N}, moreover it is extremal. The corresponding groupoid is

G = N × N × ℝ × ℝ_{∗}^{+} ⇉ N × ℝ

obtained by making the product of the pair groupoid over N with the groupoid of the action of ℝ_{∗}^{+} on ℝ by multiplication.

The point: Suppose now that M is a manifold of dimension n ≥ 2 and let N be a point of M. Let f : M → ℝ be a smooth function which vanishes only at N. We consider

\begin{aligned}p_{f} : & A_{f} := T M & → & T M \\ & (x, v) & ↦ & (x, f(x)v). \end{aligned}

There is a unique structure of Lie algebroid on A_{f} whose anchor is f. Moreover A_{f} is almost injective and it defines F_{N}. Thus F_{N} admits holonomy groupoids.

Let A be a Lie algebroid which defines F_{N}. The anchor of A induces a smooth map P : U → M_{n}(ℝ), where U is a neighborhood of N in M, such that P (N) = 0 and P (x) is invertible for x ≠ N.

If A is extremal then the map P must be such that for all smooth function f : U → ℝ which vanishes only at N, the map from U \ {N} to M_{n}(ℝ) definined by x ↦ f(x)P^{−1}(x) can be smoothly extended to U. Such a map P does not exist, so there are no extremal Lie algebroids and no universal holonomy groupoids for F_{N}.

5. Composed foliation: Let M be a manifold equipped with two regular foliations F_{1} and F_{2} where F_{1} is a subfoliation of F_{2} and let F be the foliation on M × ℝ defined by F_{1} × {0} and F_{2} × {t} for t ≠ 0.

Let G_{1} (resp. G_{2}) be the holonomy groupoid of F_{1} (resp. F_{2}) and φ : G_{1} → G_{2} the immersion induced by the inclusion of F_{1} into F_{2}.

We take a Euclidean metric on T F_{2} and we denote by N = (T F_{1})^{⊥} the orthogonal bundle of T F_{1} in T F_{2}. Then T F_{2} can be identified with T F_{1} ⊕ N. We equip A = (T F_{1} ⊕ N) × ℝ with the unique structure of Lie algebroid over M × ℝ whose anchor is:

\begin{aligned}p : & A := (T F_{1} ⊕ N) × ℝ & → & T (M × ℝ) = T M × T ℝ \\ & (v_{1}, v_{2}, t) & ↦ & (v_{1} + tv_{2}, (t, 0)). \end{aligned}

This Lie algebroid is almost injective, it defines F and it is extremal for F. The corresponding Lie groupoid is the normal groupoid of φ defined by M. Hilsum and G. Skandalis[17]. More precisely there is an action of G_{1} on N = T F_{2}/T F_{1}, the normal bundle of the inclusion of T F_{1} in T F_{2}.

Thus G_{1} ×_{M} N = {(γ, X) ∈ G_{1} × N | X ∈ N_{r(γ)}} is equipped with a groupoid structure:

(γ_{1}, X_{1}) · (γ_{2}, X_{2}) = (X_{1} + γ_{1}(X_{2}), γ_{1}γ_{2})

when (γ_{i}, X_{i}) ∈ G_{1} ×_{M} N and r(γ_{2}) = s(γ_{1}). At the set level, the normal groupoid of φ is the union groupoid

G_{2} × ℝ_{∗} ∪ (G_{1} ×_{M} N) × {0} ⇉ M × ℝ.

An extreme case is when F_{1} is the foliation of M by the points and F_{2} is the foliation of M with M as unique leaf. In this case we recover the tangent groupoid of M of A. Connes [7].

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