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


Authors: Stefan Friedl and Stefano Vidussi
Journal: J. Amer. Math. Soc. 21 (2008), 597-610
MSC (2000): Primary 57R17, 57M27
DOI: https://doi.org/10.1090/S0894-0347-07-00577-2
Published electronically: August 28, 2007
MathSciNet review: 2373361
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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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Additional Information

Stefan Friedl
Affiliation: Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada
Email: sfriedl@gmail.com

Stefano Vidussi
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: svidussi@math.ucr.edu

DOI: https://doi.org/10.1090/S0894-0347-07-00577-2
Received by editor(s): August 2, 2006
Published electronically: August 28, 2007
Additional Notes: The second author was partially supported by NSF grant #0629956.
Dedicated: Dedicated to the memory of Xiao-Song Lin
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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