Cohomology of flat connections

Jan Kubarski Institute of Mathematics, Technical University of Łódź PL-90-924 Łódź, al. Politechniki 11, Poland E-mail: kubarski@ck-sg.p.lodz.pl

April 25, 2001

Abstract

We study flat connections in spherical Lie algebroids (over oriented compact manifolds) defined everywhere but a finite number of points. Under some assumptions concerning dimensions with any such isolated singularity we join a real number called an index. For ℝ-spherical Lie algebroids, this index cannot be an integer. We prove the index theorem saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indexes are given.

1 Introduction

Lie algebroids arise in many subjects of differential geometry and play a role analogous to that of Lie algebras for Lie groups (i.e. compose an infinitesimal invariant). For example, they arise in the theory of differential groupoids, principal bundles and vector bundles [ P], [ L2], [ K-S], [ M1], [ K2], [ N1], [ N2], transversally complete and transversally parallelizable foliations [ MO1], [ MO2], nonclosed Lie subgroups [ MO2], [ K4], Poisson manifolds [ C-D-W], [ D-S], [ G1], [ V1], [ V2], [ Ko1], [ Ko2] and others (see the survey article by K.Mackenzie [ M2], and also see J.Kubarski [ K6]). Connections – splittings of the Atiyah sequences of Lie algebroids (in regular case) – correspond in the majority of the above categories to well known geometric objects such as distributions or differential systems.

On the ground of Lie algebroids, we observe an interesting analogy of the theory of sphere bundles, namely, it turns out that, in some sense, the roles of flat connections for Lie algebroids and of cross-sections for sphere bundles correspond mutually. The common ideas are the index at a singularity and the theorem of Euler-Poincaré-Hopf type as well as some technique methods. This analogy was first noticed for Lie algebroids with 1-dimensional isotropy Lie algebras in the geometry of regular Poisson manifolds over ℝ-Lie foliations [ K8]. The main purpose of our work is to research this phenomenon in the domain of transitive Lie algebroids without a restriction of the dimensions of isotropy Lie algebras.

This paper is based on [ K7] and [ K10]. In [ K7] the idea of the fibre integral is adopted to regular Lie algebroids: the integration operator over the adjoint bundle of Lie algebras is defined, the class of Lie algebroids for which this operator commutes with differentials (giving then a homomorphism on cohomology) is characterized and many families of examples coming from principal bundles, TC-foliations and Poisson manifolds are given. Paper [ K10] deals with a subclass of the above class which contains so-called s-Lie algebroids, defined as the transitive ones with spherical isotropy Lie algebras i.e. cohomologically looking like a sphere (Lie algebras ℝ, sk (3, ℝ), sl (2, ℝ) are examples). For an s-Lie algebroid there is constructed a long exact sequence of cohomology (Gysin types) and the Euler class.

In this paper 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. In the context of Lie algebroids coming from S^{1} or Spin (3) principal bundles (over M^{2} or M^{4}, respectively) the above theorem generalizes the classical E-P-H theorem since cross-sections of these bundles determine flat connections (but not every connection determines a cross-section). In the end, some integral formulae for indexes are obtained. We add that in general the index can not be an integer and the sum of indexes has nothing in common with the Euler-Poincaré characteristic of the Lie algebroid understood as an alternative sum of the dimensions of the cohomology groups of this Lie algebroid.

2 Fibre integral and Gysin sequence

By a Lie algebroid on a manifold M [ P] we mean a system A = (A, [[·, ·]], γ) consisting of a vector bundle A on M and mappings [[·, ·]] : Sec A × Sec A → Sec A, γ : A → T M, such that (1) (Sec A, [[·, ·]]) is an ℝ-Lie algebra, (2) γ, called the anchor, is a homomorphism of vector bundles, (3) Sec γ : Sec A → 𝔛(M), ξ ↦ γ ∘ ξ, is a homomorphism of Lie algebras,

(4) [[ξ, f · η]] = f · [[ξ, η]] + (γ ∘ ξ)(f) · η, ξ, η ∈ Sec A, f ∈ C^{∞}(M). The properties make the space of global cross-sections Sec A a Lie-Rinehart algebra [called also a Lie module, a Lie pseudoalgebra, etc, according to particular authors] over the commutative algebra C^{∞} (M) [ H1]. A Lie algebroid A is said to be transitive if γ is an epimorphism of vector bundles, and regular if γ is of constant rank. When the isotropy Lie algebras 𝔤_{|x} of A are isomorphic to a given Lie algebra 𝔤 then A is shortly called a 𝔤-Lie algebroid. In the sequel, the notions and the notations from [ P], [ M1], [ K3], [ M-R], [ K1], [ M1] are adopted. Among them, the adjoint bundle of Lie algebras ggg := Ker γ, the Atiyah sequence 0 → ggg → A ⟶^{γ} F → 0, the notions of representations, connections, homomorphisms (strong or non-strong) of Lie algebroids as well as the exterior derivative d_{A} and the pullback of A-differential forms, are used.

In [ K7] we introduce the notion of a vertically oriented Lie algebroid as a pair (A, ε) consisting of a regular Lie agebroid A and a non-singular cross-section ε of ⋀^{n} ggg, n = rankggg. By a homomorphism of vertically oriented Lie algebroids (A, ε) → (A′, ε′) , assuming rankggg = rankggg′ = n, we mean a non-strong, in general, homomorphism T : A → A′, inducing t : M → M′, of Lie algebroids, such that (⋀^{n} T^{+}) (ε_{x}) = ε′_{tx}, x ∈ M (where T^{+} : ggg → ggg′ is the restriction of T to adjoint bundles). We write (T, t) : (A, ε) → (A′, ε′).

An operator of the fibre integral ∫̸_{A} in a vertically oriented Lie algebroid (A, ε) is introduced. For a transitive Lie algebroid (the case considered in this work), the fibre integral ∫̸_{A} : Ω_{A}^{⋆} (M) → Ω^{⋆−n} (M) is defined in the following way: ∫̸_{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 real A-differential forms, i.e. the space Sec ⋀ A^{⋆} of cross-sections of the bundle ⋀ A^{⋆}. The following are basic properties (a) ∫̸_{A} ∘ γ^{⋆} = 0, (b) ∫̸_{A}γ^{⋆}ψ ∧ Φ = ψ ∧ ∫̸_{A}Φ for arbitrary forms ψ ∈ Ω (M) and Φ ∈ Ω_{A} (M) , (c) ∫̸_{A} is an epimorphism [ K7].

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. The transitive Lie algebroid A fulfilling properties (a1) and (a2) above is shortly called a TUIO-Lie algebroid. In [ K7] and [ K9] many examples can be found. The paper [ K10] deal with the subcategory of TUIO-Lie algebroids for which isotropy Lie algebras 𝔤 are spherical, i.e. satisfy conditions H^{k} (𝔤) = ℝ for k = 0, dim 𝔤, and H^{k} (𝔤) = 0 for 1 ≤ k ≤ dim 𝔤 − 1. Such Lie algebroids are called briefly s-Lie algebroids. Many examples for principal bundles and TC-foliations are given. For an s-Lie algebroid (A, ε) , there is constructed a long exact sequence of cohomology (Gysin sequence)

⋯ ⟶ 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}} ⋯  (2.1)

The class χ_{(A,ε)} := D^{0} (1) ∈ H^{n+1} (M) is called the Euler class of (A, ε) and χ_{(A,ε)} = [Ψ] where γ^{∗}Ψ = d_{A}Φ for ∫̸_{A}Φ = −1. The equality D (α) = α ∧ χ_{(A,ε)} holds. The Euler class χ_{(A,ε)} can be computed via the Chern-Weil homomorphism of A (introduced in [ K3]).

3 Difference class

By a connection in a transitive Lie algebroid A we mean a splitting λ : T M → A of the Atiyah sequence 0 → ggg ↪ A →^{γ} T M → 0, i.e. a homomorphism of vector bundles such that γ ∘λ = id. If λ is a homomorphism of Lie algebroids λ∘[X, Y] = [[λ∘X, λ∘Y]], X, Y ∈ 𝔛 (M) , then λ is called flat. In this situation, the pullback of differential forms λ^{⋆} : Ω_{A} (M) → Ω (M) commutes with differentials λ^{⋆} ∘ d_{A} = d_{M} ∘ λ^{⋆}, giving — on cohomology — a homomorphism of algebras λ^{#} : H_{A} (M) → H (M) . Let λ be a flat connection in an s-Lie algebroid (A, ε). According to the exactness of sequence (2.1), [ K7, Prop. 4.2.1 (b)] and [ K10, Cor. 3.4] we can easily show that

H_{A} (M) = ker∫̸_{A}^{#} ⨁ ker λ^{#},  (3.1)

∫̸_{A}^{#}| ker λ^{#} : ker λ^{#} ⟶^{≅} H (M) .  (3.2)

By the above, there exists a uniquely determined cohomology class

ω_{λ} ∈ ker λ^{#n} ⊂ H_{A}^{n} (M)

such that ∫̸_{A}^{#}ω_{λ} = 1 ∈ H^{0} (M). ω_{λ} depends on the mapping λ^{#} : H_{A} (M) → H (M) only. The class ω_{λ} is called the cohomology class of a flat connection λ.

For two flat connections λ, σ : T M → A, their cohomology classes ω_{λ}, ω_{σ} ∈ H_{A}^{n} (M) satisfy the equality ∫̸_{A}^{#} (ω_{λ} − ω_{σ}) = 0. Thanks to the exactness of sequence (2.1) there exists a cohomology class [λ, σ] ∈ H^{n} (M) such that

ω_{λ} − ω_{σ} = γ^{#} [λ, σ] .

Definition 3.1 The class [λ, σ] is called the difference class of flat connections λ and σ in the s-Lie algebroid (A, ε) .

Proposition 3.2 For flat connections λ and σ in an s-Lie algebroid (A, ε) , we have λ^{#}α − σ^{#}α = − (∫̸_{A}^{#}α) ∧ [λ, σ] , α ∈ H_{A} (M) .

Since γ^{#} is a monomorphism for a flat Lie algebroid A, we obtain

Corollary 3.3 Let λ and σ be two flat connections in an s-Lie algebroid A. Then the following conditions are equivalent: (a) λ^{#} = σ^{#}, (b) ω_{λ} = ω_{σ}, (c) [λ, σ] = 0. If H^{n} (M) = 0, then, clearly, H^{n} (M) ∋ [λ, σ] = 0, therefore λ^{#} = σ^{#}.

Lemma 3.4 Let (A, ε) be an arbitrary s-Lie algebroid with rankggg = n. For a representative Ψ of the Euler class χ_{(A,ε)} and an n-form Φ ∈ Ω_{A}^{n} (M) such that ∫̸_{A}Φ = −1 and d_{A}Φ = γ^{⋆}Ψ, for any open subset U ⊂ M and two flat connections λ, σ : T U → A_{|U}, the following equalities hold:

(1) ω_{σ} = [γ_{|U}^{⋆}σ^{⋆} (Φ_{|U}) − Φ_{|U}],

(2) [λ, σ] = [λ^{⋆}Φ_{|U} − σ^{⋆}Φ_{|U}].

Theorem 3.5 (The naturality of the difference class) Let (T, t) : (A, ε) → (A′, ε′) be a homomorphism of s-Lie algebroids such that T_{x} : A_{∣x} → A′_{∣tx}, x ∈ M, are isomorphisms.

(a) Assume that σ, σ′ are flat connections in A and A′, respectively, such that T ∘σ = σ′∘t_{∗}. Then T^{#}ω_{σ}′ = ω_{σ}.

(b) For any two pairs of such flat connections (λ, λ′) , (σ, σ′) , we obtain t^{#}([λ′, σ′]) = [λ, σ] .

Proof. (a): To check (a), it is sufficient to notice that σ^{#} (T^{#}ω_{σ}′) = (T ∘ σ)^{#} ω_{σ}′ = (σ′ ∘ t_{⋆})^{#} ω_{σ}′ = t^{#}σ^{′#}ω_{σ}′ = 0 and (see [ K7]) ∫̸_{A}^{#}T^{#}ω_{σ}′ = t^{#}∫̸_{A′}^{#}ω_{σ}′ = 1.

(b): According to the fact that γ^{#} is a monomorphism, we only need to notice

γ^{#}t^{#} [λ′, σ′] = (t_{⋆} ∘ γ)^{#} [λ′, σ′] = (γ′ ∘ T )^{#} ([λ′, σ′]) = T^{#} (ω_{λ}′ − ω_{σ}′)
= ω_{λ} − ω_{σ} = γ^{#} [λ, σ] .

The folowing theorem gives a relationship between the Euler class and the difference class (compare with the classical theorem for sphere bundles, for example [ G-H-V]).

Theorem 3.6 Let {U, V } be an open covering of M and let λ_{U} : T U → A_{|U} and σ_{V} : T V → A_{|V} be flat connections in (A, ε) over U and V, respectively (U, V need not be connected). Consider the Mayer-Vietoris sequence of the triad {M, U, V } for the usual real de Rham cohomology and let ∂ : H (U ∩ V) → H (M) be the connecting homomorphism. Then

χ_{(A,ε)} = ∂ [λ, σ]

where λ = λ_{U}|_{U∩V} and σ = σ_{V} |_{U∩V} .

Proof. For the inclusions j_{1} : U ∩ V ↪ U and j_{2} : U ∩ V ↪ V according to Lemma 3.4, [λ, σ] = [λ^{⋆}Φ_{|U∩V} − σ^{⋆}Φ_{|U∩V} ] = [j_{1}^{⋆}(λ_{U}^{⋆}Φ_{|U}) − j_{2}^{⋆}(σ_{V}^{⋆} Φ_{|V})]. Since d(λ_{U}^{⋆}Φ_{|U}) = λ_{U}^{⋆}d_{A_{|U}} Φ_{|U} = λ_{U}^{⋆}γ_{|U}^{⋆}Ψ_{|U} = Ψ_{|U}, analogously d(σ_{V}^{⋆} Φ_{|V}) = Ψ_{|V} , we get — via the construction of ∂ — ∂ [λ, σ] = [Ψ] = χ_{(A,ε)}.

4 The index of a flat connection at an isolated singular point and the Euler number

By a local connection with singularity at a point a ∈ M in a Lie algebroid A we mean the connection

σ : T U̇ → A_{|U̇}, a ∈ U ⊂ M (U is open), U̇ = U \ {a} .

Let (A, ε) be an arbitrary s-Lie algebroid over an n + 1-dimensional oriented manifold M (n ⩾ 1) with n = rankggg and let σ : T U̇ → A_{|U̇} be a local connection with singularity at a ∈ U ⊂ M. Take additionally a neighbourhood V ∋ a such that V ⊂ U and V ≅ ℝ^{n+1}. A_{|V} possesses, [ M1], a global flat connection λ : T V → A_{|V} . Denote λ|V̇ (V̇ = V \ {a})

by λ̇ and consider the difference class [λ̇, σ|V̇] ∈ H^{n}(V̇). Let α_{V} : H^{n}(V̇) ⟶^{≅} ℝ be the canonical mapping [ G-H-V, Vol.I] (V̇ has the orientation induced from M). By analogous reasoning as in the theory of sphere bundles [ G-H-V, Vol.I] and due to Corollary 3.3 we check that the number α_{V} ([λ̇, σ|V̇]) is independent of the auxiliary flat connection λ and of the neighbourhood V. This means α_{V} ([λ̇, σ|V̇]) depends only on the choice of σ.

Definition 4.1 The number α_{V} ([λ̇, σ|V̇]) is called the index of σ at a and denoted by

j_{a} (σ) .

Proposition 4.2 (Naturality of the index) Let (Â, ε̂) be another s-Lie algebroid over an oriented n + 1-dimensional manifold M̂ and (T, t) : (Â, ε̂) → (A, ε) be a homomorphism of s-Lie algebroids fulfilling conditions

T_{x} : Â_{∣x} → A_{∣tx}, x ∈ M, is an isomorphism,

t : M̂ → M is a diffeomorphism onto an open subset.

Let a ∈ M, â ∈ M̂, t \binom{}{â} = a. Take a local flat connection σ : T U̇ → A_{|U̇} with singularity at a. Then the mapping T^{#}σ : T Ẇ → Â_{|Ẇ} , W = t^{−1} [U] , Ẇ = W \ \left\{{ \atop â}\right\} , defined by (T^{#}σ) (v) = T_{|pv}^{−1} (σ (t_{⋆}v)) is a flat connection in Â with singularity at â, and jâ (T^{#}σ) = j_{a} (σ) .

The main goal of this article is a theorem joining the index sum ∑ j_{a_{v}} (σ) of any flat connection with a finite number of singularities {a_{1}, …, a_{k}} to the Euler class of Lie algebroid.

Theorem 4.3 (The Euler-Poincaré-Hopf theorem for flat connections) Let (A, ε) be an s-Lie algebroid of rank n over an oriented compact manifold M of dimension n + 1 and let σ : T (M \ {a_{1}, …, a_{k}}) → A be a flat connection with singularities at points a_{1}, …, a_{k}. Then the Euler class χ_{(A,ε)} ∈ H^{n+1} (M) is given by the formula

χ_{(A,ε)} = (∑_{v=1}^{k} j_{a_{v}} (σ)) · ω_{M}

where ω_{M} is the orientation class of M; equivalently, ∫_{M}^{#} χ_{(A,ε)} = ∑_{v=1}^{k} j_{a_{v}} (σ). In particular, the index sum ∑_{v=1}^{k} j_{a_{v}} (σ) is independent of the choice of the connection.

Proof. For each v = 1, …, k, choose a neighbourhood U_{v} ∋ a_{v} diffeomorphic to ℝ^{n+1} and such that the sets U_{v} are pairwise disjoint. Put U = ⋃ U_{v}, V = M \ {a, …, a_{k}} . Then M = U ∪ V and U ∩ V = ⋃ U̇_{v} where U̇_{v} = U_{v} \ {a_{v}} . Take arbitrary flat connections λ̃_{v} : T U_{v} → A_{|U_{v}}, v = 1, …, k. The family {λ̃_{v}} determines one flat connection λ̃ : T U → A_{|U} such that λ̃|_{U_{v}} = λ̃_{v}. Define λ̌ = λ̃|_{U∩V} and σ̌ = σ|_{U∩V} . According to Theorem 3.6, χ_{(A,ε)} = ∂[λ̌, σ̌]. In the sequel, put λ_{v} = λ̃_{v}|U̇_{v} and σ_{v} = σ|U̇_{v}. Then [λ̌, σ̌] = ⊕_{v} [λ_{v}, σ_{v}] . By [ G-H-V, Prop.VII Chap.VI, Vol.I] ∫_{M}^{#} ∘∂ = α, where α : ⨁_{v} H^{n}(U̇_{v}) → ℝ is equal to ⊕β_{v} ↦ ∑ α_{U_{v}} (β_{v}) . Therefore we get

∫_{M}^{#} χ_{(A,ε)}_{A} = ∫_{M}^{#} ∂[λ̌, σ̌] = ∫_{M}^{#} ∂ (⊕_{ν} [λ_{v}, σ_{v}]) = α (⊕_{ν} [λ_{v}, σ_{v}])

= ∑_{v=1}^{k} α_{U_{v}} [λ_{v}, σ_{v}] = ∑_{v=1}^{k} j_{a_{v}} (σ) .

The sum

j_{(A,ε)} = ∑_{v=1}^{k} j_{a_{v}} (σ)

is called the Euler number of the s-Lie algebroid (A, ε) . According to Theorem 5.4 from [ K10], the Euler number j_{(A,ε)} is not – in general – an invariant of the cohomology algebra of A and 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 [ K9].

5 Integral formulae

Proposition 5.1 For a trivial s-Lie algebroid A = T M × 𝔤 vertically oriented by a tensor 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤 (n = dim 𝔤) and equipped with a ”standard” flat connection τ_{0} : T M → T M×𝔤, v ↦ (v, 0) , we have

(a) if σ : T M → A is a flat connection, then its cohomology class ω_{σ} is given by ω_{σ} = −σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}] where σ̂ = pr_{2} ∘σ : T M → 𝔤 and φ_{o} ∈ ⋀^{n} 𝔤^{⋆} is a tensor such that ι_{ε_{o}}φ_{o} = 1. In particular ω_{τ_{0}} = 1 × [φ_{o}] .

(b) The difference class [τ_{0}, σ] is equal to [τ_{0}, σ] = σ̂^{#} [φ_{o}] .

Proof. Using the fact that the projection pr_{2} : T M → 𝔤 is a nonstrong homomorphism of Lie algebroids [ K5], we can easily see that

σ^{#} (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = 0,
∫̸_{A}^{#} (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = 1.

Now (a) follows from the definition of the cohomology class ω_{σ} whereas (b) may be obtained from the definition of the difference class and the equality

ω_{τ_{0}} − ω_{σ} = 1 × [φ_{o}] − (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = γ^{#}σ̂^{#} [φ_{o}] .

Corollary 5.2 For arbitrary flat connections λ, σ : T M → T M × 𝔤,

[λ, σ] = (σ̂^{#} − λ̂^{#}) [φ_{o}] .

Put ℝ^{.}^{n+1} = ℝ^{n+1} \ {0} and let 𝔤 be any n-dimensional unimodular Lie algebra 𝔤. Take tensors 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤, φ_{o} ∈ ⋀^{n} 𝔤^{⋆} joined by the relation ι_{ε_{o}} (φ_{o}) = 1. Fix a flat connection σ : T ℝ^{.}^{n+1} → T ℝ^{.}^{n+1} × 𝔤 in the trivial Lie algebroid A = T ℝ^{n+1} × 𝔤 (oriented by the tensor ε_{o}). Let i : S^{n} ↪ ℝ^{.}^{n+1} be the inclusion.

Proposition 5.3 The index j_{0} (σ) of σ is given by the formula

j_{0} (σ) = ∫_{S^{n}} σ_{S}^{⋆} (φ_{o})  (5.1)

where σ_{S} is a nonstrong homomorphism of Lie algebroids defined as the composition σ_{S} :

T S^{n} ↪^{i_{⋆}} T ℝ^{.}^{n+1} ⟶^{σ̂} 𝔤.

Treat now σ̂ : T ℝ^{.}^{n+1} → 𝔤 as a 1-form on ℝ^{.}^{n+1} with values in 𝔤 and take the exterior

n-product σ̂ ∧ … ∧ σ̂ ∈ Ω^{n}(ℝ^{.}^{n+1}; ⋀^{n} 𝔤). We have σ̂^{⋆} (φ_{o}) = \frac{1}{n!} (φ_{o})_{⋆} (σ̂ ∧ … ∧ σ̂) . Therefore

j_{0} (σ) = \frac{1}{n!}φ_{o} (∫_{S^{n}} i^{⋆} (σ̂ ∧ … ∧ σ̂)) .  (5.2)

Example 5.4 Each trivial Lie algebroid A = T ℝ^{n+1} × 𝔤 is integrable: A = A (P) for P = ℝ^{n+1}×G where G is an arbitrary Lie group with the Lie algebra 𝔤. A connection σ : T ℝ^{.}^{n+1} → A induces a connection H ⊂ T (ℝ^{.}^{n+1}×G) in the principal bundle ℝ^{.}^{n+1}×G, and the flatness of σ means the integrability of H. Assume a leaf L of the foliation H is the graph of some function f : ℝ^{.}^{n+1} → G. (If n ≥ 2, then such a function always exists which follows from the simple connectedness of ℝ^{.}^{n+1} and the reduction theorem [ K-N]). Therefore σ̂ (v) = R_{f(x)⋆}^{−1} (f_{⋆}v) , R_{f(x)} which is the right translation on G by f(x), and f^{⋆} (Δ_{R}) = ⟨φ_{o}, \frac{1}{n!} (σ̂ ∧ … ∧ σ̂)⟩ for the right-invariant n-form Δ_{R} ∈ Ω_{R}^{n} (G) equalling φ_{o} at the unity e of G.

(A) If G is compact, n-dimensional, oriented by Δ_{R} and the Lie algebra of G is spherical, then as a consequence of (5.1) and (5.2) we have

j_{0} (σ) = ∫_{S^{n}} (f_{|S^{n}})^{⋆} Δ_{R} = deg (f_{|S^{n}}) · ∫_{G} Δ_{R} .  (5.3)

As a corollary (taking any mapping f : ℝ^{.}^{n+1} → S^{n} such that f_{|S^{n}} = id_{S^{n}}), we obtain the existence of a local, flat singular connection having a nonzero index at the singularity.

Formula (5.3) yields that the set of real numbers being the indexes at a given point of singular local, flat connections coming from functions is discrete (more exactly, is equal to the set of multiples of ∫_{G} Δ_{R}). Such a situation takes place, for example, for all flat connections in any sk (3, ℝ)-Lie algebroid over M^{4} (since we can take G = SO(3)).

(B) If G is not compact, then Δ_{R} = d (Θ) for some Θ and

j_{0} (σ) = ∫_{S^{n}} (f_{|S^{n}})^{⋆} Δ_{R} = ∫_{S^{n}} d (f_{|S^{n}}^{∗}Θ) = 0.

Such a situation takes place, for example, in any sl (2, ℝ)-Lie algebroid over M^{4} (since we can take G = SL (2, ℝ)). Clearly, this fact can be noticed immediately by using a base e, f, g of sl (2, ℝ) such that [e, f] = g, [f, g] = 2f, [g, e] = 2e.

For an ℝ-Lie algebroid, not every local, singular and flat connection comes from a function, see the below example.

Example 5.5 In any ℝ-Lie algebroid over M^{2} we can construct a local, flat and singular connection whose index is a preassigned real number. Indeed, since ℝ is abelian, therefore the flatness of σ is equivalent to the closedness of the 1-form σ̂ on M^{2}. In this case, the product k · σ̂, k ∈ ℝ, also gives a flat connection. Therefore, if σ : T ℝ^{.}^{2} → T ℝ^{.}^{2} × ℝ, v ↦ (v, σ̂ (v)) , is a flat connection with a nonzero index at 0, j_{0} (σ) ≠ 0, then, for an arbitrary real number k ∈ ℝ, the mapping

τ : T ℝ^{.}^{2} ⟶ T ℝ^{.}^{2} × ℝ, v ⟼ (v, \frac{k}{j_{0} (σ)} · σ̂ (v)) ,

is a flat connection with j_{0} (τ) = k. Except for a discrete set of real numbers, this connection does not come from a function. More implicitly, considering ε_{0} = 1 ∈ ℝ and taking τ̂ = \frac{k}{2π} (\frac{x}{x^{2}+y^{2}} dy − \frac{y}{x^{2}+y^{2}} dx) , k ∈ ℝ, we have j_{0} (τ) = ∫_{S^{1}} τ̂ = k.

Example 5.6 In the Hopf S^{1}-bundle P = (S^{3} → S^{2}), for two different points p_{1}, p_{2} ∈ S^{2} and for any real number k ∈ ℝ, there exists a global flat connection σ_{k} with two singularities at {p_{1}, p_{2}} , such that the index j_{p_{1}} (σ_{k}) is equal to k. Indeed, since the Euler class of P is equal to the orientation class of S^{2}, any flat connection λ with a singularity at {p_{1}} has the index at p_{1} equal to 1 (assuming ∫_{G} Δ_{R} = 1). Take p_{2} ≠ p_{1} and M = S^{2} \ {p_{2}}. Since M is contractible, P_{|M} is trivial P_{|M} ≅ M × S^{1}. The connection λ determines a connection λ̄ : T (M\ {p_{1}}) → A ((M\ {p_{1}}) × S^{1}) = T (M\ {p_{1}}) × ℝ. Take σ̂ = pr_{2} ∘λ̄ : T (M\ {p_{1}}) → ℝ. For an arbitrary real number k ∈ ℝ,

σ̄_{k} : T (M\ {p_{1}}) ⟶ T (M\ {p_{1}}) × ℝ, v ⟼ (v, k · σ̂ (v)),

is a flat connection. σ̄_{k} determines a flat connection σ_{k} in P with a singularity at {p_{1}, p_{2}} , such that j_{p_{1}} (σ_{k}) = k.

In the end we give some remarks concerning the existence of a connection with a finite number of singularities. We start with the case 𝔤 = ℝ.

Proposition 5.7 In each invariantly oriented ℝ-Lie algebroid over an arbitrary manifold M for which H^{2} (M) = 0 there exists a flat connection, in particular, when M is 2-dimensional non-compact.

Proof. According to [ K7], each invariantly oriented ℝ-Lie algebroid A over M is isomorphic to (M × ℝ) ⨁ T M with pr_{2} : (M × ℝ) ⨁ T M → T M as the anchor and the bracket [[·, ·]] defined via some closed real 2-form Ω in the following way: [[(f, X) , (g, Y)]] = (−Ω (X, Y) + ∂_{X}g − ∂_{Y} f, [X, Y]) , X, Y ∈ 𝔛 (M) , f, g ∈ C^{∞} (M). Each connection λ : T M → (M × ℝ) ⨁ T M has the form λ (v) = (λ̄ (v) , v) for a 1-differential form λ̄ ∈ Ω^{1} (M). A simple calculation shows that λ is flat if and only if d \binom{}{λ̄} = Ω. If H^{2} (M) = 0, such a 1-form exists.

As a corollary we get

Corollary 5.8 In each s-Lie algebroid of rank 1 over a compact 2-manifold M there exists a flat connection with a beforehand finite non-empty set of isolated singularities.

If a sk (3, R)-Lie algebroid over a compact 4-manifold comes from a Spin (3)-principal bundle, then - of course - it possesses a flat connection with one singularity (since such a cross-section of the sphere bundle exists [ G-H-V, Vol. I]). In the general case, the problem is open.

The problem for sl (2, ℝ)-Lie algebroids looks differently. Namely, by the main theorem (4.3) and Example 5.4 (B) we have that χ_{(A,ε)} = 0 for any invariantly oriented sl (2, ℝ)-Lie algebroid (A, ε) over a compact connected oriented manifold M^{4}, admitting a flat connection with a finite number of isolated singularities. Really, such a Lie algebroid is flat since locally we can remove a singularity: if σ is a flat connection in T ℝ^{.}^{4} × sl (2, ℝ) then σ is given by a function f : ℝ^{.}^{4} → SL (2, ℝ). Using the fact that the third group of homotopy of SL (2, ℝ) is zero, π_{3} (SL (2, ℝ)) = 0, we can find f̄: ℝ^{4} → SL (2, ℝ) such that f (x) = f̄(x) for ∥x∥ ≥ ε for a given small ε. This implies that we may remove the singularity at 0.

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