DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Proceedings of the Conference

August 24-30, 1986, Brno, Czechoslovakia

CHARACTERISTIC CLASSES OF SOME PRADINES-TYPE GROUPOIDS AND A GENERALIZATION OF THE BOTT VANISHING THEOREM

JAN KUBARSKI

Abstract. This paper contains an application of characteristic classes of some Pradines-type groupoids over foliations, constructed by the author in [5]. Using these characteristic classes, we obtain a generalization of the Bott Vanishing Theorem to a flag {F, F′} of foliations F and F′. The classical Bott Theorem follows from the above generalization if we put F = {V }.

1. Pradines-type groupoids Φ^{F} and their Lie algebroids

There is a well-known definition of a \underline{Lie groupoid} (see [8]) as a transitive groupoid

Φ = (Φ, α, β, V, ·)

in which Φ and V are Hausdorff C^{∞}-manifolds, the mappings α, β : Φ → V (called a \underline{source} and a \underline{target} ) are submersions, and ^{−1} : Φ → Φ, u : V → Φ and · : Φ ∗ Φ → Φ — defined by the formulae: −1 (h) = h^{−1}, u (x) = u_{x} (u_{x} – the unit over x), · (g, h) = g · h (Φ ∗ Φ = {(g, h) ∈ Φ × Φ; αg = βh} is a proper submanifold of Φ × Φ) are of C^{∞}-class.

Any vector bundle F over V determines the Lie groupoid

GL (F) = (GL (F) , α, β, V, ·)

of all linear isomorphisms between fibres of F in which α, β and · are defined by α (h) = x and β (h) = y iff h : F_{|x} ⟶^{≈} F_{|y}, and g · h = g ∘ h if α (g) = β (h).

Date: October 21, 1986.

2000 Mathematics Subject Classification. 57R32, 57R30, 57R20, 55N99, 53C05.

Key words and phrases. the Bott Vanishing Theorem, the Chern-Weil homomorphism, Lie groupoid, Pradines-type groupoid over foliation, Lie algebroid.

Let Φ be any Lie groupoid over a manifold V and F – any foliation of V. Take a subgroupoid

Φ^{F} = (Φ^{F}, α^{F}, β^{F}, V, ·)

consisting of all elements of Φ such that the source and the target lie on the same leaf of F. More precisely, Φ^{F} = (α, β)^{−1} [R] where R ⊂ V ×V is the equivalence relation given by xRy iff y ∈ L_{x} (L_{x} – the leaf of F through x). If F = {V }, then Φ^{F} = Φ. In general Φ^{F} is not a submanifold of Φ. Denote by C the set of all real-valued functions defined on Φ^{F} which can be locally extended to C^{∞}-functions on Φ (i.e. C = C^{∞} (Φ)_{Φ^{F}}, see [9]). C is a differential structure on Φ^{F} and the pair (Φ^{F}, C) (further denoted briefly by Φ^{F}) is a \underline{differential} \underline{space} in the sense of R.Sikorski (see [9]). All operations in the groupoid Φ^{F} are smooth in the category of differential spaces.

Because of the submersivity of α, β : Φ → V, the set α^{−1} (x), x ∈ V, forms a proper C^{∞}-submanifold of Φ denoted by Φ_{x}. Φ_{x} constitutes a principal fibre bundle (for brevity p.f.b.) over V with the projection β_{x} : Φ_{x} → V, h ↦ βh, the isotropy Lie group G_{x} = β_{x}^{−1} (x) as the structural Lie group, and the action Φ_{x} × G_{x} → Φ_{x}, (h, a) ↦ h · a.

For the leaf L_{x} of F through x, on the set

Φ_{x}^{F} := β_{x}^{−1} [L_{x}]

there exists exactly one C^{∞}-manifold structure such that if U is open in L_{x} and L_{x|U} is a proper submanifold of V, then β_{x}^{−1} [U] is open in Φ_{x}^{F} and Φ_{x|β_{x}^{−1}[U]}^{F} is a proper submanifold of Φ_{x}. Of course, Φ_{x}^{F} is an immerse submanifold of Φ_{x} and β_{x}^{F} : Φ_{x}^{F} → L_{x}, h ↦ βh, is a submersion. Besides. Φ_{x}^{F} forms a p.f.b. over L_{x} analogously. For each h ∈ Φ^{F}, the mapping D_{h} : Φ_{βh}^{F} → Φ_{αh}^{F}, g ↦ g · h, is a diffeomorphism.

With the groupoid Φ^{F} we associate a vector bundle

(A (Φ^{F}) , p, V )

where A (Φ^{F}) = ⋃_{x∈V} T_{u_{x}}Φ_{x}^{F} ⊂ T Φ and p (v) = x iff v ∈ T_{u_{x}}Φ_{x}^{F}, x ∈ V. Moreover,

β̃_{∗}^{F} : A (Φ^{F}) → T F, v → β_{∗}v,

is an epimorphism. Therefore, it is not difficult to see that Φ^{F} is a \underline{Pradines-type} \underline{groupoid} \underline{over} \underline{the} \underline{foliation} F (see [4], [5]).

A smooth vector field X on the differential space Φ^{F} (see [9]) is called \underline{right-invariant} if

(a) X_{h} ∈ T_{h}Φ_{αh}^{F}, h ∈ Φ^{F},

(b) (D_{h})_{∗g} X_{g} = X_{gh}, g, h ∈ Φ^{F}, αg = βh.

Each right-invariant vector field X on Φ^{F} determines a C^{∞}-section X_{0} : V → A (Φ^{F}), x ↦ X (u_{x}), of p. Conversely (see [5]),

Proposition 1. For any C^{∞}-section ξ : V → A (Φ^{F}) of p, there exists exactly one right-invariant vector field ξ′ on Φ^{F} such that ξ′ (u_{x}) = ξ (x), x ∈ V. The bracket [[ξ, η]] := [ξ′, η′]_{0} defines in the vector space Sec A (Φ^{F}) of all C^{∞}-sections of p a real Lie algebras structure.

Proposition 2. The system

A (Φ^{F}) = (A (Φ^{F}) , [[·, ·]], β̃_{∗}^{f})

is a \underline{Lie algebroid} (in the sense of J.Pradines [6], [7]).

With the Lie algebroid A (Φ^{F}) we associate a short sequence of vector bundles over V

0 → 𝔤 ↪^{j} A (Φ^{F}) →^{γ} T F → 0

where γ denotes, for brevity, the mapping β̃_{∗}^{f}, and 𝔤 := ker γ. Of course, 𝔤 is independent of the choice of F. Each fibre 𝔤_{|x}, x ∈ V, is a Lie algebra with respect to the bracket [v, w] := [[ξ, η]] (x) where ξ, η ∈ Sec A (Φ^{F}) are such that ξ (x) = v, η (x) = w. Moreover, 𝔤_{|x} is the Lie algebra of the isotropy Lie group G_{x}.

If Φ = GL (F), then 𝔤 is canonically isomorphic to Hom (F; F).

2. Connections in A (Φ^{F})

By a \underline{connection} in A (Φ^{F}) (see [5]) we mean any mapping

λ : T F →A (Φ^{F})

such that γ ∘ λ = id_{T F}.

Proposition 3. Connections in A (Φ^{F}) are in one-to-one corresponding with partial connections in the p.f.b. Φ_{x} projectable onto T F (for definition of a partial connection see [3]).

Proof. A connection λ determines a partial connection H^{λ} in Φ_{x} by the formula H_{h}^{λ} := Im ((D_{h})_{∗u_{βh}} ∘ λ_{|βh}). The correspondence λ ↦ H^{λ} is the sought-for bijection. □

For a connection λ in A (Φ^{F}), the uniquely determined morphism

ω : A (Φ^{F}) → 𝔤

fulfilling ω|𝔤 = id and ω| Im λ = 0 is called a \underline{connection} \underline{form} of λ. Let F be any vector bundle over V. By a \underline{C^{∞}-form} of degree q on T F with values in F we shall mean each C^{∞}-section of the bundle

⋀^{q} (T F)^{∗} ⨂ F.

The set

Ω (T F; F)

of all such forms is a graded module over C^{∞} (V). Moreover, it has a structure of a module over the algebra

Ω (T F; ℝ)

of all real-valued C^{∞}-forms on T F.

By a \underline{curvature} \underline{base-form} (or a \underline{curvature} \underline{tensor}) of a connection λ we shall mean the form

Ω_{B} ∈ Ω^{2} (T F; 𝔤)

defined by the formula Ω_{B} (X, Y) = −ω ([[λ ∘ X, λ ∘ Y]]), X, Y ∈ Sec T F.

Proposition 4. [[λ ∘ X, λ ∘ Y]] = λ ∘ [X, Y] − Ω_{B} (X, Y).

3. The Chern-Weil homomorphism for Φ^{F}

The groupoid Φ^{F} acts on the bundle 𝔤 by the \underline{adjoint} \underline{representation} Ad^{F} defined by

Ad^{F} (h) = (τ_{h})_{∗u_{x}} : 𝔤_{|x} ⟶^{≈} 𝔤_{|y}, h ∈ Φ^{F},

where τ_{h} : G_{x} → G_{y}, a ↦ hah^{−1}, x = αh, y = βh.

Let ⋁^{k} 𝔤^{∗} be the k-symmetric power of 𝔤^{∗}. Denote by (Ad^{F})^{∨} the action of Φ^{F} on ⋁^{k} 𝔤^{∗} induced by Ad^{F}. A section

Γ ∈ Sec ⋁^{k} 𝔤^{∗}

is called \underline{Ad^{F}-invariant} if (Ad^{F})^{∨} (h) (Γ_{αh}) = Γ_{βh} for each h ∈ Φ^{F}. The set of all Ad^{F}-invariant sections of the bundle ⋁^{k} 𝔤^{∗} is denoted by

(Sec ⋁^{k} 𝔤^{∗})_{I}^{F} .

Of course, ⨁^{k} (Sec ⋁^{k} 𝔤^{∗})_{I}^{F} forms an algebra.

If F = {V }, then the letter F in the symbols Ad^{F}, (Ad^{F})^{∨} and

(Sec ⋁^{k} 𝔤^{∗})_{I}^{F} will be omitted.

We have (Sec ⋁^{k} 𝔤^{∗})_{I} ⊂ (Sec ⋁^{k} 𝔤^{∗})_{I}^{F}.

Proposition 5. Each Ad^{F}-invariant section of ⋁^{k} 𝔤^{∗} is equal to ∑_{i} f^{i}· Γ_{i} for some C^{∞}-functions f^{i} constant along the leaves of F and for some Ad-invariant sections Γ_{i}.

Proposition 6. The algebra ⨁^{k} (Sec ⋁^{k} 𝔤^{∗})_{I} of all Ad-invariant sec-

tions is canonically isomorphic to the algebra (⋁ 𝔤_{|x}^{∗})_{I} of all invariant polynomials on 𝔤_{|x} with respect to adjoint representation of G_{x} on 𝔤_{|x}, x ∈ V. This isomorphism is built with the help of the family of isomorphisms Ad (h), h ∈ Φ_{x}.

Let λ : T F → A (Φ^{F}) be any connection in A (Φ^{F}) and Ω_{B} ∈ Ω^{2} (T F; 𝔤) — its curvature base-form. We define the following homomorphism of algebras

γ^{F} : ⨁^{k} (Sec ⋁^{k} 𝔤^{∗})_{I}^{F} ⟶ Ω (T F; ℝ) , Γ ⟼ Γ_{∗} (Ω_{B}, ..., Ω_{B}) ,

where Γ ∈ Sec ⋁^{k} 𝔤^{∗} is treated as a symmetric k-linear homomorphism 𝔤 × ... × 𝔤 → ℝ via the isomorphism ⋁^{k} 𝔤^{∗} ≅ L_{s}^{k} (𝔤; ℝ),

t_{1} ∨ ... ∨ t_{k} ⟼ ((v_{1}, ..., v_{k}) ⟼ \frac{1}{k!} ∑_{σ} t_{σ(1)} (v_{1}) · ... · t_{σ(k)} (v_{k}))

and Γ_{∗} (Ω_{B}, ..., Ω_{B}) ∈ Ω^{2k} (T F; ℝ) is defined by the formula

Γ_{∗} (Ω_{B}, ..., Ω_{B}) (x; v_{1}, ..., v_{2k})
= \frac{1}{2^{k}} ∑_{σ} sgn σ · Γ_{x} (Ω_{B} (x; v_{σ(1)}, v_{σ(2)}) , ..., Ω_{B} (x; v_{σ(2k−1)}, v_{σ(2k)})) .

Now, we define a differential operator

d^{T F} : Ω (T F; ℝ) → Ω (T F; ℝ)

by (for a form Θ of degree q)

d^{T F}Θ (X_{0}, ..., X_{q}) = ∑_{j=0}^{q} (−1)^{j} X_{j} (Θ (X_{0}, ..., X̂_{j}, ..., X_{q}))

+ ∑_{i<j} (−1)^{i+j} Θ ([X_{i}, X_{j}] , ...., X̂_{i}, ..., X̂_{j}, ...) .

Let H (T F; ℝ) be the cohomology algebra of the complex

(Ω (T F; ℝ) , d^{T F}) .

We have the following two theorems which are particular cases of general theorems on the theory of cohomology of Pradines-type groupoids over foliations (see [6]).

Theorem 1. d^{T F} ∘ γ^{F} = 0.

This theorem allows us to define the following homomorphism of algebras

h_{Φ^{F}} : ⨁^{k} (Sec ⋁^{k} 𝔤^{∗})_{I}^{F} ⟶ H (T F; ℝ) , Γ ⟼ [γ^{F} (Γ)] .

Theorem 2. h_{Φ^{F}} is independent of the choice of connection.

Definition 1. We shall call the homomorphism h_{Φ^{F}} the \underline{Chern-Weil} \underline{homomorphism} for Φ^{F}. Its image Im h_{Φ^{F}} is called the \underline{Pontryagin} \underline{algebra} for Φ^{F} and denoted by

Pont (Φ^{F}) .

Remark 1. If F = {V }, then the superposition

(⋁ 𝔤_{|x}^{∗})_{I} ≅ ⨁^{k} (Sec ⋁^{k} 𝔤^{∗})_{I} ⟶ H (T V; ℝ) = H_{dR} (V)

is the classical Chern-Weil homomorphism for the p.f.b. Φ_{x}.

4. A generalization of the Bott Vanishing Theorem

As an application of the characteristic classes described above we give

Theorem 3 (A generalization of the Bott Vanishing Theorem). Let {F, F′} be a flag of foliations F and F′ on V. If

T F = T F′ ⨁ F

then Pont^{k} (GL (F)^{F}) = 0 for k > 2 · rank F.

Proof. A connection λ in GL (F)^{F} defines (see [5]) an operator

(1)  ∇^{λ} : Sec F ⟶ Ω^{1} (T F; F)

by the formula

(2)  ∇_{v}^{λ}σ = (λv) \binom{}{σ̃_{x}} , v ∈ (T F)_{|x} , σ ∈ Sec F

where σ̃_{x} := σ̃|GL (F)_{x} : GL (F)_{x} → F_{|x} and σ̃ : GL (F) → F, h ↦ h^{−1} (σ_{βh}).

∇^{λ} is linear and fulfils the conditions (a) ∇_{fX}^{λ}σ = f · ∇_{X}^{λ}σ, (b) ∇_{X}^{λ}fσ = X (f) σ + f∇_{X}^{λ}σ where X ∈ Sec T F, σ ∈ Sec F, f ∈ C^{∞} (V).

Any linear operator (??) such that (a) and (b) hold is called a \underline{covariant} \underline{derivative} in (F, F) (or, after [3], a \underline{partial} \underline{connection} in F with respect to F).

Lemma 1. The correspondence λ ↦ ∇^{λ} establishes a bijection between connections in A (GL (F)^{F}) and covariant derivatives in (F, F).

Proof. It is sufficient to show that

(i) for v ∈ (T F)_{|x}, x ∈ V, a vector λv ∈ A (GL (F)^{F})_{|x} satisfying

γ (λv) = v is, by (??), uniquely determined, (ii) the mapping λ : T F →A (GL (F)^{F}), v ↦ λv, is of the C^{∞}-

class.

(i) and (ii) easily follow from local calculations. □

By a \underline{curvature} \underline{tensor} of a covariant derivative ∇ in (F, F) we mean a tensor

R ∈ Ω^{2} (T F; Hom (F; F))

defined by the formula R_{X,Y} σ = ∇_{X}∇_{Y} σ − ∇_{Y} ∇_{X}σ − ∇_{[X,Y]}σ for X, Y ∈ Sec T F, σ ∈ Sec F.

Lemma 2. The curvature tensor R of ∇^{λ} is equal to the curvature base-form Ω_{B} of λ.

Proof. By proposition (??), we have

R_{X,Y} σ = λX ((λY) \binom{}{σ̃}^{}˜ ) − λY ((λX) \binom{}{σ̃}^{}˜ ) − (λ [X, Y] \binom{}{σ̃})
= ([[λX, λY]] − λ [X, Y]) \binom{}{σ̃} = −Ω_{B} (X, Y) \binom{}{σ̃}
= Ω_{B} (X, Y) (σ) .
□

Continuing the proof of theorem (??), we construct (analogously to Bott [1]) a covariant derivative in (F, F) whose curvature tensor R has the property R_{X,Y} = 0 for all X, Y ∈ Sec T F′. For the purpose,,take any covariant derivative ¯∇ in (F, F). For X ∈ Sec T F = Sec T F′ ⊕ Sec F, write X = X_{F′} + X_{F} where X_{F′} ∈ Sec T F′ and X_{F} ∈ Sec F.

Then define ∇_{X}σ = π [X_{F′}, σ] + ¯∇_{X_{F}} σ for X ∈ Sec T F, σ ∈ Sec F, where π : T F′ ⊕ F → F is the projection onto the second factor. It is not difficult to see that this formula defines a covariant derivative in (F, F) which fulfils the requirement condition. By lemma (??), there exists a connection λ in A (GL (F)^{F}) such that ∇^{λ} = ∇. By lemma (??), the curvature base-form Ω_{B} of λ has the property Ω_{B} (X, Y) = 0 for all X, Y ∈ Sec T F′.

Using the decomposition (T F)_{|x} = (T F′)_{|x}⊕F_{|x}, we see that γ^{F} (Γ) =

0 for Γ ∈ (Sec ⋁^{k} 𝔤^{∗})_{I}^{F} such that k > rank F. □

Remark 2. Let Φ = GL (F) and let F be as in theorem (??). By remark 33 from [5], we have that the Chern-Weil homomorphisms h_{Φ_{x}}, h_{Φ_{x}^{F}} of p.f.b.’s Φ_{x} and Φ_{x}^{F}, respectively, and h_{Φ^{F}} of the Pradines-type groupoid Φ^{F} are connected by the commuting diagram

(⋁^{k} 𝔤_{|x}^{∗})_{I} ⟶^{h_{Φ}} H_{dR}^{2k} (V)
mono ↓  ↓
id ↓ (⋁^{k} 𝔤^{∗})_{I}^{F} ⟶^{h_{Φ^{F}}} H^{2k} (T F; ℝ)
epi ↓  ↓
(⋁^{k} 𝔤_{|x}^{∗})_{I} ⟶^{h_{Φ^{F}_{x}}} H_{dR}^{2k} (L_{x}) .

For k > rank F, the bottom row is zero by the classical Bott Vanishing Theorem applied to the foliation F′|L_{x} of L_{x}, x ∈ V, which – of course – also follows from the vanishing of the middle row.

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Institute of Mathematics, Technical University of Łódź, al.Politechniki 11, Poland