  Let M be a closed oriented 3-manifold such that S^1 x M admits a symplectic
structure w. The goal of this paper is to show that M is a fiber bundle over
S^1. The basic idea is to use the obvious S^1-action on S^1 x M by rotating the
first factor, and one of the key steps is to show that the S^1-action on S^1 x
M 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 of
Giroux about one-to-one correspondence between open book decompositions of M up
to positive stabilization and co-oriented contact structures on M up to contact
isotopy.
