On the asphericity of a symplectic 𝑀³×𝑆¹

McCarthy, John

doi:10.1090/S0002-9939-00-05571-4

On the asphericity of a symplectic $M^3\times S^1$
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by John D. McCarthy PDF

Proc. Amer. Math. Soc. 129 (2001), 257-264 Request permission

Abstract:

C. H. Taubes asked whether a closed (i.e. compact and without boundary) connected oriented three-dimensional manifold whose product with a circle admits a symplectic structure must fiber over a circle. An affirmative answer to Taubes’ question would imply that any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and aspherical. In this paper, we prove that this implication holds up to connect sum with a manifold which admits no proper covering spaces with finite index. It is pointed out that Thurston’s geometrization conjecture and known results in the theory of three-dimensional manifolds imply that such a manifold is a three-dimensional sphere. Hence, modulo the present conjectural picture of three-dimensional manifolds, we have shown that the stated consequence of an affirmative answer to Taubes’ question holds.

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