  Let N be a closed, oriented 3-manifold. A folklore conjecture states that S^1
x N admits a symplectic structure only if N admits a fibration over the circle.
The purpose of this paper is to provide evidence to this conjecture studying
suitable twisted Alexander polynomials of N, and showing that their behavior is
the same as of those of fibered 3-manifolds. In particular, we will obtain new
obstructions to the existence of symplectic structures and to the existence of
symplectic forms representing certain cohomology classes of S^1 x N. As an
application of these results we will show that S^1 x N(P) does not admit a
symplectic structure, where N(P) is the 0-surgery along the pretzel knot P =
(5,-3,5), answering a question of Peter Kronheimer.
