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Let M be a closed oriented 3-manifold such that S^1 x M admits a symplecticstructure w. The goal of this paper is to show that M is a fiber bundle overS^1. The basic idea is to use the obvious S^1-action on S^1 x M by rotating thefirst factor, and one of the key steps is to show that the S^1-action on S^1 xM is actually symplectic with respect to a symplectic form cohomologous to w.We achieve it by crucially using the recent result or its relative version ofGiroux about one-to-one correspondence between open book decompositions of M upto positive stabilization and co-oriented contact structures on M up to contactisotopy.

Giroux correspondence, confoliations, and symplectic structures on S^1 x M