A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

Abstract

In this paper we show that given any 3-manifold $N$ and any non-fibered class in $H^1(N;\mathbb{Z})$ there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of $3$-manifold groups. This result allows us to completely classify symplectic $4$-manifolds with a free circle action, and to determine their symplectic cones.