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Lecture during the Workshop on Dynamical Systems and Geometry PUC - Rio de Janeiro, Brasil, 31 July - 9 August 1996

INDEX THEOREM FOR FLAT CONNECTIONS WITH SINGULARITIES

JAN KUBARSKI

1. Introduction — Euler-Poincaré-Hopf index theorem

Well known is the index theorem of Euler-Poincaré-Hopf:

Theorem 1. Let E be an n-sphere bundle with connected compact oriented base manifold M of dimension n + 1, such that E is given the local product orientation. Let σ be a cross-section of E with finitely many singularities a_{1}, …, a_{k}. Then the index sum

∑_{ν=1}^{k} j_{a_{ν}} (σ) ,

where j_{a_{ν}} (σ) is the index of σ at a_{ν}, is independent of the choice of a cross-section σ and the Euler class χ_{E} of E is given by

χ_{E} = ∑_{ν=1}^{k} j_{a_{ν}} (σ) · ω_{M},

ω_{M} is the orientation class of M.

Consider now one of the following G-principal bundles P (M, G) : (1): G = S^{1}, dim M = 2, (2): G = Spin (3) (= S^{3}) , dim M = 4. Then P is, in fact, an n-sphere over an n + 1-dimensional manifold

(n = 1 or n = 3). Any global cross-section σ : M → P determines a flat connection H^{σ} ⊂ T P by the formula

H_{|σ(x)·a}^{σ} = R_{a⋆} [Im σ_{⋆x}] , x ∈ M, a ∈ G.

We shall investigate a flat connection H in P with finitely many singularities, i.e. a flat connection in the bundle P restricted to M \

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

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1991 Mathematics Subject Classification. Primary 57 R 19, 55 R 25; Secondary 57 R 20,

Key words and phrases. Lie algebroid, Euler class, index theorem, integration over the fibre, flat connection with singularity.

{a_{1}, …, a_{k}} . For example, a cross-section σ : M \ {a_{1}, …, a_{k}} → P with singularities at a_{1}, …, a_{k} determines a flat connection H^{σ} in P over M \ {a_{1}, …, a_{k}} . For an arbitrary flat connection H with a singularity at a ∈ M, we define a real number j_{a} (H) , called the index of H at a, and prove that the index sum

∑_{ν=1}^{k} j_{a_{ν}} (H)

(for M being connected, compact and oriented) is independent of the choice of the connection and the Euler class χ_{P} of P is equal to χ_{P} = ∑_{ν=1}^{k} j_{a_{ν}} (H) · ω_{M}.

We add that j_{a} (H) need not be an integer!

As an example we can construct a flat connection with two singularities in the Hopf bundle S^{3} → S^{2}, such that the index at one of the singular points is a preassigned real number.

Locally, any principal bundle P is a product P = ℝ^{m} × G. Let dim G = n. Any connection H in P with a singularity at 0 ∈ ℝ^{m} is described via a mapping

σ̂ : T (ℝ^{m} \ {0}) ⟶ 𝔤

(𝔤 is the Lie algebra of G) such that

H_{|(x,e)} = {(v, σ̂ (v)) ; v ∈ T_{x} (ℝ^{m} \ {0})} , x ≠ 0,

and H is flat if and only if σ̂ as a 𝔤-valued 1-form on ℝ^{m} \ {0} has the property

(dσ̂) (X, Y) = − [σ̂X, σ̂Y] .

For an abelian Lie algebra 𝔤, this condition is equivalent to the closedness of the 1-form σ̂.

Assume the compactness of G. Consider now a tensor Δ ∈ ⋀^{n} 𝔤^{⋆} such that ∫_{G} Δ_{R} = 1 for the right-invariant n-form Δ_{R} on G equalling Δ at the unit e ∈ G. Take the tensor ε_{0} ∈ ⋀^{n} 𝔤 fulfilling ⟨Δ, ε_{0}⟩ = 1.

Assumption: m = n + 1, i.e. P = ℝ^{n+1} × G, dim G = n.

For a unimodular Lie algebra 𝔤 and a flat connection H, the nexterior product

σ̂ ∧ … ∧ σ̂ ∈ Ω^{n}(ℝ^{n+1} \ {0}; ⋀^{n} 𝔤)

has the following property: the real n-form

Δ_{⋆} (σ̂ ∧ … ∧ σ̂)

INDEX THEOREM FOR FLAT CONNECTIONS WITH SINGULARITIES 3

is closed and the index j_{0} (H) is given by the following integral formula

j_{0} (H) = \frac{1}{n!} ∫_{S^{n}} Δ_{⋆} (σ̂ ∧ … ∧ σ̂) .

Example. For n = 1, j_{0} (H) = ∫_{S} Δ_{⋆}σ̂ is the ”measure” of the nonexactness of Δ_{⋆}σ̂. The following is an example of a closed 1-form on ℝ^{2} \ {0}

Δ_{⋆}σ̂ = \frac{k}{2π} (\frac{x}{x^{2} + y^{2}} dy − \frac{y}{x^{2} + y^{2}} dx)

determining a flat connection H and such that

j_{0} (H) = ∫_{S} Δ_{⋆}σ̂ = k.

k is here a preassigned real number.

Theorem 2. For an arbitrary principal bundle P (M, G) such that (1) dim M = dim G + 1, M — connected, compact and oriented, G

— compact and not necessarily connected,

(2) Ad_{G} (a) = +1 for a ∈ G,

(3) the Lie algebra 𝔤 of G looks cohomologically as an n-sphere, i.e. H^{p} (𝔤) = 0 for 1 ⩽ p ⩽ n−1, H^{p} (𝔤) = ℝ for p = 0, n (n = dim G) .

[Remark: ℝ and 𝔰𝔬(3) and 𝔰𝔩(2) are examples of such Lie algebras. The following are examples of the Lie groups considered : S^{1}, Spin (3) , SO (3) and a disconnected O (3)].

Then, for any flat connection H in P with finitely many singularities {a_{1}, …, a_{k}} , the index sum

∑_{ν=1}^{k} j_{a_{ν}} (H)

is independent of the choice of the connection H and the so-called Euler class χ_{P} of P is given by

χ_{P} = ∑_{ν=1}^{k} j_{a_{ν}} (H) · ω_{M}.

We add that every principal bundle P possesses the so-called Lie algebroid A (P) = T P/G and let

0 → ggg ↪ A (P) →^{γ} T M → 0, ggg ≅ P ×_{Ad_{G}} 𝔤,

be its Atiyah sequence. Connections H in P correspond bijectively to splittings A (P) ←^{λ} T M of this short sequence (i.e. γ ∘ λ = id) and H is flat if and only if λ is a homomorphism of Lie algebroids λ ∘ [X, Y] = [[λ ∘ X, λ ∘ Y]], X, Y ∈ 𝔛 (M) .

The results presented above are particular cases of the results on the ground of suitable Lie algebroids. The last objects need not be integrable Lie algebroids, i.e. need not come from principal bundles. Therefore our theory has a meaning also outside the theory of principal bundles (for example, on the ground of TC-foliations and, in particular, for nonclosed Lie subgroups). An example of such a Lie algebroid on the ground of which the index theorem holds is:

— the Lie algebroid of the TC-foliation of left cosets of a compact connected Lie group G by a nonclosed Lie subgroup H ⊂ G such that H̄ ≠ G and dim H̄/H = 1. When G is, in addition, simply connected, then this Lie algebroid is nonintegrable.

2. Vertically oriented Lie algebroids

Consider a transitive Lie algebroid A with the Atiyah sequence 0 → ggg ↪ A (P) →^{γ} T M → 0. rankggg (= dimggg_{|x}) is called the rank of A. Let rankggg = n. Take a nowhere vanishing cross-section ε of ⋀^{n} ggg. Clearly, ε gives an orientation of ggg. The pair

(A, ε)

is called a vertically oriented Lie algebroid.

Remark 1. For each vertically oriented Lie algebroid (A, ε) of rank 1, the adjoint bundle ggg is trivial. There exist transitive Lie algebroids of rank 1 which cannot be vertically oriented. It is enough to demand the nontriviality of ggg.

Example 1. Let ggg be any nontrivial vector bundle of rank 1 on a manifold M and let us take an arbitrary flat covariant derivative ∇ on it. In A := ggg ⨁ T M we introduce a structure of a Lie algebroid in the following way:

• the bracket: [[ (σ, X) , (η, Y) ]] = (∇_{X}η − ∇_{Y} σ, [X, Y]) , σ, η ∈ C^{∞} (M,ggg) , X, Y ∈ 𝔛 (M) ,

• the anchor γ = pr_{2} .

This Lie algebroid cannot be vertically oriented.

Example 2. For a Lie group G such that

det (Ad_{G}a) = +1, a ∈ G,

[for example: (a) G — unimodular and connected, in particular (b) G — compact and connected, (c) G = O (2k + 1) — an example of a compact and disconnected Lie group], the Lie algebroid A (P) of any G-principal bundle P (M, G) is vertically oriented. The construction of a cross-section ε of ⋀^{n} ggg : for 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤, 𝔤 = 𝔤𝔩 (G) , the following determines correctly a nowhere vanishing cross-section ε ∈ Sec ⋀^{n} ggg

ε_{x} = (⋀^{n} ẑ) (ε_{o}) , z ∈ P_{|x}, x ∈ M,

where ẑ : 𝔤 →ggg_{|x}, h ↦ [(A_{z})_{⋆} h] ∈ ggg_{|x} ⊂ (T P/G)_{|x} , (A_{z} : G → P, z ↦ za) is an isomorphism of the right Lie algebra of G onto the isotropy Lie algebra ggg_{|x}.

Example 3. Each Lie algebroid A (G; H) of the TC-foliation of left cosets of a Lie group G by a nonclosed Lie subgroup H ⊂ G can be vertically oriented because the Lie algebra bundle ggg of this Lie algebroid is a trivial bundle of abelian Lie algebras

ggg ≅ G/H̄ × 𝔥̄/𝔥, 𝔥 = 𝔤𝔩 (H) , 𝔥̄ = 𝔤𝔩 \binom{}{H̄}

(then ⋀^{n} ggg is also trivial and possesses a global nonvanishing crosssection).

Notation.

Ω (M) := Sec ⋀ T^{⋆}M — the algebra of differential forms on M, d_{M} — the exterior derivative of differential forms giving the de Rham cohomology algebra H (M).

Ω_{A} (M) := Sec ⋀ A^{⋆} — the algebra of A-differential forms. In Ω_{A} (M) the exterior derivative d_{A} works giving also a cohomology algebra H_{A} (M) . d_{A} is defined by

d_{A} (Φ) (ξ_{0}, …, ξ_{p}) = ∑_{i=0}^{p} (−1)^{i} (γ ∘ ξ_{i}) (Φ (ξ_{0}, … i^{∧} …, ξ_{p})) +

+ ∑_{i<j} (−1)^{i+j} Φ ([[ξ_{i}, ξ_{j}]], ξ_{0}, … i^{∧} … j^{∧} …, ξ_{p}) .

Definition 1. Fibre integral. By an integration operator over the isotropy Lie algebra bundle ggg in a vertically oriented Lie algebroid (A, ε) of rank n (shortly: a fibre integral) we mean the operator of degree −n

∫̸_{A} : Ω_{A} (M) ⟶ Ω (M)

defined uniquely by the formula:

∫̸_{A}Φ = 0  if  deg Φ < n,
γ^{⋆} (∫̸_{A}Φ) = (−1)^{nk} ι_{ε}Φ if deg Φ = n + k, k ⩾ 0

(ι_{ε}Φ (ξ_{1}, …, ξ_{k}) = Φ (ε, ξ_{1}, …, ξ_{k}) , Φ ∈ Ω_{A}^{n+k} (M) , — a substitution operator, (γ^{⋆}ψ) (ξ_{1}, …, ξ_{p}) = ψ (γξ_{1}, …, γξ_{p}) , ψ ∈ Ω (M) , — the pullback of differential forms via γ).

It is easy to see that ∫̸_{A}γ^{⋆}ψ = 0.

Theorem 3. The fibre integral ∫̸_{A} in a vertical oriented Lie algebroid (A, ε) commutes with the exterior derivatives d_{A} and d_{M}

d_{M} ∘ ∫̸_{A} = ∫̸_{A} ∘ d_{A}

if and only if (1): every isotropy Lie algebra ggg_{|x} is unimodular, (2): ε is an invariant cross-section of ⋀^{n} ggg with respect to the ad-

joint representation of A on ⋀^{n} ggg, i.e. ⋀^{n} ad_{A} (ξ) (ε) = 0, ξ ∈ Sec A (equivalently when in any open subset U ⊂ M on which ε is of the form ε_{|U} = (h_{1} ∧ … ∧ h_{n})_{|U} , h_{i} ∈ Secggg, we have

⋀^{n} ad_{A} (ξ) (ε)_{|U} = ∑_{i=1}^{n} (h_{i} ∧ … ∧ [[ξ, h_{i}]] ∧ … ∧ h_{n})_{|U} = 0).

Definition 2. A vertically oriented Lie algebroid (A, ε) fulfilling (1) and (2) above is called unimodular invariantly oriented.

Example 4. The Lie algebroids described in the examples above:

(1): A (P) for a principal bundle P (M, G) (G compact and fulfils

the condition det (Ad_{G} a) = +1), (2): A (G; H) for dim H̄/H = 1, are unimodular invariantly oriented. Among these last there are

nonintegrable Lie algebroids.

Theorem 4. ker ∫̸_{A} is a d_{A}-stable space if and only if A is unimodular, i.e. fulfills condition (1) above.

3. Spherical Lie algebroids and Euler class

In the sequel, fix a transitive unimodular invariantly oriented Lie algebroid (A, ε) . Now, we examine the short exact sequence of graded differential spaces

0 → ker∫̸_{A} ↪ Ω_{A} (M) ⟶^{∫̸_{A}} Ω (M) → 0

and the corresponding canonical long exact sequence in cohomology

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

Recall that, in the classical theory for sphere bundles, H (ker ∫̸_{A}) ≅ H (M) (via π^{#}). In analogy to this, we need to assume that the isotropy Lie algebras ggg_{|x} look cohomologically as the n-sphere S^{n}, i.e.

H^{i} (ggg_{|x}) = \{ \begin{matrix}0 & 1 ⩽ i ⩽ n − 1, \\ ℝ & i = 0, n. \end{matrix}

A Lie algebra having these properties is called an s-Lie algebra and a transitive unimodular invariantly oriented Lie algebroid whose isotropy Lie algebras ggg_{|x} are s-Lie algebras is called an s-Lie algebroid.

Remark 2. ℝ and 𝔰𝔬(3) are examples of s-Lie algebras.

Theorem 5. For any s-Lie algebroid (A, ε)

γ^{#} : H (M) ⟶ H (ker∫̸_{A}) , [ψ] ⟼ [γ^{⋆}ψ] ,

is an isomorphism of graded algebras.

Definition 3. Define the so-called Gysin operator

\begin{matrix}D : H (M) ⟶ H (M) \\ D [ψ] = (−1)^{deg ψ+1} (γ^{#})^{−1} (∂ [ψ]) \end{matrix}

and the Gysin exact sequence

⋯ → H_{A} (M) ⟶^{∫̸_{{# \atop A}}} H (M) ⟶^{D} H (M) ⟶^{γ^{#}} H_{A} (M) → ⋯

The cohomology class

D (1) ∈ H^{n+1} (M)

is called the Euler class of (A, ε) .

Theorem 6. Let P be a principal bundle with the structure Lie group G = S^{1} or G = Spin (3) . Take a tensor ε_{o} ∈ ⋀^{n} 𝔤, n = dim G such that ∫_{G} Δ_{R} = 1 for the right-invariant n-form on G equalling Δ at the unit where ⟨Δ, ε_{o}⟩ = 1. Consider the Lie algebroid A (P) invariantly oriented by the cross-section ε defined in one of the examples above: ε_{x} = (⋀^{n} ẑ) (ε_{o}) , z ∈ P_{|x}. Then the Euler classes χ_{P} of the sphere bundle P and χ_{A(P)} of the Lie algebroid (A (P) , ε) are equal to each other

χ_{A(P)} = χ_{P} .

Remark 3. The Euler class χ_{A} can be calculated via the Chern-Weil homomorphism h_{A} of A.

4. Index of a flat connection with singularity

If λ : T M → A is a flat connection, then, by definition, λ is a homomorphism of Lie algebroids. In this situation, γ^{⋆} : Ω_{A} (M) → Ω (M) commutes with exterior derivatives, giving on cohomology a homomorphism of algebras λ^{#} : H_{A} (M) → H (M) and if (A, ε) is an s-Lie algebroid, then H_{A} (M) = ker ∫̸_{A}^{#} ⨁ ker λ^{#} and ∫̸_{A}^{#}| ker λ^{#} : ker λ^{#} → H (M) is an isomorphism. The uniquely determined cohomology class ω_{λ} ∈ ker λ^{#n} ⊂ H_{A}^{n} (M) for which ∫̸_{A}^{#}ω_{λ} = 1 ∈ H^{0} (M) is called the cohomology class of a flat connection λ. For two flat connections λ, σ : T M → A, we have ∫̸_{A}^{#} (ω_{λ} − ω_{σ}) = 0. According to the exactness of the Gysin sequence

⋯ → H (M) ⟶^{γ^{#}} H_{A} (M) ⟶^{∫̸_{{# \atop A}}} ⋯

there exists a cohomology class [λ, σ] ∈ H^{n} (M) such that ω_{λ} − ω_{σ} = γ^{#} [λ, σ] . [λ, σ] is called the difference class of λ and σ.

Definition 4. By a local connection with singularity at a point a ∈ M in a Lie algebroid A we mean a connection

σ : T U̇ ⟶ A_{|U̇}, a ∈ U ⊂ M, U̇ = U \ {a} .

From now on we shall assume that

dim M = rank A + 1.

Therefore our theory of s-Lie algebroids concerns two cases: (1): ℝ-Lie algebroids over 2-manifolds, (2): 𝔰𝔬(3)-Lie algebroids over 4-manifolds.

Theorem 7. Let (A, ε) be an arbitrary s-Lie algebroid of rank n over an n + 1-dimensional oriented manifold M, n ⩾ 1, let σ : T U̇ → A_{|U̇} be a local connection with singularity at a ∈ U ⊂ M. Take auxiliarily a neighbourhood V ∋ a such that V ⊂ U and V ≅ ℝ^{n+1}. A_{|V} possesses a global flat connection λ : T V → A_{|V |}. Consider the difference class [λ|_{T V̇} , σ|_{T V̇}] ∈ H^{n} \binom{}{V̇} and take the canonical isomorphism

α_{V} : H^{n} \binom{}{V̇} →^{≅} ℝ. Then the real number

j_{a} (σ) := α_{V} ([λ|_{T V̇} , σ|_{T V̇}])

is independent of the auxiliary flat connection λ and a neighbourhood V ∋ a.

Definition 5. j_{a} (σ) is called the index of σ at a.

5. Index Theorem for flat connections with finitely many singularities

Theorem 8. Main Theorem. Let (A, ε) be an s-Lie algebroid of rank n over an oriented compact connected 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_{a}, …, a_{k}. Then the Euler class χ_{A} ∈ H^{n+1} (M) is given by the formula

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

equivalently,

∫̸_{A}^{#}χ_{A} = ∑_{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 a connection σ.

The integral formula for indexes (locally, each transitive Lie algebroid is integrable A_{|U} ≅ A (U × G) if U is contractible) and an application of this theorem to some principal bundles are given at the beginning of my talk.

Remark 4. The natural question is: does in each s-Lie algebroid over a compact orientable manifold there exist a flat connection with a finite number of singularities? In many cases this is obviously the case: namely, for Lie algebroids of S^{1}- or S^{3}-principal bundles over suitable manifolds, because, de facto, these bundles are sphere bundles, and such bundles have global cross-sections with a finite number of singularities. Except this set of points, the principal bundle as the one having a global cross-section is trivial, thus flat. The problem exists for nonintegrable s-Lie algebroids and for SO(3)- and O (3)-principal bundles.

Example 5. In the Hopf S^{1}-bundle P = (S^{3} → S^{2}), for any real number r ∈ ℝ, there exists a global flat connection σ_{r} with two singularities at {p_{1}, p_{2}} , such that the index j_{p_{1}}σ_{r} is equal to r. Indeed, since the Euler class of P is equal to the orientation class of S^{2}, any flat connection λ with a singularity at a point {p_{1}} has the index at p_{1} equalling 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 r ∈ ℝ,

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

is a flat connection [see the last remark]. σ̄_{r} determines a flat connection σ_{r} in P with a singularity at {p_{1}, p_{2}} , such that j_{p_{1}}σ_{r} = r.

Institute of Mathematics, Technical University of Łódź POLAND