Miscellaneous

Abstract

Connected sum of Euler structures. Let M₁, M₂ be closed connected oriented 3-manifolds. Any pair of Euler structures e₁ ∈ Eul(M₁), e₂ ∈ Eul(M₂) gives rise to an Euler structure e₁#e₂ on the connected sum M = M₁#M₂. We give a description of e₁#e₂ in terms of vector fields. Pick a closed 3-ball B _s ⊂ M _s and a closed 2-disc D _s ⊂ ∂B _s for s = 1, 2. Provide the 2-sphere ∂B _s = ∂(M _s \ Int B _s ) with orientation induced by the given orientation in B _s ⊂ M _s . Pick an orientation reversing diffeomorphism ψ: D₁ → D₂. Gluing M₁ \ Int B₁ to M₂ \ Int B₂ along ψ we obtain a compact 3-manifold M⁰ obtained from M by removing a small open 3-ball. Next, we choose a non-singular tangent vector field v₁ on D₁ and let v₂ = dψ(v₁) be the corresponding vector field on D₂. For s = 1, 2, we can represent e _s by a non-singular vector field u _s on M _s such that u _s |D _s = v _s . Gluing u₁|M₁ \ Int B₁ to u₂|M₂ \ Int B₂ along ψ, obtain a non-singular vector field on M⁰. It extends to a non-singular vector field on M representing e₁#e₂. It is easy to check that e₁#e₂ does not depend on the auxiliary choices.