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Publ. Math. Debrecen 59 / 3-4 (2001), 245-269

GYSIN SEQUENCE AND EULER CLASS OF SPHERICAL LIE ALGEBROIDS

JAN KUBARSKI

Research work supported by KBN Grant PB 173/PO/97/13

Abstract. An n-dimensional Lie algebra 𝔤 will be called a spherical Lie algebra if it is cohomologically equivalent to the n-sphere, i.e. H^{i} (𝔤) = 0 for 1 ⩽ i ⩽ n − 1 and H^{i} (𝔤) = ℝ for i = 0, n. The 1-dimensional abelian Lie algebra ℝ and the 3-dimensional Lie algebras sl (2, ℝ) and sk (3, ℝ) are only such Lie algebras. This work deals with the invariantly oriented transitive Lie algebroids having spherical isotropy Lie algebras. Lie algebroids of some principal bundles and of some TC-foliations are such algebroids. The aim of this work is to construct and investigate the Gysin sequence and the Euler class of such Lie algebroids by generalizing these notions introduced for ℝ-Lie algebroids in work [ K5].

1. Introduction

A) Let (E, π, M) be an oriented n-sphere bundle. Then the fibre integral ∫̸ : Ω^{⋆} (E) → Ω^{⋆−n} (M) has the following important property [ G-H-V]: the space of cohomology of its kernel H (ker ∫̸) is isomorphic to the space of cohomology of M

π^{#} : H (M) ⟶^{≅} H (ker∫̸) . (1.1)

Due to this fact, the Gysin sequence and the Euler class of E is easily defined, see, for example, [ G-H-V].

On the other hand, in the theory of Lie algebroids we can observe an interesting analogy to the theory of sphere bundles, namely, it turns out that, in some sense, flat connections for Lie algebroids play the same role as global cross-sections for sphere bundles. This was observed in [ K6] for regular Lie algebroids with the trivial one-dimensional adjoint bundle of Lie algebras admitting a non-singular global cross-section invariant with respect to the adjoint representation (for special cases of Lie algebroids of Poisson manifolds over ℝ-Lie foliations see [ K5]). Among transitive Lie algebroids fulfilling these properties we find:

• the Lie algebroids of S^{1}-principal bundles,

• the Lie algebroid A (G; H) of the TC-foliation of left cosets of a nonclosed Lie

subgroup H in a Lie group G, such that dim H̄ − dim H = 1,

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

Received by the editors February 2, 1999.

2000 Mathematics Subject Classification. Primary 57R19, 55R25; Secondary 57R20.

Key words and phrases. Lie algebroid, Gysin sequence, Euler class.

• the Lie algebroid A (M; F) of a TP-foliation F on a compact and simply connected manifold with 1-dimensional structural Lie algebras.

Among regular non-transitive ones we have:

• the Lie algebroids of Poisson manifolds over ℝ-Lie foliations.

The fact analogous to (1.1) is their fundamental property which enables us to observe the ’flat connections – cross-sections’ analogy. What follows appears as common ideas for cross-sections of sphere bundles and flat connections for Lie algebroids: the Gysin sequence, the notion of the Euler class as well as the notion of an index at singularity and the theorem of Euler-Poincaré-Hopf.

It turns out that this analogy – for transitive Lie algebroids – takes place in a wider class of the so-called spherical Lie algebroids in which (a) the isotropy Lie algebras 𝔤 are cohomologically equivalent to the n-sphere S^{n}, i.e. H^{k} (𝔤) = 0 for 1 ≤ k ≤ n − 1 and H^{k} (𝔤) = ℝ for k = 0, n, (b) the n-exterior power ⋀^{n} ggg of the adjoint bundle ggg possesses a non-singular global cross-section ε ∈ Sec ⋀^{n} ggg invariant with respect to the adjoint representation. Therefore, except for ℝ-Lie algebroids, this analogy holds for some sl (2, ℝ) and sk (3, ℝ)-Lie algebroids, as well.

The generalization of the Euler class and the Gysin sequence for spherical Lie algebroids is the aim of this paper. The Lie algebroids of G-principal bundles for 3-dimensional Lie groups G such that 𝔤𝔩 (G) = sk (3, ℝ) (for example, G = SO (3) , O (3) , Spin (3)) or 𝔤𝔩 (G) = sl (2, ℝ) (in the case, G = SL (2)) are the first examples of spherical Lie algebroids with 3-dimensional structural Lie algebras. The case G = O (3) gives principal bundles with compact and disconnected fibres, whereas the case G = SL (2) – with non compact fibres. Therefore the classical theory of the Gysin sequence and the Euler class for oriented bundles with fibres cohomologically equivalent to a sphere [ G-H-V, Vol. II, s.5.23] does not apply to these two classes of G-principal bundles. In the next paper [ K8] the cohomology theory of flat connections in s-Lie algebroids is developed. If the dimension of the base manifold is equal to n + 1, where n is the dimension of the isotropy Lie algebras, the index at a isolated singularity is defined. A version of the Euler-Poincaré-Hopf theorem joining the sum of indexes to the Euler class is given. This generalizes the results from [ K5]

The paper is organized as follows. Sections 2 and 3 are devoted to the fibre integral for Lie algebroids (introduced in ([ K4])) and its fundamental properties. In section 4 we study the Gysin sequence and the Euler classs of a spherical Lie algebroid A. In section 5 this class is calculated via the Chern-Weil homomorphism of A. In section 6 we prove how we can calculate the cohomology algebra of A via H (M) and the Euler class of A. We also show that two Lie algebroids on the same manifold, having isomorphic cohomology algebras, can possess different Euler classes.

B) This work (as regards techniques) is based on our work [ K4] where the notion of the fibre integral ∫̸_{A} in a regular Lie algebroid (A, [[·, ·]], γ) with respect to a non-singular cross-section ε of ⋀^{n} ggg is introduced (ggg = ker γ is the adjoint bundle of Lie algebras), n = rankggg. The pair (A, ε) is called a vertically oriented Lie algebroid and, for the transitive case (considered in this work), the fibre integral ∫̸_{A} : Ω_{A}^{⋆} (M) → Ω^{⋆−n} (M) is defined as follows: ∫̸_{A}Φ = 0 if deg Φ < n and γ^{⋆} (∫̸_{A}Φ) = (−1)^{nk} ι_{ε}Φ if deg Φ = n + k, k ⩾ 0. We recall that Ω_{A} (M) denotes the space of A-differential forms, i.e. the space Sec ⋀ A^{⋆} of cross-sections of the bundle ⋀ A^{⋆}. In Ω_{A} (M) the exterior derivative d_{A} works and gives a cohomology algebra H_{A} (M) . We also add that by a homomorphism of vertically oriented Lie algebroids (T, t) : (A, ε) → (A′, ε′) of the same rank n we mean a homorphism T : A → A′ (inducing t : M → M′) of Lie algebroids [ K3], such that (⋀^{n} T^{+}) (ε_{x}) = ε′_{tx}, x ∈ M, where T^{+} : ggg → ggg′ is the restriction of T to adjoint bundles.

The basic properties of ∫̸_{A} are given in the following theorems [ K4]:

Theorem 1.1. (a) If (T, t) : (A, ε) → (A′, ε′) is a homomorphism of vertically

oriented Lie algebroids, then t^{⋆} ∘ ∫̸_{A′} = ∫̸_{A} ∘ T^{⋆},

(b) ∫̸_{A} ∘ γ^{⋆} = 0,

(c) ∫̸_{A}γ^{⋆}ψ ∧ Φ = ψ ∧ ∫̸_{A}Φ for arbitrary forms ψ ∈ Ω (M) and Φ ∈ Ω_{A} (M) ,

(d) ∫̸_{A}Φ ∧ γ^{⋆}ψ = (−1)^{nk} (∫̸_{A}Φ) ∧ ψ for ψ ∈ Ω^{k} (M) , Φ ∈ Ω_{A}^{⩾n} (M) ,

(e) ∫̸_{A} is an epimorphism.

Theorem 1.2. The operator ∫̸_{A} commutes with the exterior derivatives d_{A} and d_{M} if and only if (a1) the isotropy Lie algebras ggg_{∣x} are unimodular, and (a2) the cross-section ε is invariant with respect to the adjoint representation of A

on ⋀^{n} ggg.

We recall [ K4] that the invariance of ε is equivalent to the following property: for any open subset U ⊂ M on which ε_{|U} = (h_{1} ∧ … ∧ h_{n})_{|U} , h_{i} ∈ Secggg, the equality (∑_{i=1}^{n} h_{i} ∧ … ∧ [[ξ, h_{i}]] ∧ … ∧ h_{n})_{|U} = 0 holds for each ξ ∈ Sec A.

Theorem 1.3. The kernel ker ∫̸_{A} is a d_{A}-stable space if and only if ggg_{∣x} are unimodular.

Theorem 1.4. For a trivial Lie algebroid A = T M × 𝔤 over a connected manifold M a cross-section ε of the vector bundle ⋀^{n} ggg = M × ⋀^{n} 𝔤 is invariant if and only if it is a constant one ε (x) = ε_{o} for a ⋀^{n} ad_{𝔤}–invariant element ε_{o} ∈ ⋀^{n} 𝔤. □

The transitive Lie algebroid A fulfilling properties (a1) and (a2) from Theorem 1.2 is called a transitive unimodular invariantly oriented Lie algebroid (briefly, TUIO-Lie algebroid).

In [ K4] and [ K7] three sources of such Lie algebroids can be found:

• the Lie algebroids of G-principal bundles for a structure Lie group G not necessarily compact or connected but satisfying det (Ad_{G} a) = +1, a ∈ G,

• the Lie algebroid of the TC-foliation of left cosets of a nonclosed Lie subgroup H in a Lie group G,

• the Lie algebroid A (M, F) of a TP-foliation on a compact and simply connected manifold.

In [ K7] one can find the Poincaré duality theorem for TUIO-Lie algebroids over oriented manifolds establishing an isomorphism H_{A}^{p} (A) ≅ H_{A,c}^{n+m−p} (M)^{⋆} (n = rankggg, m = dim M, H_{A,c}^{n+m−p} (M) denotes the (n + m − p)-cohomology space of A-forms with compact support). This theorem implies the duality between γ_{c}^{#} : H_{c} (M) → H_{A,c} (M) and ∫̸_{A}^{#} : H_{A} (M) → H (M) from which we obtain (as a corollary)

Theorem 1.5. If A is a TUIO-Lie algebroid over an oriented manifold M, then the following conditions are equivalent:

(a) ∫̸_{A}^{#n} : H_{A}^{n} (M) → H^{0} (M) is an epimorphism,

(b) γ_{c}^{#m} : H_{c}^{m} (M) → H_{A,c}^{m} (M) is a monomorphism.

C) Let 𝔤 be any n-dimensional unimodular Lie algebra. For a tensor 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤, take the substitution operator i_{ε_{o}} : ⋀ 𝔤^{⋆} → ℝ. Then ker i_{ε_{o}} = ⨁_{i=0}^{n−1} ⋀^{i} 𝔤^{⋆}. Consider the cochain complex (ker i_{ε_{o}}, δ)

0 → ℝ ⟶^{0} 𝔤^{⋆} ⟶^{δ^{1}} ⋀^{2} 𝔤^{⋆} ⟶^{δ^{2}} ⋯ ⟶^{δ^{n−2}} ⋀^{n−1} 𝔤^{⋆} ⟶^{0} 0.

The last arrow is also the Chevalley-Eilenberg differential δ^{n−1} since, in a unimodular Lie algebra 𝔤, we have δ^{n−1} = 0 (this can easily be noticed from the definition or Corollary 3.2 in [ K4]). This yields that

H^{p} (ker i_{ε_{o}}) = \{ \begin{matrix}H^{p} (𝔤) & if & 0 ⩽ p < n, \\ 0 & if & p = n. \end{matrix} (1.2)

Take now a trivial unimodular Lie algebroid A = T M × 𝔤 equipped with a constant cross-section ε ∈ Sec (M × ⋀^{n} 𝔤), ε (x) = ε_{o}, 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤. ε is ⋀^{n} ad_{A}- invariant [ K4, Ex.3.1]. Consider the integration operator ∫̸_{A} corresponding to ε. The Künneth homomorphism κ : Ω (M)×⋀ 𝔤^{⋆} → Ω_{T M×𝔤} (M) , ψ⊗φ ↦ γ^{⋆}ψ∧π^{⋆}φ, (γ : T M × 𝔤 → T M and π : T M × 𝔤 → 𝔤 are projections) is an isomorphism of graded differential algebras, see [ K7, Lemma 6.1].

Lemma 1.1. The following diagram commutes for a uniquely determined isomorphism κ_{1} of graded differential algebras:

0 ⟶ Ω (M) ⨂ ker ι_{ε_{o}} ⟶ Ω (M) ⨂ ⋀ 𝔤^{⋆} ⟶^{id ⊗ι_{εo}} Ω (M) ⟶ 0
≅↓^{κ_{1}}  ≅↓^{κ}  ∥
0 ⟶ ker ∫̸ _{T M×𝔤} ⟶ Ω_{T M×𝔤} (M) ⟶^{∫̸_{T M×𝔤}} Ω (M) ⟶ 0 .

Proof. The right square commutes: (a) if deg Φ < n, Φ ∈ Ω (M) ⊗ ⋀ 𝔤^{⋆}, then ∫̸_{T M×𝔤} ∘ κ (Φ) = 0 = id ⊗i_{ε_{o}} (Φ). (b) if Φ = ψ ⊗ φ, ψ ∈ Ω (M), φ ∈ ⋀ 𝔤^{⋆} and deg Φ = n + k, k ⩾ 0, we have, for ε_{o} = h_{1} ∧ … ∧ h_{n} (h_{i} ∈ 𝔤),

(∫̸_{T M×𝔤} ∘ κ) (ψ ⊗ φ) (v_{1} ∧ … ∧ v_{k})

= ∫̸_{T M×𝔤} (γ^{⋆}ψ ∧ π^{⋆}φ) (v_{1} ∧ … ∧ v_{k})

= (−1)^{nk} (γ^{⋆}ψ ∧ π^{⋆}φ) (h_{1} ∧ … ∧ h_{n} ∧ v_{1} ∧ … ∧ v_{k})

= (π^{⋆}φ ∧ γ^{⋆}ψ) (h_{1} ∧ … ∧ h_{n} ∧ v_{1} ∧ … ∧ v_{k})

=^{⋆} ι_{ε_{o}}φ · ψ (v_{1} ∧ … ∧ v_{k})

= (id ⊗ι_{ε_{o}}) (ψ ⊗ φ) (v_{1} ∧ … ∧ v_{k}) .

The equality ”=^{⋆}” holds independently of the degree of φ (if deg φ < n, then both sides of ”=^{⋆}” are 0). Clearly, there exists a uniquely determined homomorphism κ_{1} of graded differential algebras for which the above diagram commutes, and κ_{1} is an isomorphism.

Corollary 1.1. The mapping

κ_{1#} : H (M) ⨂ H (ker ι_{ε_{o}}) ⟶^{≅} H (ker∫̸_{T M×𝔤})

induced by κ_{1} on cohomology (after its composition with the algebraic Künneth isomorphism) is an isomorphism of graded algebras.

It is easy to see that the mapping

ρ : Ω (M) ⟶ Ω (M) ⨂ ker ι_{ε_{o}}, ψ ⟼ ψ ⊗ 1,

is a homomorphism of graded differential algebras for which the following diagram

Ω (M) ⟶^{ρ} Ω (M) ⨂ ker ι_{ε_{o}}
∥  ≅↓^{κ_{1}}
Ω (M) ⟶^{γ^{⋆}} ker (∫̸_{T M×𝔤}) .

(1.3)

commutes. [The fact that Im γ^{⋆} ⊂ ker ∫̸_{T M×𝔤} follows from Th. 1.1 (b)].

Finally, consider a transitive Lie algebroid Aover a manifold Mof dimension m, with the Atiyah sequence 0 → ggg ↪ A →^{γ} T M → 0. Let ε ∈ Sec ⋀^{n} ggg, n = rank ggg, be a non-singular cross-section. Denote by ∫̸_{A} : Ω_{A}^{⋆} (M) → Ω^{⋆−n} (M)the fibre integral with respect to ε.In the present work we examine the short exact sequence of graded spaces

0 ⟶ ker ∫̸_{A} ↪^{i} Ω_{A} (M) ⟶^{∫̸_{A}} Ω (M) ⟶ 0. (1.4)

The homomrphism i is of degree 0and ∫̸_{A}of degree −n. We assume that (A, ε)is a TUIO-Lie algebroid, i.e. that Lie algebras ggg_{∣x}are unimodular and εis ⋀^{n} ad_{A}- invariant. Due to Theorems 1.2-3, sequence (1.4) is a sequence of differential spaces.

2. H (ker ∫̸_{A}) for spherical Lie algebroids

Definition 2.1. An n-dimensional real Lie algebra 𝔤 is called an spherical Lie algebra (s-Lie algebra for short) if it is cohomologically equivalent to the n-sphere S^{n}, i.e. if

H^{k} (𝔤) = \{ \begin{matrix}0 & for & 1 ⩽ k ⩽ n − 1, \\ ℝ & for & k = 0, n . \end{matrix}

Every s-Lie algebra is unimodular.

Example 2.1. The following are examples of s-Lie algebras:

• ℝ — the 1-dimensional abelian Lie algebra,

• sl (2, ℝ) — the Lie algebra of real 2 × 2-matrices with trace zero,

• sk (3, ℝ)— the Lie algebra of real 3 × 3 skew symmetric matrices. The last two Lie algebras are 3-dimensional.

Proposition 2.1. The above are only s-Lie algebras.

Proof. It is sufficient to check that • for arbitrarily taken real Lie algebra 𝔤 if H^{1} (𝔤) = 0 then H^{3} (𝔤) ≠ 0. The assumption H^{1} (𝔤) = 𝔤/ [𝔤, 𝔤] = 0 implies that the Levi factor, 𝔩, of 𝔤 is

nontrivial. Let 𝔤 = 𝔩 ⊕ 𝔪 be the Levi decomposition of 𝔤. The rest trivially follows from the fact H^{3} (𝔩) ≠ 0 and H^{0} (𝔪)^{𝔤} = ℝ and the Hochschild-Serre formula [ H-S, Th. 13] (for a = 3):

H^{a} (𝔤) = ⨁^{r} H^{r} (𝔩) ⊗ H^{a−r} (𝔪)^{𝔤} .

Definition 2.2. A TUIO-Lie algebroid is called an spherical Lie algebroid (s-Lie algebroid for short) if its isotropy Lie algebras ggg_{∣x} are s-Lie algebras.

Example 2.2. (1) Lie algebroids of S^{1}-, SL (2)-, SO (3)-, O (3)- or Spin (3)- principal bundles are s-Lie algebroids.

(2) The Lie algebroid of the TC-foliation of left-cosets of a nonclosed Lie subgroup H in a Lie group G, such that dim H̄ − dim H = 1, is an s-Lie algebroid ([ K4, Th. 3.5]).

Theorem 2.1. For any s-Lie algebroid (A, ε) over a manifold M,

γ^{#} : H (M) ⟶ H (ker∫̸_{A})

is an isomorphism of graded algebras.

Proof. M (paracompact and not necessarily connected) can be covered by a finite family of open (not necessarily connected) subsets {U_{1}, …, U_{k}} such that A_{|U_{i}} is isomorphic to a trivial Lie algebroid. Indeed each point has a neighbourhood over which A is isomorphic to a trivial Lie algebroid [ A-M], [ M], next we need to use Th.I from [ G-H-V, Vol.I, p.17].

Our theorem will be proved inductively with respect to k.

Step 1. k = 1. In this situation, A ≅ T M × 𝔤. According to (1.3), the theorem follows from the assumption about 𝔤 to be an s-Lie algebra: H (ker ι_{ε_{o}}) = ℝ.

Step 2. Assume the theorem is true for positive numbers ⩽ k. We prove this theorem for the number k + 1. Let M = U_{1} ∪ … ∪ U_{k+1} (the Lie algebroid A over U_{i} is isomorphic to a trivial one). Put U = U_{1}, V = ⋃_{i=2}^{k+1} U_{i}. Then U ∪ V = M, U ∩ V = ⋃_{i=2}^{k+1} U_{1} ∩ U_{i}. Of course, A_{|U_{1}∩U_{i}} is isomorphic to a trivial one, too. Consider the Mayer-Vietoris sequence for the covering {U, V } [ K7, sect.3]

0 → Ω_{A} (M) →^{α} Ω_{A_{|U}} (U) ⨁ Ω_{A_{|V}} (V) →^{β} Ω_{A_{|U∩V}} (U ∩ V) → 0

and the following diagram in which Ω_{1} := Ω_{A_{|U}} , Ω_{2} := Ω_{A_{|V}} , Ω_{12} := Ω_{A_{|U∩V}} and the integration operators are determined with respect to the restrictions of ε

0  0  0
↓  ↓  ↓
0 ⟶ ker ∫̸_{A} ⟶ ker ∫̸_{A_{|U}} ⨁ ker ∫̸_{A_{|V}} ⟶ ker ∫̸_{A_{|U∩V}} ⟶ 0
↓  ↓  ↓
0 ⟶ Ω_{A} (M) ⟶^{α} Ω_{1} (U) ⨁ Ω_{2} (V) ⟶^{β} Ω_{12} (U ∩ V) ⟶ 0
↓∫̸_{A}  ↓^{∫̸_{A_{|U}} ⊗∫̸_{A_{|V}}}  ↓∫̸_{A_{|U∩V}}
0 ⟶ Ω (M) ⟶^{α} Ω (U) ⨁ Ω (V) ⟶^{β} Ω (U ∩ V) ⟶ 0 .
↓  ↓  ↓
0  0  0

The bottom rows and all columns are exact. Therefore, by the nine-lemma, the top row is exact, too. This yields the following diagram with exact rows:

0 ⟶ Ω (M) ⟶ Ω (U) ⨁ Ω (V) ⟶ Ω (U ∪ V) ⟶ 0
↓^{γ_{M}^{⋆}}  ↓^{γ_{U}^{⋆} ⊕γ_{V}^{⋆}}  ↓^{γ_{U∩V}^{⋆}}
0 ⟶ ker ∫̸_{A} ⟶ ker ∫̸_{A_{|U}} ⨁ ker ∫̸_{A_{|V}} ⟶ ker ∫̸_{A_{|U∩V}} ⟶ 0 .

By the induction assumptions, γ_{U}^{#}, γ_{V}^{#} and γ_{U∩V}^{#} are isomorphisms. Writing the long cohomology sequences and using the five-lemma, we assert that γ_{M}^{#} is an isomorphism, too.

3. Gysin sequence and Euler class of s-Lie algebroids

Let us come back to sequence (1.4) of graded differential spaces for an arbitrary s-Lie algebroid (A, ε) and consider the corresponding canonical long exact sequence in cohomology

⋯ ⟶ H_{A} (M) ⟶^{∫̸_{{# \atop A}}} H (M) ⟶^{∂} H (ker∫̸_{A}) ⟶^{i_{#}} H_{A} (M) ⟶ ⋯

with the connecting homomorphism ∂ of degree n + 1. In the following diagram

H (M) ⟶^{γ^{#}} H_{A} (M)
≅↑_{γ^{#−1}}  ∥
H (ker ∫̸_{A}) ⟶^{i_{#}} H_{A} (M)
↑^{∂}  ∥
H (M) ⟵^{∫̸_{{# \atop A}}} H_{A} (M) ,

the bottom square commutes. Define the operator D : H (M) → H (M) as the composition

D : H (M) ⟶^{ω̃} H (M) ⟶^{∂} H (ker∫̸_{A}) ⟶^{γ^{#−1}} H (M)

where ω̃ is the involution ω̃ (ψ) = (−1)^{deg ψ+1} ψ. D is a homomorphism of degree n + 1, giving the exactness of the triangle

\begin{aligned}H (M) \\ & ↘^{γ^{#}} \\ D ↑ & H_{A} (M) . \\ & ↙_{}∫̸{# \atop A} \\ H (M) \end{aligned} (3.1)

Definition 3.1. D is called the Gysin homomorphism of the s-Lie algebroid (A, ε) . The long exact sequence

⋯ ⟶^{D=0} H^{k} (M) ⟶^{γ^{#k}} H_{A}^{k} (M) \underrightarrow{∫̸_{A}^{#k}=0} 0 ⟶ ⋯ (0 ⩽ k < n)

⋯ 0 ⟶ H^{n} (M) ⟶^{γ^{#n}} H_{A}^{n} (M) \underrightarrow{∫̸_{A}^{n}} H^{0} (M) ⟶^{D^{0}} ⋯

⋯ H^{p} (M) ⟶^{D^{p}} H^{p+n+1} (M) ⟶^{γ^{#}} H_{A}^{p+n+1} (M) \underrightarrow{∫̸_{A}^{#}} H^{p+1} (M) ⟶^{D^{p+1}} ⋯ (p ⩾ 0),

corresponding to (3.1), is called the Gysin sequence of (A, ε). The cohomology class

χ_{A} := D^{0} (1) = γ^{#−1} (∂ (−1)) ∈ H^{n+1} (M)

is called the Euler class of (A, ε).

Remark 3.1. The construction of the Gysin homomorphism and the Euler class. Take a closed form ψ ∈ Ω (M) . There exists a form Φ ∈ Ω_{A} (M) such that ∫̸_{A}Φ = ψ (see Th. 1.1 (e)). Then, by Th. 1.2, d_{A}Φ ∈ ker ∫̸_{A} and ∂ [ψ] = [d_{A}Φ]_{ker ∫̸_{A}} . Therefore

D [ψ] = (−1)^{deg ψ+1} γ^{#−1} [d_{A}Φ]_{ker ∫̸_{A}} .

In particular, for ψ = 1 ∈ H^{0} (M) and an n-form Φ̃ such that ∫̸_{A}Φ̃ = −ψ = −1, we have [d_{A}Φ̃]_{ker ∫̸_{A}} = γ^{#}χ_{A}. For any representative Ψ of χ_{A}, χ_{A} = [Ψ] , there exists

a form Φ ∈ Ω_{A}^{n} (M) such that

∫̸_{A}Φ = −1 and d_{A}Φ = γ^{∗}Ψ.

Indeed, [d_{A}Φ̃]_{ker ∫̸_{A}} = [γ^{∗}Ψ] , so d_{A}Φ̃ − γ^{⋆}Ψ = d_{A}Θ for a form Θ ∈ ker ∫̸_{A}. Thus

γ^{⋆}Ψ = d_{A}(Φ̃ − Θ), and Φ := Φ̃ − Θ is the searched-for form.

Proposition 3.1. D (α ∧ β) = α ∧ D (β) for arbitrary classes α, β ∈ H (M) .

Proof. Take a form Ω ∈ Ω_{A}^{n} (M) such that ∫̸_{A}Ω = 1. Let α = [φ] , β = [ψ]. By Th. 1.1 (c), we have ∫̸_{A} (γ^{⋆} (φ ∧ ψ)) ∧ Ω = φ ∧ ψ, so this equality gives

D (α ∧ β) = (−1)^{deg φ+deg ψ+1} γ^{#−1} (∂ [φ ∧ ψ])
= (−1)^{deg φ+deg ψ+1} γ^{#−1} [d_{A} (γ^{⋆} (φ ∧ ψ) ∧ Ω)]_{ker ∫̸_{A}}
= (−1)^{deg ψ+1} γ^{#−1} (γ^{#} [φ] ∧ [d_{A} (γ^{⋆}ψ ∧ Ω)]_{ker ∫̸_{A}})
= [φ] ∧ (−1)^{deg ψ+1} γ^{#−1} (∂ [ψ])
= [φ] ∧ D [ψ] = α ∧ Dβ.

Corollary 3.1. D (α) = α ∧ χ_{A}.

From this and the Gysin sequence we easily obtain

Proposition 3.2. For an s-Lie algebroid (A, ε) , the following conditions are equivalent:

(1) ∫̸_{A}^{#n} : H_{A}^{n} (M) → H^{0} (M) is an epimorphism,

(2) χ_{A} = 0,

(3) ∫̸_{A}^{#} : H_{A} (M) → H (M) is an epimorphism,

(4) γ^{#n+1} : H^{n+1} (M) → H_{A}^{n+1} (M) is a monomorphism,

(5) γ^{#} : H (M) → H_{A} (M) is a monomorphism.

In particular, if H^{n+1} (M) = 0, then χ_{A} = 0.

If additionally M is oriented then according to Th. 1.5 the above conditions are equivalent to

(6) γ_{c}^{#m} is a monomorphism.

Corollary 3.2. If an s-Lie algebroid (A, ε) is flat, then χ_{A} = 0.

Proof. Let λ : T M → A be a flat connection. Then γ ∘ λ = id and λ is a homomorphism of Lie algebroids. This implies the relation λ^{#} ∘ γ^{#} = id^{#}, which yields the monomorphy of γ^{#}. The proposition above gives the equality χ_{A} = 0.

In particular, the trivial Lie algebroid T M × 𝔤 has the vanishing Euler class. Immediately from the Gysin sequence we have

Corollary 3.3. If χ_{A} = 0, then the following short sequences are exact:

0 ⟶ H^{n+p} (M) ⟶^{∂^{#}} H_{A}^{n+p} (M) \underrightarrow{∫̸_{A}^{#}} H_{A}^{p} (M) ⟶ 0, p ⩾ 0. (3.2)

In particular,

H_{A}^{n+p} (M) ≅ H^{n+p} (M) ⨁ H^{p} (M) , p ⩾ 0.

Corollary 3.4. The Euler class χ_{A} vanishes for any ℝ-s-Lie algebroid A over a noncompact 2-manifold M and for any sl (2, ℝ) or sk (3, ℝ)-s-Lie algebroid A over a noncompact 4-manifold M.

Remark 3.2. It can easily be checked that χ_{A} = 0 in the case of the evenness of rank ggg; however, probably (see Remark (2.1)) such a situation never holds.

The comparison with principal bundles having a structure Lie group cohomologically equivalent to a sphere is given in the following theorem [we recall that each bundle with the fibre cohomologically equivalent to a sphere possesses the Euler class defined analogously as for sphere bundles [ G-H-V, Vol.II, 5.23]].

Theorem 3.1. Let P be a principal bundle with the structure Lie group G cohomologically equivalent to a sphere, invariantly oriented by ε_{o} ∈ ⋀^{n} 𝔤 (𝔤 is the Lie algebra of G, n = dim G), such that ∫_{G} Δ_{R} = 1 and Δ_{R} is the right-invariant nform on G equalling ε_{o} at the unit. Then the Euler classes of the bundle P and the invariantly oriented Lie algebroid A (P) are equal to each other:

χ_{A(P)} = χ_{P} .

The Gysin homomorphisms are identical and the Gysin sequences are ”equivalent”:

\begin{matrix}⋯ H (M) & ⟶^{D} & H (M) & ⟶^{γ^{#}} & H_{A} (M) & ⟶^{∫̸_{{# \atop A}}} & H (M) ⋯ \\ ∥ & & ∥ & & ≅↓^{τ_{P}^{#}} & & ∥ \\ ⋯ H (M) & ⟶^{D} & H (M) & ⟶^{π^{#}} & H (P) & ⟶^{∫̸_{{# \atop P}}} & H (M) ⋯ \end{matrix} (3.3)

Proof. Let χ_{A(P)} = [Ψ] . Take an n-form Φ ∈ Ω_{A}^{n} (M) such that ∫̸_{A}Φ = −1 and d_{A}Φ = γ^{⋆}Ψ. Denote by Φ̃ the right-invariant n-form on P corresponding to Φ, Φ̃ = τ_{P} (Φ) , where τ_{P} : Ω_{A(P)} (M) ≅ Ω^{R} (P) is the canonical isomorphism (see [ K2], [ K4, (1.6)]). Then, by the definition of fibre integrals, ∫̸_{P} Φ̃ = −1 and γ^{⋆}Ψ = d_{A}τ_{P}^{−1}Φ̃ = τ_{P}^{−1}dΦ̃. Therefore π^{⋆}Ψ = τ_{P} γ^{⋆}Ψ = dΦ̃, which implies that χ_{P} = [Ψ] = χ_{A(P)} and, in the consequence, the commutativity of diagram 3.3.

Remark 3.3. The last theorem does not refer to the non-compact case G = SL (2) .

Theorem 3.2. (Naturality of the Gysin sequence) A homomorphism (T, t) : (A, ε) → (A′, ε′) of s-Lie algebroids induces a homomorphism of the long exact Gysin sequences

⋯ H^{k} (M′) ⟶^{γ^{′#}} H_{A′}^{k} (M′) ⟶^{∫̸_{{# \atop A′}}} H^{k−n} (M′) ⟶^{D_{A′}} H^{k+1} (M′) ⋯
↓_{t^{#}}  (1)  ↓_{T^{#}}  (2)  ↓_{t^{#}}  (3)  ↓_{t^{#}}
⋯ H^{k} (M) ⟶^{γ ^{#}} H_{A}^{k} (M) ⟶^{∫̸_{{# \atop A}}} H^{k−n} (M) ⟶^{D_{A}} H^{k+1} (M) ⋯

Proof. The commutativity of (1) follows immediately from the equality t_{⋆} ∘ γ = γ′ ∘ T.

(2) — immediately from the equality t^{⋆} ∘ ∫̸_{A′} = ∫̸_{A} ∘ T^{⋆} Th. 1.1 (a) (passing to cohomology).

(3) From the general fact [ G-H-V, Vol.I] we have the commuting diagram

H (M′) ⟶^{∂′} H(ker ∫̸_{A′})
↓_{t^{#}}  ↓T̃^{#}
H(M) ⟶^{∂} H(ker ∫̸_{A})

where T̃ : ker ∫̸_{A′} → H(ker ∫̸_{A}) is the restriction of T^{⋆}. Hence it appears that in the diagram below all small squares commute:

D_{A′} :H (M′) ⟶^{ω̃} H (M′) ⟶^{∂′} H(ker ∫̸_{A′}) ⟶^{(γ^{′#})^{−1}} H (M′)
↓_{t^{#}}  ↓_{t^{#}}  ↓T̃^{#}  ↓_{t^{#}}
D_{A} : H (M) ⟶^{ω̃} H (M) ⟶^{∂} H(ker ∫̸_{A}) ⟶^{(γ^{#})^{−1}} H (M) ,

which easily implies the commutativity of (3).

As a consequence we obtain

Corollary 3.5. (Naturality of the Gysin class). If (T, t) : (A, ε) → (A′, ε′) is a homomorphism of s-Lie algebroids over manifolds M, M′ then χ_{A} = t^{#}χ_{A′}.

4. Gysin class via the Chern-Weil homomorphism

In [ K3], the Chern-Weil homomorphism h_{A} for any regular Lie algebroid A is constructed. We use this homomorphism for an s-Lie algebroid (A, ε) (therefore for a transitive one)

h_{A} : ⨁^{k} (Sec ⋁^{k} ggg^{⋆})_{I^{0}} ⟶  H (M)
Γ ⟼ \frac{1}{k!}[⟨Γ, \underbrace{Ω_{b} ∨ … ∨ Ω_{b}}_{k times}⟩]

where (Sec ⋁^{k} ggg^{⋆})_{I^{0}} denotes the space of invariant cross-sections of the bundle ⋁^{k} ggg^{⋆} with respect to the adjoint representation ad_{A}^{∨} of A on ⋁^{k} ggg^{⋆} and Ω_{b} ∈ Ω^{2} (M;ggg) is the curvature form of any connection λ : T M → A in the Lie algebroid A, Ω_{b} (X, Y) = −ω ([[λX, λY]]) , X, Y ∈ 𝔛 (M) (ω : A → ggg is the connection form of λ). We recall that Γ ∈ Sec ⋁^{k} ggg^{⋆} is invariant if and only if, for any ξ ∈ Sec A and h_{1}, …, h_{k} ∈ Secggg,

(γ ∘ ξ) (⟨Γ, h_{1} ∨ … ∨ h_{k}⟩) = ∑_{i=1}^{k}⟨Γ, h_{1} ∨ … ∨ [[ξ, h_{i}]] ∨ … ∨ h_{k}⟩.

Theorem 4.1. Let 0 → ggg ↪ A →^{γ} T M → 0 be the Atiyah sequence of A. The form γ^{⋆}⟨Γ, Ω_{b} ∨ … ∨ Ω_{b}⟩ for invariant Γ ∈ (Sec ⋁^{k} ggg^{⋆})_{I^{0}} is closed, more precisely,

γ^{⋆}⟨Γ, \underbrace{Ω_{b} ∨ … ∨ Ω_{b}}_{k times}⟩ = d_{A}Φ

for

Φ = k! ∑_{i+j=k−1} \frac{1}{k + j} ⟨Γ, ω ∨ \frac{1}{i!} (d_{A}ω)^{i} ∨ \frac{1}{j!} (−\frac{1}{2}[[ω, ω]])^{j}⟩. (4.1)

[Remark: This is a generalization of the standard formula [ G-H-V, Vol.II]. In the standard formula concerning principal bundles, the sign ”–” does not occur because there is a left Lie algebra of a structure Lie group there, not right].

Proof. Each point x ∈ M has a connected neighbourhood U such that A_{|U} is the Lie algebroid of a trivial principal bundle π : U × G → U for some connected Lie group G, A_{|U} = A (U × G) . Let H̄ ⊂ T (U × G) be the connection in U × G corresponding to the restriction of λ to U λ_{|U} : T U → A_{|U} and let Ω̄_{U} ∈ Ω^{2} (U, 𝔤) be the curvature form of H̄. Clearly, Γ_{|U} ∈ Sec ⋁^{k} ggg_{|U}^{⋆} is ad_{A_{|U}}^{∨} -invariant. According to the connectedness of U × G, we have the isomorphism of vector spaces

ν : (⋁^{k} 𝔤^{⋆})_{I} ⟶ (Sec ⋁^{k}ggg_{|U}^{⋆})_{I^{o}}

[ K3, p.43] where (⋁^{k} 𝔤^{⋆})_{I} denotes the space of Ad_{G}-invariant vectors from ⋁^{k} 𝔤^{⋆}.

ν is defined by the formula ν (w) (x) = ⋁^{k} \binom{}{ẑ}^{−1⋆} (w) , z ∈ (U × G)_{|x} ẑ : 𝔤 →ggg_{∣x} is an isomorphism of the right Lie algebra 𝔤 of G onto the Lie isotropy Lie algebra ggg_{∣x} at x ([ K2, p.11]). Therefore Γ_{|U} = ν (w) for some Ad_{G}-invariant element

w ∈ (⋁^{k} 𝔤^{⋆})_{I}. Let ω̄ be the connection form of H̄. By equality (15) from [ K3] and the standard formula [ G-H-V, Vol.II ], we have

π^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨ … ∨ Ω_{b|U}⟩ = ⟨w, Ω̄_{U} ∨ … ∨ Ω̄_{U}⟩ = dΦ̄

for

Φ̄ = k! ∑_{i+j=k−1} \frac{1}{k + j} ⟨w, ω̄ ∨ \frac{1}{i!} (dω̄)^{i} ∨ \frac{1}{j!} (\frac{1}{2}[[ω̄, ω̄]]^{L})^{j}⟩

= k! ∑_{i+j=k−1} \frac{1}{k + j} ⟨w, ω̄ ∨ \frac{1}{i!} (dω̄)^{i} ∨ \frac{1}{j!} (−\frac{1}{2}[[ω̄, ω̄]]^{R})^{j}⟩

([·, ·]^{L} and [·, ·]^{R} denote the left and right structures of a Lie algebra 𝔤 of the Lie group G). On the other hand, the real form π^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨…∨Ω_{b|U}⟩ on P = U ×G is right-invariant and

γ_{|U}^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨ … ∨ Ω_{b|U}⟩ = τ_{P}^{−1}π^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨ … ∨ Ω_{b|U}⟩

for the canonical isomorphism τ_{P} : Ω_{A(P)} (U) →^{≅} Ω^{R} (P) [ K2]. Since τ_{P} commutes with exterior derivatives, we have

γ_{|U}^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨ … ∨ Ω_{b|U}⟩ = τ_{P}^{−1} (dΦ̄) = d_{A_{|U}} (τ_{P}^{−1}Φ̄) .

ω̄ is Ad_{G}-equivariant, therefore dω̄ and [ω̄, ω̄]^{R} are Ad_{G}-equivariant too. Considering the adjoint representations ad_{A(P)} and Ad_{G}, we have the analogous isomorphism

τ_{P}^{∨} : Ω_{A(P)}^{ad} (M; ⋁^{s}ggg) ⟶^{≅} Ω^{Ad} (P; ⋁^{s} 𝔤)

between the spaces of equivariant forms [ M, App.A, Prop.4.12] (see also [ K1] for the language of differential groupoids). The following observations holds:

(1) τ_{P}^{∨} (ω) = ω̄ (i.e. ẑ(ω̄_{∣z} (v)) = ω_{∣x} [v]), see, for example [ K2, p.39],

(2) τ_{P}^{∨} commutes with exterior derivatives, in particular, τ_{P}^{∨} (d_{A}ω) = dω̄,

(3) τ_{P}^{∨}[[ω, ω]] = [ω̄, ω̄]^{R} ,

(4) τ_{P}^{∨}⟨ν (w) , φ⟩ = ⟨w, τ_{P}^{∨} (φ)⟩ for any invariant vector w ∈ (⋁^{s} 𝔤)_{I} and equivariant form φ ∈ Ω_{A(P)}^{ad} (M; ⋁^{s} ggg) .

From this we have

γ_{|U}^{⋆}⟨Γ_{|U}, Ω_{b|U} ∨ … ∨ Ω_{b|U}⟩ = d_{A_{|U}} (τ_{P}^{−1}Φ̄)

= d_{A_{|U}} (k! ∑_{i+j=k−1} \frac{1}{k + j} ⟨Γ_{|U}, ω ∨ \frac{1}{i!} (d_{A}ω)^{i} ∨ \frac{1}{j!} (−\frac{1}{2}[[ω, ω]])^{j}⟩) .

The arbitrariness of x yields the assertion of our theorem.

As a corollary we get

γ^{⋆}⟨Γ, Ω_{b}⟩ = d_{A}⟨Γ, ω⟩, Γ ∈ (Secggg^{⋆})_{I^{0}} , (4.2)

γ^{⋆}⟨Γ, Ω_{b} ∨ Ω_{b}⟩ = d_{A} (⟨Γ, ω ∨ d_{A}ω⟩ − \frac{1}{3}⟨Γ, ω ∨ [[ω, ω]]⟩) , Γ ∈ (Sec ⋁^{2}ggg^{⋆})_{I^{0}} . (4.3)

• Problem: Prove formula (4.1) using only the category of Lie algebroids.

4.1. s-Lie algebroids of rank 1. Consider an s-Lie algebroid (A, ε) of rank 1. Then ggg is a trivial rank 1 LAB with global ad_{A}-invariant cross-section ε ∈ Secggg, i.e. the one satisfying the relation [[ξ, ε]] = 0 for each ξ ∈ Sec A. Take

ε^{⋆} ∈ Secggg^{⋆} (4.4)

such that

ι_{ε}ε^{⋆} = 1.

Then ε^{⋆} is ad_{A}^{♮}-invariant, i.e. ad_{A}^{♮} (ξ) (ε^{⋆}) = 0 for ξ ∈ Sec A. Indeed, since rankggg = 1 and ε_{x} ≠ 0 for each x ∈ M, it is enough to notice the equality

ad_{A}^{♮} (ξ) (ε^{⋆}) (ε) = (γ ∘ ξ) ⟨ε^{⋆}, ε⟩ − ⟨ε^{⋆}, [[ξ, ε]]⟩ = 0.

Therefore ε^{⋆} belongs to the domain of the Chern-Weil homomorphism h_{A} of A.

Theorem 4.2. Under the above assumptions,

χ_{A} = h_{A}^{(2)} (−ε^{⋆}) .

Proof. According to the definition of h_{A}, we have h_{A} (−ε^{⋆}) = [⟨−ε^{⋆}, Ω_{b}⟩]. To prove our theorem, let us take the 1-form Φ = ⟨−ε^{⋆}, ω⟩ ∈ Ω_{A}^{1} (M) . According to (4.2), d_{A}Φ = γ^{⋆}⟨−ε^{⋆}, Ω_{b}⟩, therefore it is sufficient to check that ∫̸_{A}Φ = −1. Clearly, this follows from the equalities

γ^{⋆} (∫̸_{A}Φ) = ι_{ε}Φ = ⟨−ε^{⋆}, ω (ε)⟩ = ⟨−ε^{⋆}, ε⟩ = −1.

Corollary 4.1. There exist nonintegrable s-Lie algebroids (A, ε) of rank 1 having the nonzero Euler class χ_{A}.

Indeed, take the s-Lie algebroid of rank 1 of the TC-foliation of left cosets of a nonclosed Lie subgroup H in a Lie group G, such that dim H̄−dim H = 1, (Example 2.2 and [ K4, Th. 3.5.3]). The invariant cross-section ε ∈ Secggg corresponds via the isomorphism φ to an element 0 ≠ ε_{o} = [w] ∈ 𝔥̄/𝔥. Assume G to be a compact, connected and semisimple Lie group. According to the proof of Theorem 7.4.3 from [ K3], h_{A}^{(2)} ≠ 0. Since the domain of h_{A}^{(2)}, (Secggg^{⋆})_{I^{o}}, is a 1-dimensional vector space [more exactly, (Secggg^{⋆})_{I^{o}} is canonically isomorphic to (𝔥̄/𝔥)^{⋆} , see [ K3, Prop.7.4.1 and the next remarks], and via this isomorphism the invariant cross-section ε^{⋆} ∈ (Secggg^{⋆})_{I^{o}} is mapped into the element ε_{o}^{⋆} ∈ (𝔥̄/𝔥)^{⋆} such that ι_{ε_{o}}ε_{o}^{⋆} = −1], we have h_{A}^{(2)} (ε^{⋆}) ≠ 0. Adding the simple connectedness to the properties of G, we obtain the looked-for objects.

4.2. s-Lie algebroids of rank 3. Consider an s-Lie algebroid (A, ε) of rank 3, ε ∈ (Sec ⋀^{3} ggg)_{I^{0}}. According to remark after Th. 1.2, after writting ε = h_{1} ∧ h_{2} ∧ h_{3} (locally), h_{i} ∈ Secggg, we have

[[ξ, h_{1}]] ∧ h_{2} ∧ h_{3} + h_{1} ∧ [[ξ, h_{2}]] ∧ h_{3} + h_{1} ∧ h_{2} ∧ [[ξ, h_{3}]] = 0, ξ ∈ Sec A.

Analogously as in the previous section, take ε^{⋆} ∈ Sec ⋀^{3} ggg^{⋆} such that ι_{ε}ε^{⋆} = 1. Then ε^{⋆} is ⋀^{3} ad_{A}^{♮}-invariant. Indeed, it is enough to notice that

(γ ∘ ξ) ⟨ε^{⋆}, h_{1} ∧ h_{2} ∧ h_{3}⟩ −

− ⟨ε^{⋆}, [[ξ, h_{1}]] ∧ h_{2} ∧ h_{3} + h_{1} ∧ [[ξ, h_{2}]] ∧ h_{3} + h_{1} ∧ h_{2} ∧ [[ξ, h_{3}]]⟩ = 0

Since the first and the second cohomology groups of the Lie algebra ggg_{∣x} are 0, we have the canonical isomorphism [ G-H-V, Vol.III]

ρ_{x} : (⋁^{2}ggg_{∣x}^{⋆})_{I} ⟶^{≅} ⋀^{3}ggg_{∣x}^{⋆}

(each element of ⋀^{3} ggg_{∣x}^{⋆} is invariant here) given by the formula

⟨ρ_{x} (Ψ) , x ∧ y ∧ z⟩ = ⟨Ψ, [x, y] ∨ z⟩.

We recall that Ψ ∈ ⋁^{2} ggg_{∣x}^{⋆} is invariant if and only if ⟨Ψ, [x, y]∨z⟩+⟨Ψ, [x, z]∨y⟩ = 0

for any x, y, z ∈ ggg_{∣x}. By this and the fact that ggg is an LAB, the sum (⋁^{2} ggg^{⋆})_{I} :=

⋃_{x∈M} (⋁^{2} ggg_{∣x}^{⋆})_{I} ⊂ ⋁^{2} ggg^{⋆} is a smooth subbundle of ⋁^{2} ggg^{⋆}. Of course, any ⋁^{2} ad_{A}^{♮}-

invariant cross-section Γ ∈ (Sec ⋁^{2} ggg^{⋆})_{I^{0}} is a cross-section of this subbundle. All the isomorphisms ρ_{x} induce the linear smooth homomorphism

ρ : (⋁^{2}ggg^{⋆})_{I} ⟶ ⋀^{3}ggg^{⋆}.

of vector bundles.

Proposition 4.1. The homomorphism Sec ρ induced by ρ on invariant cross-sec-tions is an isomorphism

Sec ρ : (Sec ⋁^{2}ggg^{⋆})_{I^{0}} ⟶^{≅} (Sec ⋀^{3}ggg^{⋆})_{I^{0}} .

Proof. Let Γ ∈ Sec ⋁^{2} ggg^{⋆}. Then ρ ∘ Γ is invariant if and only if so is Γ. Indeed, since ggg is an LAB and ggg_{∣x} = ggg_{∣x}^{2}, then each cross-section h ∈ Secggg is locally a sum of cross-sections of the form [h_{α}, h_{β}] . The rest follows trivially from the Jacobi identity in Sec A.

Take Γ ∈ (Sec ⋁^{2} ggg^{⋆})_{I^{0}} such that

ρ (Γ) = ε^{⋆}.

Theorem 4.3. Under the above assumptions,

χ_{A} = h_{A}^{(4)} (−2Γ) .

Proof. According to the definition of h_{A} we have

h_{A}^{(4)} (−2Γ) = [⟨−Γ, Ω_{b} ∨ Ω_{b}⟩].

To prove our theorem, let us take the 3-form Φ ∈ Ω_{A}^{3} (M) defined by

Φ = ⟨−Γ, ω ∨ d_{A}ω⟩ − \frac{1}{3}⟨−Γ, ω ∨ [[ω, ω]]⟩.

According to (4.3),

γ^{⋆}⟨−Γ, Ω_{b} ∨ Ω_{b}⟩ = d_{A}Φ.

Therefore it is suffices to check that ∫̸_{A}Φ = −1, i.e. γ^{⋆} (∫̸_{A}Φ) = ι_{ε}Φ = −1. Since d_{A}ω (h_{i}, h_{j}) = [h_{i}, h_{j}] and [[ω, ω]] (h_{i}, h_{j}) = 2 [h_{i}, h_{j}] and ω (h_{i}) = h_{i}, we obtain

ι_{ε} (ω ∨ d_{A}ω − \frac{1}{3}ω ∨ [[ω, ω]])

= (ω ∨ d_{A}ω − \frac{1}{3}ω ∨ [[ω, ω]]) (h_{1} ∧ h_{2} ∧ h_{3})

= \frac{1}{3} (h_{1} ∨ [h_{2}, h_{3}] − h_{2} ∨ [h_{1}, h_{3}] + h_{3} ∨ [h_{1}, h_{2}]) .

Therefore

ι_{ε}Φ = ⟨−Γ, ι_{ε} (ω ∨ d_{A}ω − \frac{1}{3}ω ∨ [[ω, ω]])⟩

= \frac{1}{3}⟨−Γ, [h_{2}, h_{3}] ∨ h_{1} − [h_{1}, h_{3}] ∨ h_{2} + [h_{1}, h_{2}] ∨ h_{3}⟩

= \frac{1}{3}⟨−ρ (Γ) , h_{2} ∧ h_{3} ∧ h_{1} − h_{1} ∧ h_{3} ∧ h_{2} + h_{1} ∧ h_{2} ∧ h_{3}⟩

= ⟨−ε^{⋆}, h_{1} ∧ h_{2} ∧ h_{3}⟩

= −1.

5. Computation of the algebra H_{A} (M)

For an arbitrarily taken element Ω ∉ ℝ to which we assign the degree n, we form the anticommutative graded algebra (⋀ Ω, ∧) ≅ H (S^{n}) as the exterior algebra over the 1-dimensional graded vector space Lin Ω (homogeneous of degree n) spanned by Ω.

Below, n = 1 or n = 3, therefore n is odd (see Prop. 2.1).

Theorem 5.1. Let (A, ε) be any s-Lie algebroid of rank n and let Ψ ∈ Ω^{n+1} (M) be a representative of the Euler class χ_{A} = [Ψ] . According to Remark 3.1, we can take an n-form Ω ∈ Ω_{A}^{n} (M) such that ∫̸_{A}Ω = −1 and d_{A}Ω = γ^{⋆}Ψ. Consider the skew tensor product Ω (M) ⨂ ⋀ Ω of anticommutative graded algebras and define the operator d : Ω (M) ⨂ ⋀ Ω → Ω (M) ⨂ ⋀ Ω by the formulae

\begin{aligned}d (ψ ⊗ 1) & = & d_{M}ψ ⊗ 1 \\ d (ψ ⊗ Ω) & = & (−1)^{deg ψ} (ψ ∧ Ψ) ⊗ 1 + d_{M}ψ ⊗ Ω. \end{aligned}

Then d is an antiderivation and (Ω (M) ⨂ ⋀ Ω, d) is a graded differential algebra.

Proof. Clearly, d is a differential of degree +1, such that d (x ∧ y) = dx ∧ y + (−1)^{deg x} x ∧ dy for x = ψ_{1} ⊗ 1, y = ψ_{2} ⊗ 1 or for x = ψ_{1} ⊗ 1, y = ψ_{2} ⊗ Ω. The remaining third case: x = ψ_{1} ⊗ Ω, y = ψ_{2} ⊗ Ω holds (n is odd), giving zero on both sides of this equality:

d (ψ_{1} ⊗ Ω ∧ ψ_{2} ⊗ Ω) = d ((−1)^{n·deg ψ_{2}} ψ_{1} ∧ ψ_{2} ⊗ Ω ∧ Ω) = d (0) = 0,

d (ψ_{1} ⊗ Ω) ∧ ψ_{2} ⊗ Ω + (−1)^{deg ψ_{1}+n} ψ_{1} ⊗ Ω ∧ d (ψ_{2} ⊗ Ω)

= (−1)^{deg ψ_{1}} (ψ_{1} ∧ Ψ) ⊗ 1 ∧ ψ_{2} ⊗ Ω + (−1)^{deg ψ_{1}+deg ψ_{2}+n} ψ_{1} ⊗ Ω ∧ ψ_{2} ∧ Ψ ⊗ 1

= (−1)^{deg ψ_{1}} (1 + (−1)^{n^{2}}) ψ_{1} ∧ Ψ ∧ ψ_{2} ⊗ Ω

= 0 .

Theorem 5.2. The mapping

μ : Ω (M) ⨂ ⋀ Ω ⟶ Ω_{A} (M)

defined by μ (ψ ⊗ 1) = γ^{⋆}ψ, μ (ψ ⊗ Ω) = γ^{⋆}ψ ∧ Ω, is a degree 0 homomorphism of graded algebras.

Proof. The homogeneity of μ and the commutativity of μ with differentials d_{A} and d, as well as the condition μ (x ∧ y) = μ (x) ∧ μ (y) for x = ψ_{1} ⊗ 1, y = ψ_{2} ⊗ 1 or x = ψ_{1} ⊗ Ω and y = ψ_{2} ⊗ Ω is easy to see. This condition for x = ψ_{1} ⊗ Ω and y = ψ_{2} ⊗ Ω follows from the observation that Ω ∧ Ω = 0 in ⋀ Ω, and Ω ∧ Ω = 0 in Ω (M) (n is odd).

Theorem 5.3. For an s-Lie algebroid (A, ε) over a manifold M, the mapping induced by μ on cohomology

μ_{#} : H (Ω (M) ⨂ ⋀ Ω) ⟶ H_{A} (M)

is an isomorphism of graded algebras.

Proof. Consider the short exact sequence

0 ⟶ Ω (M) ⟶^{i} Ω (M) ⨂ ⋀ Ω ⟶^{ρ} Ω (M) ⟶ 0,

i (ψ) = ψ ⊗ 1, ρ (ψ_{1} ⊗ 1 + ψ_{2} ⊗ Ω) = −ψ_{2}. i and ρ commute with differentials. The following diagram

0 ⟶ Ω (M) ⟶^{i} Ω (M) ⨂ ⋀ Ω ⟶^{ρ} Ω (M) ⟶ 0
↓^{γ^{⋆}}  ↓^{μ}  ∥
0 ⟶ ker ∫̸_{A} ⟶ Ω_{A} (M) ⟶^{∫̸_{A}} Ω (M) ⟶ 0 ,

is commutative; indeed,

∫̸_{A} ∘ μ (ψ_{1} ⊗ 1 + ψ_{2} ⊗ Ω)
= ∫̸_{A} (γ^{⋆}ψ_{1} + γ^{⋆}ψ_{2} ∧ Ω) = 0 + ψ_{2} ∧∫̸_{A}Ω = −ψ_{2}
= ρ (ψ_{1} ⊗ 1 + ψ_{2} ⊗ Ω) ,
μ ∘ i (ψ) = μ (ψ ⊗ 1) = γ^{⋆}ψ .

Passing to the long exact sequences, we get from the five-lemma (since γ^{#} is an isomorphism [Th. 2.1]) that μ_{#} is an isomorphism.

Corollary 5.1. If χ_{A} = 0, then H_{A} (M) ≅ H (M) ⨂ H (S^{n}) as graded algebras.

Proof. Let χ_{A} = 0. Then we can choose Ψ = 0 as a representative of the Euler class χ_{A}, which gives d (ψ_{1} ⊗ 1 + ψ_{2} ⊗ Ω) = dψ_{1} ⊗ 1 + dψ_{2} ⊗ Ω, i.e. d is equal to the tensor product of the differentials d_{M} and 0_{⋀ Ω} [ G]. The Künneth theorem yields the isomorphism of graded algebras

H (Ω (M) ⨂ ⋀ Ω) ≅ H (Ω (M)) ⨂ H (⋀ Ω) ≅ H (M) ⨂ H (S^{n}) .

This result and Th.5.3 imply that H_{A} (M) ≅ H (Ω (M) ⨂ ⋀ Ω) ≅ H (M) ⨂ H (S^{n}) as graded algebras.

Theorem 5.4. If dim M = n+1 and M is compact, connected and oriented, then, for arbitrary s-Lie algebroids A and A′ over M of rank n with nonzero Euler classes χ_{A} ≠ 0 ≠ χ_{A′}, there exists an isomorphism of cohomology algebras H_{A} (M) ≅ H_{A′} (M) .

Proof. Take the orientation class [Δ] ∈ H^{n+1} (M) of M. Then χ_{A} = k · [Δ] = [k · Δ] and χ_{A′} = k′ · [Δ] = [k′ · Δ] for some reals k, k′ ∈ ℝ\ {0} . Therefore Ψ = k · Δ and Ψ′ = k′ · Δ are representatives of the Euler classes. According to Remark 3.1, there exist n-forms Ω ∈ Ω_{A}^{n} (M) and Ω′ ∈ Ω_{A′}^{n} (M) such that ∫̸_{A}Ω = −1, ∫̸_{A′}Ω′ = −1 and d_{A}Ω = γ^{⋆}Ψ, d_{A′}Ω′ = γ^{′⋆}Ω′. Theorem 5.3 says that H_{A} (M) ≅ H (Ω (M) ⨂ ⋀ Ω) and H_{A′} (M) ≅ H (Ω (M) ⨂ ⋀ Ω′) provided that in Ω (M) ⨂ ⋀ Ω and Ω (M) ⨂ ⋀ Ω′ the following differentials d_{k} and d_{k′} are defined:

d_{k} (ψ ⊗ 1) = d_{M}ψ ⊗ 1
d_{k} (ψ ⊗ Ω) = d_{M}ψ ⊗ Ω + (−1)^{deg ψ} (ψ ∧ Ψ) ⊗ 1

= \{ \begin{matrix}d_{M}ψ ⊗ Ω, & deg ψ > 0, \\ (ψ · Ψ) ⊗ 1 + d_{M}ψ ⊗ Ω, & deg ψ = 0, \end{matrix}

analogously for d_{k′}.

Put

I : Ω (M) ⨂ ⋀ Ω ⟶ Ω (M) ⨂ ⋀ Ω′,
ψ ⊗ 1 ⟼ ψ ⊗ 1,
1 ⊗ Ω ⟼ 1 ⊗ \frac{k}{k′} Ω′,
ψ ⊗ Ω ⟼ ψ ⊗ Ω′, deg ψ > 0.

Of course, I is an isomorphism of graded vector spaces. We check the commutativity of I with differentials:

• d_{k′} ∘ I (ψ ⊗ 1) = d_{k′} (ψ ⊗ 1) = dψ ⊗ 1 = I ∘ d_{k} (ψ ⊗ 1) ,

• d_{k′} ∘ I (1 ⊗ Ω) = d_{k′} (1 ⊗ \frac{k}{k′} Ω′) = \frac{k}{k′} · Ψ′ ⊗ 1 = Ψ ⊗ 1 = I ∘ d_{k} (1 ⊗ Ω) ,

• if deg ψ > 0, then d_{k′} ∘ I (ψ ⊗ Ω) = d_{k′} (ψ ⊗ Ω′) = dψ ⊗ Ω′ = I ∘ d_{k} (ψ ⊗ Ω) .

Therefore I_{#} : H (Ω (M) ⨂ ⋀ Ω) ⟶^{≅} H (Ω (M) ⨂ ⋀ Ω′) is an isomorphism of cohomology algebras which implies that H_{A} (M) ≅ H_{A′} (M) .

Remark 5.1. According to the above theorem, the Euler class χ_{A} is not – in general – an invariant of the cohomology algebra of A and according to [ K7, Prop. 7.6.] has nothing in common with the Euler-Poincaré characteristic of A; the last, when considered for TUIO-Lie algebroids (dim M + rankggg is odd), always is 0.

Acknowledgments. I would like to thank Martin Bordemann (Freiburg) for assistance, advice and some interesting helpful discussions, especially concerning Prop. 2.1 (more precisely, for the point up to me the Hochschild-Serre formula to prove this proposition).

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JAN KUBARSKI Institute of Mathematics Technical University of Łódź Al. Politechniki 11 PL-90-924 Łódź POLAND

and

Institute of Mathematics and Informatics Czȩstochowa Technical University ul. Da̧browskiego 69 PL-42-201 Czȩstochowa POLAND

   E-mail address: kubarski@ck-sg.p.lodz.pl