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POSTER

2nd European Congress of Mathematics, Budapest, Hungary 21-27 July 1996

EULER-POINCARE-HOPF THEOREM FOR FLAT CONNECTIONS

JAN KUBARSKI

There is well a known 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}} (s) is the index of σ at a_{ν} is independent on 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 a one of the following G-principal bundle P (M, G) :

• G = S^{1}, dim M = 2,

• G = Spin (3) (= S^{3}) , dim M = 4, Then P is in fact an n-sphere over 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} (σ (x) · a) = R_{a⋆} [Im σ_{⋆x}] , x ∈ M, a ∈ G.

We will investigate a flat connection H in P with finitely many singularities, i.e. a flat connection in the bundle P restricted to M \ {a_{1}, …, a_{k}} . For example a cross-section σ : M \ {a_{1}, …, a_{k}} → P

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

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Received by the editors 13 July, 1996.

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.

with singularities at a_{1}, …, a_{k} determines a flat connection H^{σ} in P over M \ {a_{1}, …, a_{k}} . With 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 to be connected, compact and oriented) is independent on the choice of a 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 a one of the singular point is a preasigned real number.

Locally any principal bundle P is a product P = ℝ^{m} × G and let dim G = n. Any connection H in P with 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 𝔤 unimodular and H flat the n-exterior product

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

has the following property: the real n-form

Δ_{⋆} (σ̂ ∧ … ∧ σ̂)

is closed and the index j_{0} (H) is given via 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 — a compact oriented,

(2) Ad_{G} (a) = +1 for a ∈ G,

(3) the Lie algebra 𝔤 of G cohomologically looks as the 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 considered Lie groups: S^{1}, Spin (3), SO (3) and a nonconnected 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 on the choice of a connection H and the Euler class χ_{P} of P is given by

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

The presented above results are particular cases of 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).

Institute of Mathematics, Technical University of Łódź POLAND