INVARIANT COHOMOLOGY OF REGULAR LIE ALGEBROIDS

Jan KUBARSKI

Institute of Mathematics, Technical University of Łódź, PL-90-924 Łódź, Al.Politechniki 11, Poland

Abstract

We look at the classical theorem dealing with invariant cohomology from the viewpoint of the theory of Lie algebroids. Its generalization concerning an action of a compact Lie group on a regular Lie algebroid is obtained. Some applications to tangential cohomology and actions on vector bundles are given.

1. Introduction and results.

Let

t : G × M → M  (1.1)

be a left action of a Lie group G on a manifold M and let Ω_{I}(M) ⊂ Ω(M) be the subalgebra of G-invariant differential forms. Ω_{I}(M) is stable under the differentiation of forms. The inclusion

i : Ω_{I}(M) ↪ Ω(M)

induces a homomorphism on cohomology:

i_{#} : H_{I}(M) → H(M).

Well known is the following theorem concerning the invariant cohomology:

1.2. Theorem. [G-H-V] If G is compact then i induces a monomorphism on cohomology. When G is, in addition, connected, then i_{#} is an isomorphism.

We look at this from the viewpoint of the theory of Lie algebroids [K2], [K3], [MA1], [MA2], [P], noticing that (1.1) induces an action T : G × T M → T M of G on the Lie algebroid T M (namely, T (a, v) = (t_{a})_{∗}(v)).

For an arbitrary action T : G × A ⟶ A of a Lie group G on a regular Lie algebroid A, the space of G-invariant A-diferential forms Ω_{A,I}(M) is stable under the differentiation d_{A}, and the inclusion i : Ω_{A,I}(M) ↪ Ω_{A}(M) induces a homomorphism on cohomology: i_{#} : H_{A,I}(M) ⟶ H_{A}(M). The main theorem of our work (Theorem 3.8) says that if T extends to some homomorphism of Lie algebroids T̂ : T G×A ⟶ A, then the sentences analogous to the classical version hold. Some applications to invariant tangential cohomology and to actions of Lie groups on vector bundles are given.

2. Algebroid’s preliminary.

2.1. Sources of Lie algebroids. The notion of a Lie algebroid comes from J.Pradines [P] and was invented for the

study of differential groupoids.

Later, differential geometry has discovered many new objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. So, the sources of Lie algebroids are, for example,

— differential groupoids,

— principal bundles,

— vector bundles,

— actions of Lie groups on manifolds,

— transversally complete foliations,

— nonclosed Lie subgroups,

— Poisson manifolds,

— some complete closed pseudogroups.

2.2. Definition of a regular Lie algebroid over a foliated manifold. Let F be a C^{∞} constant dimensional and involutive distribution on a C^{∞} Haus-

dorff paracompact connected manifold M. The pair (M, F) is then called a foliated manifold. By a regular Lie algebroid over (M, F) [K3], [MA1], [P] we mean a system

(A, [[·, ·]], γ)  (2.2.1)

consisting of a vector bundle A over M and mappings

[[·, ·]] : Sec A × Sec A ⟶ Sec A, γ : A ⟶ T M,

such that (i) (Sec A, [[·, ·]]) is a real Lie algebra, (ii) γ, called by K.Mackenzie [MA1] an anchor, is a homomorphism of vector bundles, and Imγ = F, (iii) Sec γ : Sec A ⟶ X (M), ξ ⟼ γ ∘ ξ, is a homomorphism of Lie algebras, (iv) [[ξ, f · η]] = f · [[ξ, η]] + (γ ∘ ξ)(f) · η, ξ, η ∈ Sec A, f ∈ Ω^{∘}(M).

Let A and A′ be two regular Lie algebroids on a manifold M. A strong homomorphism H : A′ ⟶ A of vector bundles (i.e. the one over id_{M} : M ⟶ M) is said to be a (strong) homomorphism of Lie algebroids if γ ∘ H = γ′ and Sec H : Sec A′ ⟶ Sec A is a homomorphism of Lie algebras.

In the case when F = T M, i.e. when γ : A ⟶ T M is a surjective homomorphism, (2.2.1) is called a transitive Lie algebroid.

Each transitive Lie algebroid (2.2.1) over a manifold M is locally trivial in the following sense: for each point x ∈ M, there exist a neighbourhood U ⊂ M of x and a strong isomorphism φ : A_{|U} ≅ T U × 𝔤 of transitive Lie algebroids where T U × 𝔤 is the so-called trivial Lie algebroid [NVQ] with the bracket [[(X, σ), (Y, η)]] = ([X, Y], L_{X}η − L_{Y} σ − [σ, η]) and the anchor γ = pr_{1} : T U × 𝔤 ⟶ T U.

From the viewpoint of this work we pay our attention to the following two examples:

2.2.2. Example. Any C^{∞} constant dimensional involutive distribution F ⊂ T M is an example of a nontransitive Lie algebroid with the Lie bracket [X, Y] of vector fields as a commutator [[X, Y]] and the inclusion F ↪ T M as an anchor.

2.2.3. Example. Ngo-Van-Que in 1967 [NVQ] investigated the Lie algebroid A(𝔣) of the Lie groupoid GL(𝔣) of all linear isomorphisms between fibres of 𝔣 (in fact, Ngo-Van-Que denoted it in a different way as 𝔤𝔩(𝔣)). He discovered an operator interpretation of global cross-sections of the Lie algebroid A(𝔣). Namely, a global cross-section ξ ∈ Sec A(𝔣) determines a covariant differential operator L_{ξ} : Sec 𝔣 ⟶ Sec 𝔣 (and vice-versa), fulfilling the condition L_{ξ}(f · ν) = f · L_{ξ}ν + (γ ∘ ξ)(f) · ν, f ∈ Ω^{∘}(M), ν ∈ Sec 𝔣, when γ : A(𝔣) ⟶ T M is the anchor. This problem was thoroughly studied by A.Kumpera in 1971 [KU] and by K.Mackenzie in 1987 [MA1] and by me [K3].

The Lie algebroid A(𝔣) can be constructed isomorphically as follows [K3]. The fibre A(𝔣)_{|x} is the space of linear homomorphisms l : Sec 𝔣 ⟶ 𝔣_{|x} such that there exists a vector u ∈ T_{x}M for which

l(f · ν) = f(x) · l(ν) + u(f) · ν_{x}, f ∈ Ω^{∘}(M), ν ∈ Sec 𝔣.

A cross-section ξ ∈ Sec A(𝔣) defines in an evident manner a covariant differential operator in 𝔣. The bracket of cross-sections of A(𝔣) is defined classically for differential operators. Clearly, each element l ∈ A(𝔣)_{|x} is factorized by some linear mapping l̃ from the space of 1-jets at x:

Sec 𝔣 ⟶ J^{1}(𝔣)_{|𝔵}
↘l ↓ _{ l̃}
𝔣_{|x}

The structure of a vector bundle in A(𝔣) can be given [MA1] as a subbundle A(𝔣) ⊂ Hom(J^{1}𝔣, 𝔣). Equivalently [K3] — by local trivializations

ψ̄ : T U × End(V) ⟶ A(𝔣)_{|U}

determined by local ones ψ : U × V ⟶ p^{−1}[U] of 𝔣 (V being the typical fibre of 𝔣) via the formula

ψ̄(v, a) ∈ A(𝔣)_{|x}, v ∈ T_{x}U, a ∈ End(V),
ψ̄(v, a)(ν) = ψ_{|x}(v(ν_{ψ})) + a(ν_{ψ}(x)), ν ∈ Sec 𝔣,

where ν_{ψ} : U ⟶ V, x ⟼ ψ_{|x}(ν_{x}).

2.2.4. Example. The Lie algebroid of the trivial line bundle 𝔣 = M × ℝ is (due to [K3; 5.4.4]) equal to the trivial Lie algebroid A(𝔣) = T M × ℝ. For a vector v ∈ T_{x}M and a real number a ∈ ℝ the element (v, a) ∈ A(𝔣)_{|𝔵} is defined by

(v, a) : Sec 𝔣 = Ω^{∘}(M) ⟶ ℝ, f ↦ v(f) + a · f(x).

2.3. The notion of a nonstrong homomorphism of Lie algebroids. This is a fundamental notion in our work.

2.3.1. Definition. Let (A, [[·, ·]], γ) and (A′, [[·, ·]]′, γ′) be two Lie algebroids on manifolds M and M′, respectively. By a (nonstrong, in general) homomorphism between them

H : (A′, [[·, ·]]′, γ′) ⟶ (A, [[·, ·]], γ)

we mean a homomorphism of vector bundles H : A′ ⟶ A, say, over f : M′ ⟶ M, such that,

(a) γ ∘ H = f_{∗} ∘ γ′,

(b) let ξ, ξ′ ∈ Sec A′ be two arbitrarily taken cross-sections of A′ and let η_{1}, …, η_{n} be cross-sections of A being a basis of Sec A over an open subset U ⊂M. Then (on f^{−1}[U]) H ∘ ξ and H ∘ ξ′ can be written in the forms

H ∘ ξ_{|f^{−1}[U]} = ∑_{i} f^{i} · (η_{i} ∘ f)_{|f^{−1}[U]},

H ∘ ξ′_{|f^{−1}[U]} = ∑_{j} f^{′j} · (η_{j} ∘ f)_{|f^{−1}[U]}

for some f^{i}, f^{′i} ∈ Ω^{∘}(M′). We postulate that

H ∘[[ξ, ξ′]]_{|f^{−1}[U]} = ( ∑_{i,j} f^{i}·f^{′j}[[η_{i}, η_{j}]]∘f +∑_{j}[(γ′ ∘ ξ)(f^{′j}) − (γ′ ∘ ξ′)(f^{j})] · η_{j} ∘ f)_{|f^{−1}[U]}.

In the case of Lie algebroids A and A′ on the same manifold M, the notion of a homomorphism H : A′ ⟶ A over id_{M} : M ⟶ M (i.e. the so-called strong homomorphism) coincide with the one described in 2.2.

2.3.2. Example. [MA2] The tangent mapping f_{∗} : F′ ⟶ F to a C^{∞} morphism f : (M′, F′) ⟶ (M, F) of the category of foliated manifolds is an example of a (nonstrong, in general) homomorphism of regular Lie algebroids.

2.3.3. Example. Let t : 𝔣′ ⟶ 𝔣 be a linear homomorphism of vector bundles over f : M′ ⟶ M. Then t induces a (nonstrong, in general) homomorphism of Lie algebroids

A(t) : A(𝔣′) ⟶ A(𝔣)

by the formula

A(t)_{|x} : A(𝔣′)_{|x} ⟶ A(𝔣)_{|f(x)},
A(t)_{|x}(v) : Sec 𝔣 ⟶ 𝔣_{|f(x)}, σ ⟼ v(σ^{#}),

where σ^{#} ∈ Sec 𝔣′ is given by σ^{#}(y) = (f_{|y})^{−1}(σ(fy)), y ∈ M′.

2.4. Inverse-image of a regular Lie algebroid. Each homomorphism of regular Lie algebroids H : A′ ⟶ A over f : (M′, F′) ⟶

(M, F) can be represented as a superposition

H : A′⟶^{H̄}f^{∧}A⟶^{χ}A

of some strong homomorphism H̄ with the canonical nonstrong one χ : f^{∧}A ⟶ A where f^{∧}A is the so-called inverse-image of A over f defined in the following way:

It is a regular Lie algebroid over (M′, F′)

(f^{∧}A, [[·, ·]], pr_{1})

in which

(1) f^{∧}A = F′ ×_{(f_{∗},γ)} A = {(v, w) ∈ F′ × A; f_{∗}(v) = γ(w)} ⊂ F′ ⊕ f^{∗}A,

(2) the bracket [[·, ·]] in Sec f^{∧}A is defined as follows: let (X_{i}, ξ̄_{i}) ∈ Sec f^{∧}A where X_{i} ∈ Sec F′, ξ̄_{i} ∈ Sec f^{∗}A. Then, locally (say on U ⊂ M′), ξ̄_{i} is of the form ∑_{j} g_{i}^{j} · ξ_{i}^{j} ∘ f for some g_{i}^{j} ∈ Ω^{∘}(M′) and ξ_{i}^{j} ∈ Sec A and we put

[[(X_{1}, ξ̄_{1}), (X_{2}, ξ̄_{2})]]_{|U}

= ([X_{1}, X_{2}], ∑_{j,k} g_{1}^{j} · g_{2}^{k} · [[ξ_{1}^{j}, ξ_{2}^{k}]] ∘ f + ∑_{k} X_{1}(g_{2}^{k}) · ξ_{2}^{k} ∘ f − ∑_{j} X_{2}(g_{1}^{j}) · ξ_{1}^{j} ∘ f)_{|U}.

Clearly, χ = pr_{2} : f^{∧}A ⟶ A is a homomorphism of regular Lie algebroids over f.

The strong homomorphism H̄ is defined by:

H̄ : A′ ⟶ f^{∧}A, v ⟼ (γ′(v), H(v)).

2.5. Product of regular Lie algebroids. By the product of two regular Lie algebroids (A, [[·, ·], γ) and (A′, [[·, ·]]′, γ′) over

foliated manifolds (M, F) and (M′, F′), respectively, we mean the Lie algebroid

(A × A′, [[·, ·]], γ × γ′)

over the foliated manifold (M × M′, F × F′) provided that the bracket [[ξ̄, η̄]] in the space Sec(A × A′) is defined in the following way (the decomposition of any vector v ∈ (A × A′)_{|(x,y)} = A_{|x} × A′_{|y} is denoted by (v^{1}, v^{2})):

[[ξ̄, η̄]]_{(x,y)}^{1} = [[ξ̄^{1}(·, y), η̄^{1}(·, y)]](x) + γ′(ξ̄_{(x,y)}^{2})(η̄^{1}(x, ·)) − γ′(η̄_{(x,y)}^{2})(ξ̄^{1}(x, ·)),

[[ξ̄, η̄]]_{(x,y)}^{2} = [[ξ̄^{2}(x, ·), η̄^{2}(x, ·)]](y) + γ(ξ̄_{(x,y)}^{1})(η̄^{2}(·, y)) − γ(η̄_{(x,y)}^{1})(ξ̄^{2}(·, y)).

2.5.1. Example. The trivial Lie algebroid T M × 𝔤 described in 2.2 is the product of the Lie algebroid T M over M with the Lie algebra 𝔤 treated as a Lie algebroid over a point (M × {point} = M).

2.5.2. Example. For any manifolds M and N, the canonical isomorphism of vector bundles T (M × N) ⟶^{≅} T M × T N is an isomorphism of Lie algebroids.

2.5.3. Example. For any regular Lie algebroids A^{1} and A^{2}, the projection pr^{i} : A^{1} × A^{2} ⟶ A^{i} is a homomorphism of Lie algebroids.

2.5.4. Theorem. If H : A ⟶ B and H′ : A′ ⟶ B′ are homomorphisms of regular Lie algebroids over f : (M, F) ⟶ (N, G) and f′ : (M′, F′) ⟶ (N′, G′), respectively, then

H × H′ : A × A′ ⟶ B × B′, (v, w) ⟼ (H(v), H′(w)),

is a homomorphism of Lie algebroids over f ×f′ : (M ×M′, F ×F′) ⟶ (N ×N′, G× G′), respectively.

Proof. According to 2.4, it is sufficient to check this for strong homomorphisms and for canonical ones, which is an easy but uphill task.

2.6. A-differential forms.

2.6.1. Definition. Let A be a regular Lie algebroid. By an A-differential form of degree p we mean a cross-section Φ ∈ Sec ⋀^{p} A^{∗}. In the space

Ω_{A}(M) := Sec ⋀ A^{∗} (:= ⨁^{p≥0} Sec ⋀^{p} A^{∗})

the differential d_{A} works giving a cohomology algebra H_{A}(M), [K1], [K4], [MA1], [MR]. We recall that

d_{A} : Ω_{A}^{∗}(M) ⟶ Ω_{A}^{∗+1}(M)

is defined by

(d_{A}Φ)(ξ_{0}, …, ξ_{p}) = ∑_{i=0}^{p}(−1)^{i}(γ ∘ ξ_{i})(Φ(ξ_{0}, …, ξ̂_{i}, …, ξ_{p})) +
+ ∑_{i<k}(−1)^{i+k}Φ([[ξ_{i}, ξ_{k}]], ξ_{0}, …, ξ̂_{i}, …, ξ̂_{k}, …, ξ_{p}),

Φ ∈ Ω_{A}^{p}(M), ξ_{i} ∈ Sec A.

If A = T M then d_{A} is the usual differentiation of differential forms.

2.6.2. Definition. Let H : A′ ⟶ A be a homomorphism of regular Lie algebroids over f : M′ ⟶ M. By a pullback H^{∗}Φ of an A-differential form Φ ∈ Ω_{A}^{p}(M) we mean an A′-differential form of degree p defined by

H^{∗}(Φ)(x; v_{1} ∧ … ∧ v_{p}) = Φ(f(x); Hv_{1} ∧ … ∧ Hv_{p}).

2.6.3. Theorem. The differential of forms commutes with the pullback of forms:

H^{∗}(d_{A}Φ) = d_{A′}(H^{∗}Φ).

Proof. According to 2.4 it is sufficient to check this for strong homomorphisms and for canonical ones.

2.7. Homotopy. Let A and A′ be regular Lie algebroids on the manifolds M and M′, respectively,

and let ϕ, ϕ′ : A′ → A be homomorphisms of Lie algebroids. By a homotopy joining ϕ to ϕ′ we mean a homomorphism of Lie algebroids

H : T ℝ × A′ ⟶ A

such that H(θ_{0}, ·) = ϕ and H(θ_{1}, ·) = ϕ′ where θ_{0} and θ_{1} are null vectors tangent to ℝ at 0 and 1, respectively.

2.7.1. Theorem. If H : T ℝ × A′ ⟶ A is a homotopy joining ϕ to ϕ′, then the operator of degree -1

h : Ω_{A}^{∗}(M) ⟶ Ω_{A′}^{∗−1}(M′)

defined by the formula

h(Φ)_{x}(v_{1} ∧ … ∧ v_{p−1}) = ∫_{0}^{1}(H^{∗}Φ)_{(t,x)}(\frac{∂}{∂t}_{|(t,x)} ∧ v_{1} ∧ … ∧ v_{p−1})dt,

Φ ∈ Ω_{A}^{p}(M) , x ∈ M′ , v_{i} ∈ A′_{|x} , p ≥ 1 , is a chain homotopy operator joining ϕ^{∗} to ϕ^{′∗} , i.e. the following equality holds

h^{p+1} ∘ d_{A}^{p} + d_{A′}^{p−1} ∘ h^{p} = ϕ^{∗p} − ϕ^{′∗p}, p ≥ 0.

Proof. Easy standard calculations.

3. Action of a Lie group on a Lie algebroid and the main theorem.

3.1. Definition. By a left action of a Lie group G on a regular Lie algebroid A over a foliated manifold (M, F) we mean an action

T : G × A ⟶ A

of G on a vector bundle A, such that, for g ∈ G,

T_{g} : A ⟶ A

is a (nonstrong, in general) homomorphism of Lie algebroids. Let the homomorphism T_{g} be over t_{g} : M ⟶ M. By definition, the following diagram commutes:

A ⟶^{T_{g}} A
↓ γ  ↓ γ
F ⟶^{(t_{g})_{∗}} F ,

therefore t_{g} is a homomorphism of foliated manifolds t_{g} : (M, F) ⟶ (M, F), and, of course, t : G × M ⟶ M, (g, x) ↦ (t_{g})(x), is an action of G on M.

3.2. Example. An action t : G × M ⟶ M of a Lie group G on a manifold M induces an action of G on the Lie algebroid T M

T : G × T M ⟶ T M, (g, v) ⟼ (t_{g})_{∗}(v).

3.3. Definition. Let T : G × A ⟶ A be an arbitrary action of G on a Lie algebroid A. An A-differential form Φ ∈ Ω_{A}(M) is called G-invariant when

T_{g}^{∗}Φ = Φ, for all g ∈ G.

By theorem 2.6.3, d_{A}(T_{g}^{∗}Φ) = T_{g}^{∗}(d_{A}Φ), therefore d_{A} maps the space of Ginvariant A-forms into itself. As a consequence we have that the space of G-invariant A-differential forms

Ω_{A,I}(M) = {Φ ∈ Ω_{A}(M); ∀_{g∈G} (T_{g}^{∗}Φ = Φ)}

is stable under d_{A} giving a complex (Ω_{A,I}(M), d_{A}) and, next, the graded cohomology algebra H_{A,I}(M). Clearly, the inclusion

i : Ω_{A,I}(M) ↪ Ω_{A}(M)  (3.4)

induces a homomorphism on cohomology

i_{#} : H_{A,I}(M) → H_{A}(M).  (3.5)

3.6. QUESTION. Does the compactness of G imply the monomorphy of i_{#} and the adding of a connectedness of G — the isomorphy ?

3.7. Remark. In the proof of classical Theorem 1.1, the pullback of forms via t : G × M ⟶ M is really used, i.e. de facto, the possibility of extending action T : G × T M ⟶ T M (see example 3.2) to a homomorphism of Lie algebroids

T̂ : T G × T M ⟶ T M

(of course T̂ = t_{∗} and T G × T M is the product of Lie algebroids, T G × T M ≅ T (G × M)).

In this work the following generalization of theorem 1.1 is proved:

3.8. MAIN THEOREM. Let T : G × A ⟶ A be an action of a Lie group G on a regular Lie algebroid A, extending to a homomorphism of Lie algebroids

T̂ : T G × A ⟶ A,

that is, to a homomorphism T̂ such that T̂|G×A = T. If G is compact, then inclusion (3.4) induces monomorphism (3.5) on cohomology. If, additionally, G is connected, then (3.5) is an isomorphism.

4. The proof of the main theorem.

Part 1. The compactness of G ⇒ i_{#} is a monomorphism.

Let dim G = n. For Φ ∈ Ω_{T G×A}^{n+p}(G × M), p ≥ 0, and x ∈ M, we define the form

Φ_{x} ∈ Ω^{n}(G; ⋀^{p} A_{|x}^{∗})

by the formula

Φ_{x}(a; η_{1a} ∧ … ∧ η_{na})(ξ_{1x} ∧ … ∧ ξ_{px}) = Φ_{(a,x)}(η_{1a} ∧ … ∧ η_{na} ∧ ξ_{1x} ∧ … ∧ ξ_{px}),

η_{ia} ∈ T_{g}G, ξ_{ix} ∈ A_{|x}.

The crucial role is played by the following operator of integration aver the fibre:

∫̸_{G} : Ω_{T G×A}(G × M) ⟶ Ω_{A}(M)

(∫̸_{G}Φ)_{x} = \{ \begin{matrix}∫_{G} Φ_{x} & degΦ ≥ n \\ 0 & degΦ < n, \end{matrix}

x ∈ M.

4.1. Lemma.

d_{A}(∫̸_{G} Φ) = (−1)^{n}∫̸_{G} d_{T G×A}Φ.

Proof will be given at Section 5.

Let Δ ∈ Ω^{n}(G) be the left-invariant n-form on G for which ∫_{G} Δ = 1. Take the operator of degree 0

I_{Δ} : Ω_{T G×A}(G × M) ⟶ Ω_{A}(M),

Φ ⟼ ∫̸_{G}(pr_{T G}^{∗}Δ) ∧ Φ,

where pr_{T G} : T G × A ⟶ A is the projection onto the first factor. I_{Δ} commutes with the differentiations

I_{Δ} ∘ d_{T G×A} = d_{A} ∘ I_{Δ}.  (4.2)

Indeed, according to Lemma 4.1 above,

I_{Δ} ∘ d_{T G×A}(Φ) = ∫̸_{G} pr_{T G}^{∗}Δ ∧ d_{T G×A}(Φ)
= ∫̸_{G}(−1)^{n}d_{T G×A}(pr_{T G}^{∗}Δ ∧ Φ)
= d_{A}∫̸_{G}(pr_{T G}^{∗}Δ) ∧ Φ
= d_{A} ∘ I_{Δ}(Φ).

I_{Δ} has also the following properties:

I_{Δ} ∘ T̂^{∗}(Φ) ∈ Ω_{A,I}(M) for Φ ∈ Ω_{A}(M),  (4.3)
I_{Δ} ∘ T̂(Φ) = Φ for Φ ∈ Ω_{A,I}(M).  (4.4)

Indeed, since

(I_{Δ} ∘ T̂^{∗}(Φ))_{x} = ∫_{G}(T_{a}^{∗}Φ)_{x}da, x ∈ M,

therefore we obtain (4.3)

(T_{b}^{∗}(I_{Δ} ∘ T̂^{∗}(Φ)))_{x} = ∫_{G}(T_{ab}^{∗}Φ)_{x}da = (I_{Δ} ∘ T̂^{∗}(Φ))_{x}, b ∈ G.

Equality (4.4) follows trivially from (4.3).

Finally, we define the operator

ρ : Ω_{A}(M) ⟶ Ω_{A,I}(M), Φ ⟼ I_{Δ} ∘ T̂^{∗}(Φ).

The operator ρ commutes with differentiations:

ρ ∘ d_{A}(Φ) = I_{Δ} ∘ T̂^{∗}(d_{A}(Φ))
= _{Δ} ∘ d_{T G×A}T̂^{∗}(Φ)
= d_{A} ∘ I_{Δ} ∘ T̂^{∗}(Φ)
= d_{A} ∘ ρ(Φ).

By the above, ρ induces a homomorphism on cohomology

ρ_{#} : H_{A}(M) ⟶ H_{A,I}(M).

According to (4.3) and (4.4), ρ fulfils the equality ρ ∘ i =id for the inclusion i : Ω_{A,I}(M) ⟶ Ω_{A}(M), therefore gives on cohomology ρ_{#} ∘ i_{#} =id, which implies the monomorphy of i_{#}.

Part II. The compactness and the connectedness of G ⇒ i_{#} is an epimorphism. The mapping

j_{e} : A ⟶ T G × A, v ⟼ (θ_{e}, v),

θ_{e} being the null vector at the unit e ∈ G, is a homomorphism of Lie algebroids.

4.5. Lemma. There exists a mapping of degree -1

h_{M} : Ω_{T G×A}^{∗}(G × M) ⟶ Ω_{A}^{∗−1}(M)

such that

I_{Δ} − (j_{e})^{∗} = d_{A} ∘ h_{M} + h_{M} ∘ d_{T G×A}.

Proof will be given at Section 6.

From this lemma we easily obtain

i ∘ ρ − id_{Ω_{A}(M)} = I_{Δ} ∘ T̂^{∗} − (j_{e})^{∗} ∘ T̂^{∗}
= (d_{A} ∘ h_{M} + h_{M} ∘ d_{T G×A}) ∘ T̂^{∗}
= d_{A} ∘ h_{M} ∘ T̂^{∗} + h_{M} ∘ T̂^{∗} ∘ d_{A}.

Let Φ ∈ Ω_{A}(M) be closed, d_{A}(Φ) = 0. According to the above,

(i ∘ ρ − id_{Ω_{A}(M)})(Φ) = d_{A}(h_{M} ∘ T̂^{∗}(Φ))

gives (on cohomology) the equality

i_{#} ∘ ρ_{#} − id_{#} = 0

from which the epimorphy of i_{#} is obvious.

5. The proof of lemma 4.1.

Clearly, it is sufficient to consider a form Φ = f · pr_{T G}^{∗}φ ∧ pr_{A}^{∗}θ, where f ∈ Ω^{∘}(G × M), φ ∈ Ω_{T G}(G) = Ω(G), θ ∈ Ω_{A}^{p}(M), p ≥ 0, in three cases: (a) deg φ ≤ n − 2, (b) deg φ = n − 1, (c) deg φ = n.

(a): Both sides of the examined equality are zero.

(b): The left-hand side is zero: ∫̸_{G} Φ = 0. To see that the right-hand side is also equal to zero, we need to observe that

∫̸_{G} Ψ ∧ pr_{A}^{∗}θ = (∫̸_{G} Ψ) ∧ θ, Ψ ∈ Ω_{T G×A}(G × M).

Then, according to this and that G is compact, we have

(∫̸_{G} d_{T G×A}(Φ))_{x} = ( ∫_{G} d_{T G}(f(·, x) · φ)) ∧ θ_{x} = 0, x ∈ M.

(c): After some calculations we obtain, for x ∈ M and a ∈ G,

(d_{T G×A}(f · pr_{T G}^{∗}φ ∧ pr_{A}^{∗}θ))_{(a,x)} = (−1)^{n}φ_{a} ⊗ d_{A}(f(a, ·) · θ)_{x}

as well as

(∫̸_{G} d_{T G×A}(f · pr_{T G}^{∗}φ ∧ pr_{A}^{∗}θ))_{x}
= (−1)^{n} ∫_{G} φ_{a} ⊗ d_{A}(f(a, ·) · θ)_{x}da
= (−1)^{n}(d_{A}(∫̸_{G} f · pr_{T G}^{∗}φ ∧ pr_{A}^{∗}θ))_{x}.

6. The proof of lemma 4.5.

Let U ∋ e be an arbitrary open neighbourhood of e in G, diffeomorphic to ℝ^{n}. Take a compact carrier n-form ψ̃ ∈ Ω_{c}(U) such that ∫_{U} ψ̃ = 1 and extend it (putting 0 outside U) to a smooth n-form ψ ∈ Ω_{c}(G); suppψ ⊂ U, and ∫_{G} ψ = 1. Clearly, ∫_{G} Δ − ψ = 0, so Δ − ψ = d_{T G}φ for some φ ∈ Ω^{n−1}(G). We define

k_{M} : Ω_{T G×A}^{∗}(G × M) ⟶ Ω_{A}^{∗−1}(M), Φ ⟼ (−1)^{n}∫̸_{G}(pr_{T G}^{∗}φ) ∧ Φ.

and

I_{ψ} : Ω_{T G×A}^{∗}(G × M) ⟶ Ω_{A}^{∗}(M), Φ ⟼ ∫̸_{G}(pr_{T G}^{∗}ψ) ∧ Φ.

Using Lemma 4.1, we assert that

(I_{Δ} − I_{ψ})(Φ) = ∫̸_{G} d_{T G×A}(pr_{T G}^{∗}φ) ∧ Φ
= ∫̸_{G} d_{T G×A}(pr_{T G}^{∗}φ ∧ Φ) + (−1)^{n}pr_{T G}^{∗}φ ∧ d_{T G×A}Φ
= (−1)^{n}d_{A}(∫̸_{G} pr_{T G}^{∗}φ ∧ Φ) + (−1)^{n}∫̸_{G} pr_{T G}^{∗}φ ∧ d_{T G×A}Φ
= d_{A} ∘ k_{M}(Φ) + k_{M} ∘ d_{T G×A}(Φ),

which means that

I_{Δ} − I_{ψ} = d_{A} ∘ k_{M} + k_{M} ∘ d_{T G×A}.  (6.1)

Take the inclusion U ×M ⟶ G×M and the homomorphism of Lie algebroids

λ : T U × A ⟶ T G × A

induced by it. For the mapping

Iψ̃ : Ω_{T U×A}(U × M) ⟶ Ω_{A}(M), Φ ⟼ ∫̸_{U} pr_{T U}^{∗}(ψ̃) ∧ Φ,

we have

Iψ̃ ∘ λ^{∗} = I_{ψ}.  (6.2)

Since U is diffeomorphic to ℝ^{n}, there exists a homotopy

H : ℝ × U ⟶ U

joining the constant mapping ê = H(0, ·) : U ⟶ U, y ⟼ e, to id_{U} = H(1, ·) : U ⟶ U. The homotopy H induces a homomorphism of Lie algebroids

H̃ = H_{∗} × id_{A} : T ℝ × (T U × A) ⟶ T U × A.

This last homomorphism joins H̃_{0} = H̃(θ_{0}, ·) = j̃_{e} ∘ p̃r_{A} : T U × A ⟶ T U × A, (v, w) ⟼ (θ_{e}, w), to H̃_{1} = H̃(θ_{1}, ·) =id_{T U×A}.

According to Theorem 2.7.1, there exists a chain homotopy h̃ : Ω_{T U×A}(U × M) ⟶ Ω_{T U×A}(U × M) joining H̃_{0} to H̃_{1}, i.e.

h̃ ∘ d_{T U×A} + d_{T U×A} ∘ h̃ = H̃_{1} − H̃_{0}.  (6.3)

After the composition of (6.3) on the left with Iψ̃ and on the right with λ^{∗} we obtain (due to (6.2))

I_{ψ} − (j_{e})^{∗} = d_{A} ∘ (Iψ̃ ∘ h̃ ∘ λ^{∗}) + (Iψ̃ ∘ h̃ ∘ λ^{∗}) ∘ d_{T G×A}.

Adding (6.1) to the above equality, we get

I_{Δ} − (j_{e})^{∗} = d_{A} ∘ h_{M} + h_{M} ∘ d_{T G×A}

for h_{M} = k_{M} + Iψ̃ ∘ h̃ ∘ λ^{∗}.

7. Examples.

7.1. Invariant tangential cohomology. Let (M, F) be a foliated manifold (without singularities) and let t : G×M ⟶ M

be an action of G on (M, F), i.e. an action of G on M such that (t_{a})_{⋆}[F] = F for all a ∈G. The mapping T : G × F ⟶ F, (a, v) ↦ (t_{a})_{⋆}(v), is an action of G on the trivial regular Lie algebroid F. Denote by Ω_{F} (M) and Ω_{F,I}(M) the spaces of tangential differential forms and of tangential invariant differential forms, respectively.

7.1.1. Theorem. If G is compact and all orbits of the action t are integral manifolds of F, then the inclusion i : Ω_{F,I}(M) ↪ Ω_{F} (M) induces a monomorphism i_{#} : H_{F,I}(M) ⟶ H_{F} (M) on cohomology. If, additionally, G is connected, then i_{#} is an isomorphism.

Proof. Indeed, by the assumptions, the homomorphism of Lie algebroids t_{⋆} maps T G × F into F, giving a homomorphism T̂ : T G × F ⟶ F – an extension of T : G × F ⟶ F .

7.2. Action on a vector bundle. Take an arbitrary action

t : G × 𝔣 ⟶ 𝔣  (7.2.1)

of G on a vector bundle 𝔣 . The linear homomorphism t_{g} : 𝔣 → 𝔣 defines a homomorphism of Lie algebroids A(t_{g}) : A(𝔣) ⟶ A(𝔣), see Example 2.3.3. The mapping

T : G × A(𝔣) → A(𝔣), (g, v) ↦ A(t_{g})(v),  (7.2.2)

is an action of G on the Lie algebroid A(𝔣). Action (7.2.2) extends to some homomorphism of Lie algebroids T̂ : T G × A(𝔣) ⟶ A(𝔣). Indeed, let pr_{2} : G × M ⟶ M denote the projection onto the second factor. We have, for the canonical homomorphism t̄: pr_{2}^{∗}𝔣 ⟶ 𝔣 induced by t,

T̂ : T G × A(𝔣) ≅ A(pr_{2}^{∗}𝔣) ⟶^{A(t̄)} A(𝔣).

Therefore we have the following

7.2.3. Theorem. If (7.2.1) is an action of a compact Lie group G on a vector bundle 𝔣, then ihe inclusion i : Ω_{A(𝔣),I}(M) ↪ Ω_{A(𝔣)}(M) induces a monomorphism on cohomology i_{#} : H_{A(𝔣),I}(M) → H_{A(𝔣)}(M) being an isomorphism when G is, in addition, connected.

References

[G-H-V] W. Greub, S. Halperin, R. Vanstone, Connections, Curvature, and Cohomology, Vol.II, Academic Press, New York and London, 1973.

[K1] J. Kubarski, Pradines-type groupoids over foliations; cohomology, connections and the Chern-Weil homomomorphism, Preprint Nr 2, Institute of Mathematics, Technical University of Lodz, August 1986.

[K2] ——, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. Universite de Lyon 1, 1/A, 1989.

[K3] ——, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. Universite de Lyon 1, in press.

[K4] ——, Characteristic classes of flat and of partially flat regular Lie algebroids over foliated manifolds, Publ. Dep. Math. University de Lyon 1, in press.

[KU] A. Kumpera, An Introduction to Lie groupoids. Nucleo de Estudos e Pasquisas Cientificas, Rio de Janeiro, 1971.

[MA1] K. Mackenzie, Lie groupoids and Lie algebroids in differential Geometry, London Mathematical Society, Lecture Note Series 124, Cambridge, 1987.

[MA2] ——, Algebraic constructions in the category of Lie algebroids, J.Algebra.

[MR] L. Maxim-Raileanu, Cohomology of Lie algebroids, An. Sti. Univ.”Al. I.Cuza” Iasi. Sect. I a Mat. XXII f 2 (1976), 289- 307.

[NVQ] Ngo-Van-Que, Du prolongement des espaces fibres et des structures infinitesimales, Ann. Inst. Fourier, (Grenoble), 17,1 (1967), 157-223.

[P] J. Pradines, Théorie de Lie pour les groupoïdes différentiables, Atti Conv Intern Geom 7 Diff. Bologne, 1967, Bologna- Amsterdam.