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

Friedl, Stefan¹ ; Vidussi, Stefano²

¹ Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada
² Department of Mathematics, University of California, Riverside, California 92521

Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 597-610

Résumé

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.

Détail

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

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