Symplectic 𝐒¹×𝐍³, subgroup separability, and vanishing Thurston norm

Friedl, Stefan; Vidussi, Stefano

doi:10.1090/S0894-0347-07-00577-2

Symplectic $\mathbf {S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm
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by Stefan Friedl and Stefano Vidussi

J. Amer. Math. Soc. 21 (2008), 597-610

DOI: https://doi.org/10.1090/S0894-0347-07-00577-2

Published electronically: August 28, 2007

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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.

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