Symplectic $\mathbf{S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm

Abstract

Let N be a closed, oriented 3-manifold. A folklore conjecture states that $S^{1} \times N$ admits a symplectic structure if and only if $N$ admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes 3-manifolds with vanishing Thurston norm, graph manifolds and 3-manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic 3-manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the 3-manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.