Abstract
A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form $S^1 \times N^3$ admits a symplectic structure, then $N$ fibers over $S^1$. In fact we will completely determine the symplectic cone of $S^1\times N$ in terms of the fibered faces of the Thurston norm ball of $N$.
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