Gysin Sequence of some Lie algebroids.

Lecture, Colloque Trajectorien, Strasbourg-Obernai, France 12-16 June 1995

JAN KUBARSKI

1. Preliminaries. For any oriented bundle (E, π, M, F), there is defined an operator of integration along the fibre

∫̸_{E} : Ω_{c}^{∗}(E) ⟶ Ω^{∗−n}(M), n = dim M,

acting from the complex Ω_{c}^{∗}(E) of forms with fibre-compact support, by the formula

(∫̸_{E}Φ)_{x} = \{ \begin{matrix}0 & degΦ < n \\ ∫̸_{E_{|x}}Φ_{x} & degΦ ≥ n + k, k ≥ 0, \end{matrix}

where, for k ≥ 0, Φ_{|x} ∈ Ω^{n}(E_{|x}; ⋀^{k} T_{x}^{∗}M) is a form such that Φ_{x}(h_{1} ∧…∧h_{n})(v_{1} ∧…∧v_{k}) = (−1)^{nk}Φ(x; h_{1} ∧…∧h_{n} ∧\overline{ν}_{1} ∧…∧\overline{ν}_{k}), π_{∗}(\overline{ν}_{i}) = v_{i}.

The important properties of ∫̸_{E}Φ are as follows:

1) ∫̸_{E}π^{∗}Ψ ∧ Φ = Ψ ∧ ∫̸_{E}Φ,

2) ∫̸_{E} is an epimorphism,

3) d_{M} ∘ ∫̸_{E} = ∫̸_{E} ∘ d_{E},

4) Ker∫̸_{E}Φ is d_{E}-stable, and many others.

The operator ∫̸_{E} takes part in many subjects of algebraic topology, for example, in the Gysin sequence of oriented sphere bundle and then in the construction of the Euler class of such bundles.

I am going to initiate the study of some Lie algebroids, using an operator of integration along the fibre of isotropy Lie algebra bundles, defined in a natural manner.

2. Introduction. Each principal bundle P (M, G) is oriented, any leftinvariant n-form Δ_{L} ∈ Ω^{L}(G), n = dim G, on a structure Lie group G defines an orientation of P. Let ∫̸_{G}Δ_{L} = 1 and take the tensor ε_{o} ∈ ⋀^{n} 𝔤, 𝔤 - the Lie algebra of G, such that

< Δ_{L}(e), ε_{o} >= 1.

We represent ε_{o} in the form ε = h_{1} ∧…∧h_{n} (h_{i} – any basis of 𝔤). Denote, as usual, by Z_{i} the fundamental vector fields on P generated by h_{i} and put

ι_{ε_{o}} = ι_{Z_{n}} ∘ … ∘ ι_{Z_{1}} : Ω^{∗}(P) ⟶ Ω^{∗−n}(P).

Remark 2.1. If

(*)  det(Ada) = +1 for each a ∈ G,

then ι_{ε_{o}} ∘ T_{a}^{∗} = T_{a}^{∗} ∘ ι_{ε_{o}} (for example, if G is unimodular and connected). Therefore, if Φ is a right-invariant form, Φ ∈ Ω^{I}(P), then ι_{ε_{o}}Φ is a base form,

ι_{ε_{o}}Φ = π^{∗}(Φ̃)

for Φ̃ ∈ Ω(M).

The following theorem is known :

Theorem 2.2. If G is compact and fulfils condition (∗) (for example, if G is compact and connected), then

Ω^{I}(P) ⟶ Ω(P)
ω∘ι_{εo} ↓  ↓^{∫̸}
Ω_{b}(P) ⟵^{π^{∗}} Ω(M)

where ω(Φ) = (−1)^{nk}Φ, Φ ∈ Ω^{n+k}(M).

Let A(P) = T P_{/G} be the Lie algebroid of a principal bundle P and let

0 ⟶ ggg ⟶ A(P) ⟶^{γ} T M ⟶ 0

be its Atiyah sequence with the bundle of isotropy Lie algebras ggg. Consider the cross-section ε ∈ Sec ⋀^{n} ggg defined by

ε_{x} = (⋀^{n} ẑ)(ε_{o}), z ∈ P_{|x},

where ẑ: 𝔤 ⟶ ggg_{|x}, h ⟼ [A_{z∗}(h)] (ẑ is an isomorphism of Lie algebras provided that in 𝔤 there is a right Lie algebra of G).

Remark 2.3. The correctness of this definition of ε (i.e. the independence of the choice of z ∈ P_{|x}) follows from the above condition ( ∗).

Theorem 2.4. The cross-section ε is invariant with respect to the adjoint representation of the Lie algebroid A(P) on the bundle ⋀^{n} ggg.

Proof. Indeed, take the adjoint representation of the principal bundle P on the vector bundle ggg

Ad_{P} : P ⟶ L(ggg), z ⟼ ẑ,

(Lggg is a GL(𝔤)-principal bundle of repers and Ad_{P} – Ad_{G}-representation). Ad_{P} induces the representation

Ad_{P}^{∧} : P ⟶ L(⋀^{n}ggg), z ⟼ ⋀^{n} ẑ, ⋀^{n} ẑ: ⋀^{n} 𝔤 ⟶^{≅} ⋀^{n}ggg_{|x}, x = πz.

ε is Ad_{P}^{∧}-invariant. Indeed, for ε_{o} ∈ ⋀^{n} 𝔤, we have - by definition -

Ad_{P}^{∧}(z)(ε_{o}) = (⋀^{n} ẑ)(ε_{o}) = ε_{x}.

Since (Ad_{P}^{∧})′ = (ad_{A(P)}^{∧})′ : A ⟶ A(⋀^{n} ggg), we obtain that ε in ad_{A(P)}^{∧}-invariant.

Remark 2.5. We recall that any vector bundle 𝔣 determines the Lie algebroid A(𝔣) whose global cross-sections are the so-called covariant differential operators. Any homomorphism T : A ⟶ A(𝔣) is called a representation of A on 𝔣. A crosssection ν ∈ Sec 𝔣 is called T -invariant if and only if T (ξ)(ν) = 0 for all ξ ∈ Sec A.

By the adjoint representation of a Lie algebroid A we mean the representation ad_{A} : A ⟶ A(ggg) defined in such a way: the CDO ad_{A} ∘ ξ, ξ ∈ Sec A, is equal to ad_{A} ∘ ξ = [[ξ, ·]] : Secggg ⟶ Secggg. ad_{A} induces the representation ad_{A}^{∧} : A ⟶ A(⋀^{n} ggg):

ad_{A}^{∧}(ξ)(ν_{1} ∧ … ∧ ν_{n}) = ∑ ν_{1} ∧ … ∧ ad_{A}(ξ)(ν_{i}) ∧ … ∧ ν_{n}.

Then, the invariantness of a cross-section ε = ν_{1} ∧ … ∧ ν_{n} ∈ Sec ⋀^{n} ggg (ν_{i} denotes here a cross-sections of ggg) means the fulfilling of the equality

∑ ν_{1} ∧ … ∧ [[ξ, ν_{i}]] ∧ … ∧ ν_{n} = 0.

3. Integration along the isotropy Lie algebra bundle in regular Lie algebroids. Let A be any regular Lie algebroid on a foliated manifold (M, F) and consider the Atiyah sequence

0 ⟶ ggg ⟶ A ⟶^{γ} F ⟶ 0

and any cross-section ε ∈ Sec ⋀^{n} ggg. If ε_{x} is nonzero at each point x ∈ M, then it is called an orientation of A. Of course, ε is an orientation of the bundle of isotropy Lie algebras ggg (in the standard way). We recall that by a real A-differential form we mean a cross-section of the bundle ⋀ A^{∗}. The space of A-differential forms is denoted by Ω_{A}(M).

Definition 3.1. Let a Lie algebroid A be oriented via ε. By the integration along ggg in Lie algebroid A we shall mean the operator

∫̸_{A} : Ω_{A}(M) ⟶ Ω_{F} (M)

defined by the formula

∫̸_{E}Φ = \{ \begin{matrix}0 & degΦ < n \\ (−1)^{nk}ι_{ε}Φ & degΦ ≥ n. \end{matrix}

3.2. ∫̸_{A} is an epimorphism such that ∫̸_{A} ∘ γ^{∗} = 0.

4. Commutativity of ∫̸_{A} with differentiations.

Proposition 4.1. 1 ) If rankF = 0 (i.e. the Lie algebroid A is completely nontransitive) then the commutativity

(∗∗)  d_{F} ∘ ∫̸_{A}Φ = ∫̸_{A} ∘ d_{A}Φ

holds for each form Φ ∈ Ω_{A}(M).

2) Let rankF > 0. Write the cross-section ε locally via ε = ν_{1} ∧ … ∧ ν_{n} (ν_{i} is a local basis of ggg in a nbh of a fixed point x). Commutativity (∗∗) holds for a form Φ ∈ Ω_{A}^{n+k}(M) (in a nbh of x) if and only if, for any ξ_{1}, ..., ξ_{k+1} ∈ Sec A and x ∈ M,

(∑_{i=1}^{k+1}(−1)^{i}tr(ad_{ν_{i}}) · ε_{x}) ∧ (ξ_{1})_{x} ∧ … ∧ (ξ_{k+1})_{x}+

+ ∑_{j=1}^{k+1}(−1)^{j+n}ad_{A}^{∧}(ξ_{jx})(ε) ∧ (ξ_{1})_{x} ∧ … ĵ … ∧ (ξ_{k+1})_{x} = 0.

Corollary 4.2. If rankF ≥ 1, then condition (∗∗) holds for every form Φ if and only if

(a) every isotropy Lie algebra ggg_{|x}is unimodular,

(b) the cross-section ε is ad_{A}^{∧}-invariant.

A Lie algebroid fulfilling (a) is called unimodular, while fulfilling (b) - invariantly oriented.

Example 4.3. For the trivial transitive Lie algebroid A = T M × 𝔤, a crosssection ε ∈ Sec M ×⋀^{n} 𝔤 is invariant if and only if ε is a constant section ε(x) = ε_{o} for ε_{o} ∈ ⋀^{n} 𝔤 being ad_{𝔤}^{∧}-invariant. Therefore, every constant cross-section ε is invariant if and only if 𝔤 is unimodular.

5. Ker∫̸_{A} . Theorem. (a) If rankF = 0, then Ker∫̸_{A} is d_{A}-stable.

(b) If rankF > 0, then Ker∫̸_{A} is d_{A}-stable if and only if A is unimodular.

6. Gysin sequence. We apply the above operator of integration along the fibres of isotropy Lie algebra bundle to a construction of Gysin sequence for some transitive Lie algebroids.

6.1. The case of a trivial Lie algebroid. Let A = T M × 𝔤 be a trivial Lie algebroid. Denote by γ ang π_{2} the projections of T M × 𝔤 onto the first and the second factor, respectively. The mapping

κ : Ω(M) ⊗ Ω(𝔤) ⟶ Ω_{A}(M), ϕ ⊗ φ ⟼ γ^{∗}ϕ ∧ π_{2}φ,

is an isomorphism of graded differential algebras, therefore, passing to cohomology, we obtain an isomorphism

κ_{#} : H(M) ⊗ H(𝔤) ⟶^{≅} H_{A}(M)

of graded algebras.

Let in the sequel, 𝔤 be a unimodular Lie algebra and let A be invariantly oriented by a constant cross-section ε ∈ Sec(M × ⋀^{n} 𝔤), ε(x) = ε_{o} ≠ 0. ε determines the operator of integration ∫̸_{A} : Ω_{A}(M) ⟶ Ω(M). Take the substitution operator

ι_{ε_{o}} : ⋀ 𝔤^{∗} ⟶ ℝ, ϕ ⟼< ϕ, ε_{o} > .

Clearly, Kerι_{ε_{o}} = ⨁_{i=0}^{n−1} ⋀^{i} 𝔤^{∗}. Since the Chevalley-Eilenberg differential d^{n−1} = 0 (because 𝔤 is unimodular), then

H^{p}(Kerι_{ε_{o}}) = \{ {H^{p}(𝔤) 0 ≤ p < n \atop 0 p = n.}

The following diagram commutes with the only homomorphism κ_{1}

0 ⟶ Ω(M) ⊗ Kerι_{ε_{o}} ⟶ Ω(M) ⊗ ⋀ 𝔤^{∗} ⟶^{id×ι_{εo}} Ω(M) ⟶ 0
↓^{κ_{1}}  ≅↓^{κ}  ↓^{∥}
0 ⟶  Ker∫̸_{A}  ⟶ Ω_{A}(M) ⟶^{∫̸_{A}} Ω(M) ⟶ 0.

Therefore

κ_{1#} : H(M) ⊗ H(Kerι_{ε_{o}}) ⟶^{≅} H(Ker∫̸_{A}), [ϕ] ⊗ [φ] ⟼ [γ^{∗}ϕ ∧ π_{2}φ],

is an isomorphism of graded algebras.

Corollary 6.1.1. If

(s)  H^{p}(𝔤) = 0 for 1 ≤ p ≤ n − 1,

then H(Kerι_{ε_{o}}) = ℝ, and

γ_{#} : H(M) ⟶^{≅} H(Ker∫̸_{A}), [ϕ] ⟼ [γ^{∗}ϕ],

is an isomorphism of graded algebras.

Example 6.1.2. 3-dimensional semisimple Lie algebras fulfils condition (s). 6.2. Let us take a transitive Lie algebroid A over M, dim M > 0, with the

Atiyah sequence

0 ⟶ ggg ⟶ A ⟶^{γ} T M ⟶ 0,

and assume that (a) ggg_{|x} are unimodular Lie algebras,

(b) A is invariantly oriented by a cross-section ε ∈ Sec ⋀^{n} ggg, n =rankggg. 6.2.1. Open question. Is such a transitive Lie algebroid integrable ? Using the above corollary 6.1.1 and the existence of a finite covering U =

{U_{1}, …, U_{m}} of M such that A_{|U_{i}} are trivial Lie algebroids (U_{i} can be nonconnected) and the standard method of homological algebras (nine and five lemmas), we assert the following

Theorem 6.2.2. If A is a transitive unimodular and invariantly oriented Lie algebroid such that the structure Lie algebras ggg_{|x} satisfy condition (s), then

γ_{#} : H(M) ⟶^{≅} H(Ker∫̸_{A}), [ϕ] ⟼ [γ^{∗}ϕ],

is an isomorphism of graded algebras.

6.3. Gysin sequence. Assume that (s) holds for Lie algebras ggg_{|x}. Consider the short exact sequence

0 ⟶ Ker∫̸_{A} ⟶^{i} Ω_{A}(M) ⟶^{∫̸_{A}} Ω(M) ⟶ 0

of graded differential spaces. Passing to cohomology, we obtain the long exact sequence

… ⟶ H_{A}^{∗}(M) ⟶^{∫̸_{A}_{#}} H^{∗−n} ⟶^{∂} H^{∗+1}(Ker∫̸_{A}) ⟶^{i_{#}} H_{A}^{∗+1}(M) ⟶ …

Taking

D = γ_{#}^{−1} ∘ ∂ ∘ ω, ω(ϕ) = (−1)^{deg ϕ+1}ϕ,

(D is called a Gysin mapping), we obtain the long exact sequence

… ⟶ 0 ⟶ H^{k}(M) ⟶^{γ_{#}^{k}} H_{A}^{k}(M) ⟶ 0 ⟶ … (k < n)

… ⟶ 0 ⟶ H^{n}(M) ⟶^{γ_{#}^{k}} H_{A}^{n}(M) ⟶^{∫̸_{A}_{#}^{n}} H^{0}(M) ⟶^{D} H^{n+1}(M) ⟶^{γ_{#}^{n+1}}

H_{A}^{n+1}(M) ⟶ …

… H^{p}(M) ⟶^{D} H^{p+n+1}(M) ⟶^{γ_{#}} H_{A}^{p+n+1}(M) ⟶^{∫̸_{A}_{#}} H^{p+1}(M) ⟶^{D} …

called the Gysin sequence of A. The cohomology class χ_{A} = D(1) ∈ H^{n+1}(M) is called the Euler class of A.