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In this paper we show that given any 3-manifold N and any non-fibered classin H^1(N;Z) there exists a representation such that the corresponding twistedAlexander polynomial is zero. This is obtained by extending earlier work of theauthors, together with results of Agol and Wise on separability of 3-manifoldgroups. This result allows us to completely classify symplectic 4-manifoldswith a free circle action, and to determine their symplectic cones.

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