Twisted Alexander Polynomials and Symplectic Structures

Let N be a closed, oriented 3-manifold. A folklore conjecture states that S¹ × 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 S¹ × N. As an application of these results we will show that S¹ × 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.