  An interesting question in symplectic topology, which was posed by C. H.
Taubes, concerns the topology of closed (i.e. compact and without boundary)
connected oriented three dimensional manifolds whose product with a circle
admits a symplectic structure. The only known examples of such manifolds are
those which fiber over a circle. Taubes asked whether these examples are the
only examples of such manifolds. An affirmative answer to Taubes' question
would have the following consequence: 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.
