Principal Fibre Bundles with Structural Lie Groupoid

Gheorghe Ivan

Abstract

This paper deals with transformation Lie groupoids and fibre bundles with structural groupoid. The aim of this paper is to prove the theorem of construction of a principal fibre bundle with structural Lie groupoid.

Mathematics Subject Classification: 22A22, 55R99, 57S99 Key words: Lie groupoid, transformation Lie groupoid, principal fibre bundle with structural Lie groupoid

Introduction

The principal fibre bundle with structural Lie groupoid introduced by A. Haefliger ([2]), is a generalization of the concept of principal fibre bundle with structural Lie group.

In the first two sections we reffer to Lie groupoids, Lie subgroupoids and to the action of a Lie groupoid on a manifold.

In the third section we study the principal fibre bundles with stuctural Lie groupoid. Also, we construct a principal fibre bundle with stuctural Lie groupoid Γ, associated of a given manifold M and of a familly of transition functions with values in Γ ( theorem 3.1.).

In the four section is introduced the fibre bundle with type fibre and structural Lie groupoid.

Let us we shall assume that the manifolds are finite dimensional, smooth of class C^{∞}, paracompact, connected and without boundary.

1 Lie groupoids

Definition 1.1. ([7]) A groupoid Γ on Γ_{0} is a pair of sets (Γ; Γ_{0}) equipped with: (i) surjections α, β : Γ → Γ_{0}, (source and target maps );

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 39-48 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(ii) map μ : Γ_{(2)} → Γ, (x, y) → μ(x, y) = x · y or xy ( partial ) multiplication, where Γ_{(2)} = {(x, y) ∈ Γ × Γ | β(x) = α(y)}; each pair (x, y) in Γ_{(2)} is said to be composable;

(iii) injection ϵ : Γ_{0} → Γ, u → ϵ(u) = ũ (inclusion map);

(iv) map i : Γ ⟶ Γ, x → i(x) = x^{−1} ( inversion map).

These maps must satysfy the following algebraic axioms generalizing those of group:

(G1) (associative law) (xy)z = x(yz) ( if one triple product is defined then so is the other );

(G2) (identities) For each x ∈ Γ we have (ϵ(α(x)), x); (x, ϵ(β(x)) ∈ Γ_{(2)} and ϵ(α(x)) · x = x · ϵ(β(x)) = x;

(G3) (inverses) For each x ∈ Γ we have (x, i(x)); (i(x), x) ∈ Γ_{(2)} and μ(x, i(x)) = ϵ(β(x)), μ(i(x), x) = ϵ(α(x)). △

We denote sometimes a groupoid Γ on Γ_{0} by (Γ, α, β; Γ_{0}) or (Γ; Γ_{0}).

The subset ϵ(Γ_{0}) of Γ is called the unity set of Γ .

For any u ∈ Γ_{0}, the sets α^{−1}(u) and β^{−1}(u) will be called a α-fibre and β-fibre over u.

A groupoid Γ on Γ_{0} is said to be transitive if the map α × β : Γ ⟶ Γ_{0} × Γ_{0}, defined by (α × β)(x) = (α(x), β(x)), (∀)x ∈ Γ is surjective.

We deduce from the axioms that, for any groupoid Γ on Γ_{0} the following rules hold:

(1.1) α ∘ ϵ = β ∘ ϵ = id_{Γ_{0}} and ϵ(u) · ϵ(u) = ϵ(u), (∀) u ∈ Γ_{0}. (1.2) α(xy) = α(x) and β(xy) = β(y) for all (x, y) ∈ Γ_{(2)}. (1.3) x^{−1} · x = ϵ(β(x)) and x · x^{−1} = ϵ(α(x)) (1.4) β(x^{−1}) = α(x) and β(x) = α(x^{−1}) (1.5) The set Γ(u) = α^{−1}(u) ∩ β^{−1}(u) is a group under the restriction of the

partial multiplication, called the isotropy group at u of Γ.

The isotropy groups Γ(α(x)) and Γ(β(x)) are isomorphic.

(1.6) If (Γ; Γ_{0}) is transitive, then the isotropy groups Γ(u), (∀) u ∈ Γ_{0} are isomorphes.

By group bundle we mean a groupoid Γ on Γ_{0} such that α(x) = β(x) for each x ∈ Γ.

If Γ is a groupoid on Γ_{0}, then Is(Γ) = {x ∈ Γ | α(x) = β(x)} is a group bundle, called the isotropy group bundle of Γ. Example 1.1. Coarse groupoid associated to a set. If B is a set, then B × B is a groupoid on B with the rules: α(x, y) = x; β(x, y) = y; ϵ(x) = (x, x), the partial multiplication is given by: μ((x, y), (y′, z)) = (x, z) iff y = y′ and the inverse of (x, y) ∈ B × B is defined by (x, y)^{−1} = (y, x). △ Definition 1.2. Let (Γ, α, β, μ; Γ_{0}) and (Γ′, α′, β′, μ′; Γ′_{0}) be two groupoids.

(i) A morphism of groupoids or groupoid morphism is a pair (f, f_{0}) of maps f : Γ ⟶ Γ′ and f_{0} : Γ_{0} ⟶ Γ′_{0} such that the following two conditions are satisfied:

(1.7) f(μ(x, y)) = μ′(f(x), f(y)), (∀) (x, y) ∈ Γ_{(2)};

(1.8) α′ ∘ f = f_{0} ∘ α and β′ ∘ f = f_{0} ∘ β.

If Γ_{0} = Γ′_{0} and f_{0} = Id_{Γ_{0}}, we say that f is Γ_{0}- morphism.

(ii) A groupoid morphism (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is be called strong morphism of groupoids, if for every (f(x), f(y)) ∈ Γ′_{(2)} we have (x, y) ∈ Γ_{(2)}. △ Example 1.2. Let (Γ, α, β; Γ_{0}) be a groupoid and Γ_{0} × Γ_{0} the coarse groupoid associated to Γ_{0}.Then α × β : Γ ⟶ Γ_{0} × Γ_{0}, (α × β)(x) = (α(x), β(x)) is a Γ_{0}- morphism. △

A groupoid morphism (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is said to be isomorphism of groupoids if the maps f and f_{0} are bijective.

We observe that a Γ_{0}- morphism of groupoids f : (Γ; Γ_{0}) ⟶ (Γ′; Γ_{0}) is an isomorphism iff the map f : Γ ⟶ Γ′ is bijective. Definition 1.3. A subgroupoid of (Γ; Γ_{0}) is a pair (Γ′, Γ′_{0}) of subsets, where Γ′ ⊆ Γ, Γ′_{0} ⊆ Γ_{0} such that the following conditions are verified:

(i) α(Γ′) ⊆ Γ′_{0}, β(Γ′) ⊆ Γ′_{0}, ϵ(Γ′_{0}) ⊆ Γ′; (ii) for every x, y ∈ Γ′ such that the product x·y is defined implies that x·y ∈ Γ′,

i.e. Γ′ is closed under the partial multiplication.

(iii) (∀)x ∈ Γ′ ⟹ x^{−1} ∈ Γ′. △ Example 1.3. Let (Γ; Γ_{0}) be a groupoid and u ∈ Γ_{0}. Then:

(a) ϵ(Γ_{0}) = {ũ | u ∈ Γ_{0}} is a subgroupoid of Γ over Γ_{0}, called the nul subgroupoid of Γ. .

(b) The isotropy group Γ(u) at u and Is(Γ) = ∪_{u∈Γ_{0}}Γ_{u}^{u} are subgroupoids of Γ. △ Remark 1.1. If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid, then not always, Imf = {f(x) | x ∈ Γ} is a subgroupoid of Γ′.

In the case when (f, f_{0}) is a strong morphism of groupoids, then Imf is a groupoid on Imf_{0} ( see, ([3]) ). △

Definition 1.4. ([7]) (a) A groupoid Γ on Γ_{0} is a Lie groupoid if Γ and Γ_{0} are manifolds, α and β are differentiable submersions so that Γ_{(2)} is a differentiable submanifold of the product manifold Γ × Γ, ϵ is a differentiable embedding and μ and i are differentiable maps.

(b) A Lie groupoid Γ on Γ_{0} is transitive if α × β : Γ_{0} × Γ_{0} is a differentiable submersion.

(c) A morphism of groupoids f : Γ ⟶ Γ^{‘} will be called a morphism of Lie groupoids if f is a differentiable map. △

We observe that any Lie groupoid G having e as unity may be considered to be a {e} - Lie groupoid. Conversely, a Lie groupoid with one unit is a Lie group. Example 1.4. (a) ( the coarse Lie groupoid ). Let Γ_{0} be a manifold. The manifold Γ_{0} × Γ_{0} endowed with the structure of coarse groupoid is a Lie groupoid.

(b) Let (P, π, M, G) be a principal bundle with the Lie group G as structural group. Let us consider the quotient Γ of P × P by the diagonal action ( in other words Γ = P ×P/ρ where ρ is the equivalence relation such that (z, z′) ∼ (zs, z′s) for any s ∈ G ). Then Γ is a transitive Lie groupoid on M, whose α - fibres are principal bundles, called the jauge groupoid of given principal bundle.

(c) (the fundamantal groupoid of a manifold ). Let B be a manifold and Π(B) = {(x, [σ], y) | x, y ∈ B, [σ] is the the homotopy classe of paths, σ(0) = x, σ(1) = y }. Then Π(B) is a groupoid on B with the rules: α(x, [σ], y) = x, β(x, [σ], y) = y, μ((x, [σ], y), (y′, [τ], z)) = (x, [σ ∘ τ], z) iff y = y′, where σ ∘ τ is the concatenation of paths σ and τ, ϵ(x) = (x, [constant], x) and (x, [σ], y)^{−1} = (y, [σ^{−1}], x), where σ^{−1}(t) = σ(1 − t), (∀) t ∈ [0, 1].

If Π(B) is equipped with the quotient topology of the compact open topology on the space of paths of B, then α × β : Π(B) → B × B is a covering map. It follows that Π(B) is a Lie groupoid on B, called it the fundamental or Poincaré groupoid associated to B. Its isotropy groups are the fundamental groups π_{1}(B, x), (∀) x ∈ B.

(d) A vector bundle E ⟶^{p} M is a group bundle on M. Here Γ = E is the total space, Γ_{0} = M is the base space, α = β = π so that Γ_{(2)} = ⊎_{x∈M}E_{x} × E_{x} ( E_{x} is the fibre at x ) and the composition law is fibrewise addition. This groupoid is not transitive. Proposition 1.1. ([7]) If Γ is a transitive Lie groupoid over Γ_{0}, then:

(i) the isotropy groups of Γ are Lie groups isomorphes.

(ii) each u ∈ Γ_{0} the fibre α^{−1}(u), is a differentiable principal bundle over Γ_{0} with the projection β and the isotropy group Γ(u) as structural group, i.e.: ξ_{α} = (α^{−1}(u),β, Γ_{0}; Γ(u)) is a differentiable principal bundle.△ Definition 1.5. Let (Γ; Γ_{0}) be a Lie groupoid. A Lie subgroupoid of Γ is a Lie groupoid (Γ′; Γ′_{0}) together with a morphism of Lie groupoids (φ, φ_{0}) : (Γ′; Γ′_{0}) → (Γ; Γ_{0}) which is an injective immersion.

A Lie subgroupoid (Γ′, φ) is an embedded differentiable subgroupoid if φ is an embedding. △ Proposition 1.2. ([1]) Let (Γ, α, β; Γ_{0}) be a Lie groupoid. Then:

(a) Is(Γ) is a closed embedded submanifold of Γ and it is a Lie subgroupoid of Γ.

(b) The α-connected component Γ_{α} ( i.e., the union of connected components of the points of Γ_{0} in the α-fibres of Γ is an open wide Lie subgroupoid of Γ.

(c) If Γ is a Lie groupoid with connected fibres, then the set Γ̃ of homotopy classes α-fibre to α-fibre of the paths contained in α-fibres having as origin the elements u,for all u ∈ Γ_{0}, has a naturally structure of Lie groupoid over Γ_{0}, with simply connected α-fibres, such that the projection ρ : Γ̃ ⟶ Γ, [σ] ⟶ ρ([σ]) := σ(1) is a morphism of Lie groupoids.△

2 The action of a Lie groupoid on a manifold

Let (Γ, α, β; Γ_{0}) be a Lie groupoid and M a manifold for which is given a differentiable map ρ : M → Γ_{0}.

We denote by M ×_{Γ_{0}} Γ = {(x, a) ∈ M × Γ | ρ(x) = α(a)} the corresponding fibre product. Definition 2.1. (i) An action of Γ on M is a differentiable map M ×_{Γ_{0}} Γ → M, (x, a) → x · a which verify the following conditions:

(2.1) ρ(x · a) = β(a), for all (x, a) ∈ M ×_{Γ_{0}} Γ;

(2.2) (x · a) · b = x · (ab), for al (a, b) ∈ Γ_{(2)};

(2.3) x · u = x, for all u ∈ Γ_{0}.

(ii) We say that a Lie groupoid Γ is a Lie transformation groupoid on a manifold M or that Γ acts ( differentiably) on M on the right, if there exists an action M ×_{Γ_{0}} Γ → M, (x, a) → x · a. △

We also write R_{a}x for x · a.

If we write a · x and assume σ(a · x) = α(a), for all (a, x) ∈ Γ ×_{Γ_{0}} M, where σ : M → Γ_{0} is a differentiable map and a · (b · x) = (ab) · x, for all (a, b) ∈ Γ_{(2)} instead (2.3), we say that Γ acts on M on the left and we use the notation L_{a}x for a · x also.

An action of Γ on M on the left is denoted by: Γ ×_{Γ_{0}} M → M, (a, x) → a · x, with Γ ×_{Γ_{0}} M = { (a, x) ∈ Γ × M | σ(x) = β(a) }.

From definitions, it follows that every element a ∈ Γ induces a transformation of M, denoted by x → x · a, where x ∈ M. In particular, R_{u} and L_{u} are identity transformations of M, for all u ∈ Γ_{0}.

We say that Γ acts effectivelly ( resp., freely ) on M if R_{a}x = x, for all x ∈ M ( resp., for some x ∈ M ) implies a = u ∈ Γ_{0}.

We say that the action of Γ on M is proper, if the map M ×_{Γ_{0}} Γ → Γ × Γ, (x, a) → (x, x · a) is proper.

Examples 2.1. Let (Γ; Γ_{0}) be a Lie groupoid and H a closed wide subgroupoid of Γ.

(i) We let H acts on Γ on the right as follows. Every a ∈ H maps x ∈ Γ into x·a ; M = Γ is manifold for which (H, α, β; Γ_{0}

is Lie groupoid and ρ = β : M = Γ → Γ_{0} is differentiable map.

In this case, Γ ×_{Γ_{0}} H = {(x, a) ∈ Γ × H | β(x) = α(a) }. The conditions (2.1)-(2.3) from Definition 2.1. are easy verified and we obtain that

Γ ×_{Γ_{0}} H → Γ, (x, a) → x · a is an action of H on Γ on the right.

(ii) We let Γ acts on Γ/H on the right as follows. a closed wide subgroupoid of Γ.

The quotient space Γ/H admits a structure of manifold and the projection Γ → Γ/H is a differentiable map.

Also, we have that ρ : Γ/H → Γ_{0}, ρ(xH) = β(x), (∀) xH ∈ Γ /H is a differentiable map and the corresponding fibre product is Γ/H ×_{Γ_{0}} Γ = {(xH, a) ∈ Γ/H × Γ | ρ(xH) = α(a) }.

Then the mapping (xH, a) → xH · a = xaH defines an action of Γ on Γ/H on the right. Indeed, we have:

- ρ(xH, a) = ρ(xaH) = β(xa) = β(a), (∀) (xH, a) ∈ Γ/H ×_{Γ_{0}} Γ;

- (xH · a) · b = (xa)H · b = ((xa) · b)H = x · (ab)H = xH · (ab), (∀) (a, b) ∈ Γ_{(2)},

- xH · u = (xu) · H = xH, (∀) u ∈ Γ_{0}, i.e. the conditions (2.1) - (2.3) from Definition 2.1. hold. Γ is a Lie transformation groupoid on Γ/H which is transitive. The action of Γ on Γ/H is effective iff H does not contains any normal

subgroupoid ≠ Γ_{0} of Γ.

(iii) We let Γ acts on Γ/H on the left as follows. The mapping σ : Γ/H → Γ_{0}, xH → σ(xH) = α(x), is a differentiable

map from the manifold Γ/H into Γ_{0}, and the corresponding fibre product is Γ ×_{Γ_{0}} Γ/H = {(a, xH) ∈ Γ × Γ × H | σ(xH) = β(a) }.

The map Γ ×_{Γ_{0}} Γ/H → Γ/H, (a, xH) → a · xH = (ax)H is an action of Γ on Γ/H on the left. Indeed, we have:

- σ(a · xH) = σ((ax)H) = α(ax) = α(a), (∀) (a, xH) ∈ Γ ×_{Γ_{0}} Γ/H;

- (a · (b · xH) = a · (bx)H = (a(bx))H = ((ab)x)H = (ab) · xH, (∀) (a, b) ∈ Γ_{(2)},

- u · xH = (ux) · H = xH, (∀) u ∈ Γ_{0}, i.e. the conditions (2.1) - (2.3) hold.△

Example 2.2. Given a Lie groupoid Γ and a manifold M, Γ acts freely on P = M×Γ on the right as follows.

We consider the mapping ρ : P = M × Γ → Γ_{0}, (x, a) → ρ(x, a) = β(a). Then P ×_{Γ_{0}} Γ = {((x, a), b) ∈ P × Γ | ρ(x, a) = α(b) } and for all (x, a) ∈ P ×_{Γ_{0}} Γ we have α(b) = β(a) and hence (a, b) ∈ Γ_{(2)}.

The action of Γ on M ×Γ is given by: (M ×Γ)×_{Γ_{0}} Γ → M ×Γ, ((x, a), b) → (x, ab). △

3 Principal fibre bundles with structural Lie groupoid

Let Γ be a Lie groupoid acting freely on a manifold P on the right.

On P define an equivalence relation:

u ∼ v ⟺ (∃) a ∈ Γ such that (u, a) ∈ P ×_{Γ_{0}} Γ and v = R_{a}u, and denote the equivalence classes by [u], u ∈ P and the set of equivalence

classes by P/Γ.

For each u ∈ P, the orbit uΓ = {R_{a}u | (∀) a ∈ Γ } is closed in P.

The quotient space P/Γ admits a structure of manifold and the projection π : P ⟶ P/Γ, u → π(u) = [u] is a differentiable map.

Definition 3.1. ([2]) A (differentiable) principal bundle over M with structural Lie groupoid Γ consists of a manifold P and an action of Γ on P satisfying the following conditions:

(3.1) Γ acts freely on P on the right: (u, a) ∈ P ×_{Γ_{0}} Γ → u · a = R_{a}u ∈ P; (3.2) M is the quotient space of P by the equivalence relation induced by Γ

i.e. M = P/Γ, and the canonical projection π : P → M is differentiable;

(3.3) P is locally trivial, that is, every point x ∈ M has a neighborhood U such that π^{−1}(U) is diffeomorphic with U ×Γ in the sense that there is a diffeomorphism ψ : π^{−1}(U) → U × Γ such that ψ(u) = (π(u), φ(u)), where φ is a mapping of π^{−1}(U) into Γ satisfying φ(u · a) = φ(u) · a, for all u ∈ π^{−1}(U) and a ∈ Γ, with u · a ∈ P. △

A principal fibre bundle with structural Lie groupoid Γ will be denoted by (P, π, M, Γ) or simply P. We call P the total space, M the base space, Γ the structural groupoid and π the projection.

For x ∈ M , π^{−1}(x) is a closed submanifold of P, which is differentiably isomorphic with Γ. It is called the fibre over x. Remark 3.1. If Γ is a Lie group ( i.e. Γ_{0} = {e} ), then (P, π, M, Γ) is a principal fibre bundle with Γ as structural group. △ Example 3.1. (i) Given a Lie groupoid Γ and a manifold B, Γ acts freely on P = B × Γ on the right ( see, Example 2.2. ) by the action:

((x, a), b) ∈ (B × Γ) × Γ → (x, ab) ∈ B × Γ, (∀) (a, b) ∈ Γ_{(2)}.

Then (B × Γ, π, B, Γ), where π : B × Γ → B is the projection on first factor, is a principal fibre bundle with Γ as structural groupoid, called the trivial principal bundle.

- (ii) Let (P, π, M, Γ) be a principal fibre bundle with structural groupoid Γ. Given an open submanifold V of M, π^{−1}(V) is an open submanifold of P on which Γ acts freely and (π^{−1}(V), π, V, Γ) is a principal fibre bundle over the base V with structural groupoid Γ.

- (iii) Let Γ be a Lie groupoid and H a closed Lie subgroupoid of Γ. H acts differentiably on Γ to the right ( see, Example 2.1.(i) ) by the action: (x, a) ∈ Γ ×_{Γ_{0}} H → x · a ∈ Γ.

Then (Γ, π, Γ/H, H) is a principal fibre bundle over the base manifold Γ/H with structural groupoid H.

- (iv) Let ξ = (P, π, M, Γ) be a principal fibre bundle with Γ as structural groupoid and H a closed Lie subgroupoid of Γ.

Being a Lie subgroupoid of Γ, H acts on P on the right. Let P/H be the quotient space of P by this action of the Lie groupoid H.

We obtain a principal fibre bundle ξ_{H} = (P, π_{H}, P/H, H) over the base P/H with structural groupoid H, where π_{H} : P → P/H is the projection. △ Proposition 3.1. Let ξ = (P, π, M, Γ) be a principal fibre bundle with structural Lie groupoid Γ. Then for each u ∈ Γ_{0}, the 5-uple

ξ_{Γ(u)} = (P, π_{Γ(u)}, P/Γ(u), Γ(u)) is a principal fibre bundle with the isotropy group Γ(u) as structural Lie group. Proof. The Lie group Γ(u) is a Lie subgroupoid of Γ and it acts on P on the right.

We denote by P/Γ(u) the quotient space of P by this action of Γ(u).

We obtain a principal fibre bundle ξ_{Γ(u)} = (P, π_{Γ(u)}, P/Γ(u), Γ(u)) over the base P/Γ(u) with structural group Γ(u) , where π_{Γ(u)} : P → P/Γ(u) is the projection . △

By condition (3.3) from Definition 3.1., for a principal fibre bundle (P, π, M, Γ) it is possible to choose an open covering {U_{i} | i ∈ I} of M, each π^{−1}(U_{i}) provided with an diffeomorphism x → (π(x), φ_{i}(x)) of π^{−1}(U_{i}) onto U_{i} × Γ such that φ_{i}(x · a) = φ_{i}(x) · a, (∀) (x, a) ∈ P ×_{Γ_{0}} Γ.

If x ∈ π^{−1}(U_{i} ∩ U_{j}), and (x, a) ∈ P ×_{Γ_{0}} Γ, then we get

φ_{j}(x · a)(φ_{i}(x · a))^{−1} = φ_{j}(x)(φ_{i}(x))^{−1}, which shows that φ_{j}(x)(φ_{i}(x))^{−1} depends only on π(x) not on x; hence the mapping

x ∈ π^{−1}(u_{i} ∩ U_{j}) → φ_{j}(x)(φ_{i}(x))^{−1} ∈ Γ is constant on each fiber.

We can define a mapping ψ_{ji} : U_{i} ∩ U_{j} → Γ by:

(3.4) ψ_{ji}(π(x)) = φ_{j}(x)(φ_{i}(x))^{−1}.

The familly of mappings ψ_{ji} for all i, j ∈ I such that U_{i} ∩ U_{j} ≠ ∅ are called transitions functions of the bundle (P, π, M, Γ) corresponding to the open covering (U_{i})_{i∈I} of M.

They enjoy the following properties:

(3.5) ψ_{ki}(x) = ψ_{kj}(x) · ψ_{ji}(x), (∀) x ∈ U_{i} ∩ U_{j} ∩ U_{k};

(3.6) ψ_{ii}(x) = u, where u ∈ Γ_{0} for all x ∈ U_{i};

(3.7) ψ_{ij}(x) · ψ_{ji}(x) = u, where u ∈ Γ_{0} for all x ∈ U_{i} ∩ U_{j}.

Conversely, given an open covering (U_{i})_{i∈I} of M and a set of differentiable mappings ψ_{ji} of U_{i} ∩ U_{j} ≠ ∅ into a Lie groupoid Γ which satisfy the relation (3.5), we shall construct a differentiable principal fibre bundle with transitions functions (ψ_{ji})_{i,j∈I}. Theorem 3.1. Let M be a manifold, (U_{i})_{i∈I} an open covering of M and Γ a Lie groupoid. For every familly of differentiable mappings (ψ_{ji})_{i,j∈I}, which satisfy the relation (3.5), where ψ_{ji} : U_{i} ∩ U_{j} → Γ, there exists a differentiable principal fibre bundle (P, π, M, Γ) with transition functions (ψ_{ji})_{i,j∈I}. Proof. For each i, consider a space X_{i} which is the direct product U_{i} × Γ and let X = ∪_{i∈I}X_{i} be the topological sum of X_{i}, i ∈ I. By taking open sets in each X_{i} as open sets in X, X becomes a differentiable manifold.

We introduce an equivalence relation ρ in X as follows: (x, a) ∈ U_{i} × Γ ⊂ X is equivalent to (x, b) ∈ U_{j} × Γ ⊂ X iff b = ψ_{ji}(x) · a in Γ.

The condition (3.5) and its consequences (3.6) and (3.7) imply that this is an veritable equivalence relation.

Now we define P as the quotient space of X by ρ, i.e. P = X/ρ.

We claim that P is the required principal fibre bundle (P, π, M, Γ).

In order to prove this, we first note that Γ acts on P on the right by the following rule:

if [(x, a)] ∈ P is the equivalence class of (x, a) ∈ U_{i}×Γ, then we define [(x, a)]·b to the equivalence class of (x, ab) ∈ U_{i} × Γ, i.e. [(x, a)] · b = [(x, ab)]. Γ acts freely on P, as we easily see.

That P admits a structure of differentiable manifold and that Γ acts differentiably on P will be seen latter on.

At any rate, the quotient of P by the equivalence relation by Γ ( two points of P are equivalent iff one is mapped into another by some element of Γ ) gives a projection π of P onto M.

Observe then that the projection of X onto P is one-to-one on the set X_{i} so that π^{−1}(U_{i}) is the set of ρ - classes of (x, a), x ∈ U_{i} and a ∈ Γ which is hence in one-to-one correspondence with product U_{i} × Γ.

We now introduce a structure of differentiable manifold in P by requiring that π^{−1}(U_{i}) is an open submanifold of P which is diffeomorphic with U_{i} × Γ. This is possible since every [(x, a)] ∈ P is contained in π^{−1}(U_{i}) for some i and since the identification (x, a) ∈ U_{i} × Γ with (x, ψ_{ji}(x) · a) ∈ U_{j} × Γ is made by means of differentiable mappings ψ_{ji}.

With this differentiable structure in P, it is no difficult to verify all the conditions so that P becomes a differentiable principal fibre bundle over the base M with Γ as structural groupoid.

Moreover, the transition functions of (P, π, M, Γ) corresponding to the covering (U_{i})_{i∈I} are precisely the given (ψ_{ji})_{i,j∈I}. △ Definition 3.2. Let (P, π, M, Γ) and (P′, π′, M′, Γ′) two principal fibre bundles with structural Lie groupoids Γ, Γ′, respectively. A homomorphism of (P, π, M, Γ) into (P′, π′, M′, Γ′) consists of a differentiable mapping f : P → P′ and a Lie groupoid morphism g : Γ → Γ′ such that f(x · a) = f(x) · g(a), for all x ∈ P and a ∈ Γ with (x, a) ∈ P ×_{Γ_{0}} Γ.

We will denote it by (f, g) : (P, π, M, Γ) → (P′, π′, M′, Γ′) or (f, g) : P → P′. △ Every homomorphism (f, g) : P → P′ maps each fibre of P into a fibre of P′

and hence induces a mapping of f_{0} : M → M′.

A homomorphism (f, g) : (P, π, M, Γ) → (P′, π′, M′, Γ′) is an isomorphism of principal fibre bundles with structural groupoids, if f : P → P′ is a diffeomorphism and g : Γ → Γ′ is an isomorphism of Lie groupoids.

Applying the Proposition 1.1.(i) and Propositionn 3.1. we have the following proposition.

Proposition 3.2. Let ξ = (P, π, M, Γ) be a principal fibre bundle with a transitive Lie groupoid Γ as structural groupoid. Then the principal fibre bundles ξ_{Γ(u)} = (P, π_{Γ(u)}, P/Γ(u), Γ(u)) , (∀) u ∈ Γ_{0} are isomorphic. △

4 Fibre bundles with type fibre and structural groupoid

Let ξ = (P, π, M, Γ) be a principal fibre bundle and F a manifold on which Γ acts on the left by (a, x) ∈ Γ ×_{Γ_{0}} F → a · x ∈ F.

We shall construct a fibre bundle ξ_{F} = (E, π_{E}, M, F, Γ) associated to ξ with fibre F and groupoid Γ, where E = P ×_{Γ} F.

On the product P × F, we let Γ acts on the right as follows.

An element a ∈ Γ maps (x, y) ∈ P × F into (x · a, a^{−1} · y) ∈ P × F, for all (x, a) ∈ P ×_{Γ_{0}} Γ and (a, y) ∈ Γ ×_{Γ_{0}} F.

The quotient space of P ×F by this groupoid action is denoted by E = P ×_{Γ} F. In E will be introduced a differentiable structure such that E becomes a

differentiable manifold. The mapping P × F → M which maps (x, y) into π(x) induces a mapping π_{E}, called the projection of E onto M. For each z ∈ M, the set π_{E}^{−1}(z) is called the fibre of E over z.

Every point z ∈ M has a neighboorhood U such that π^{1}(U) is diffeomorphic to U × Γ. Identifying π^{−1}(U) with U × Γ , we see that the action of Γ on π^{−1}(U) × F on the right is given by:

(x, a, y) → (x, a · b, b^{−1} · y), for (x, a, y) ∈ U × Γ × F and b ∈ Γ such that (a, b) ∈ Γ_{(2)} and (b, y) ∈ Γ ×_{Γ_{0}} F.

It follows that the diffeomorphism π^{−1}(U) ≅ U × Γ induces a diffeomorphism π_{E}^{−1}(U) ≅ U × F.

Now, we can therefore introduce a differentiable structure in E by requirement that π_{E}^{−1}(U) is an open submanifold of E wich is diffeomorphic with U × F under the diffeomorphism π_{E}^{−1}(U) ≅ U × F.

The projection π_{E} is then a differentiable map of E onto M.

We call (E, π_{E}, M, F, Γ) = ξ_{F} , the fibre bundle over the base M with ( type ) fibre F and ( structural ) groupoid Γ associated to principal fibre bundle ξ = (P, π, M, Γ). Example 4.1. Let ξ = (P, π, M, Γ) be a principal fibre bundle and H a closed subgroupoid of Γ.

(i) In a natural way, Γ acts on the quotient space Γ/H on the left ( see, Example 2.1.(iii) ). Then the fibre bundle over M , with fibre Γ/H and groupoid Γ which is associated to principal fibre bundle ξ = (P, π, M, Γ) is ξ_{Γ/H} = ( E = P ×_{Γ} Γ/H, π_{E}, M, Γ/H, Γ ).

(ii) Since Γ acts on Γ on the left by translations it follows that the fibre bundle over M with fibre Γ and groupoid Γ associated to ξ is identified with ξ .△

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University of West of Timişoara Department of Mathematics 4, Bd. V. Pârvan, 1900, Timişoara, Romania E-mail:ivan@hilbert.math.uvt.ro