## 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.## References

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## Bibliographic Information

**Stefan Friedl**- Affiliation: Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada
- MR Author ID: 746949
- Email: sfriedl@gmail.com
**Stefano Vidussi**- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: svidussi@math.ucr.edu
- Received by editor(s): August 2, 2006
- Published electronically: August 28, 2007
- Additional Notes: The second author was partially supported by NSF grant #0629956.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2373361

Dedicated: Dedicated to the memory of Xiao-Song Lin