Twisted Alexander polynomials detect fibered 3-manifolds. (English) Zbl 1231.57020

Friedl, Stefan; Vidussi, Stefano

Ann. Math. (2) 173, No. 3, 1587-1643 (2011).

It is well known that, if a knot \(K \subset S^3\) is fibered, then the Alexander polynomial is monic and the degree equals twice the genus of \(K.\) Various generalizations of this result have been performed, showing that twisted Alexander polynomials give necessary conditions for \((N, \phi)\) (where \(N\) is a compact, connected, oriented 3-manifolds with empty or toroidal boundary and \(\phi \in H^1(N; Z)\)) to fiber: see [C. T. McMullen, Ann. Sci. Ecol. Norm. Super. (4) 35, No. 2, 153–171 (2002; Zbl 1009.57021)], [J. C. Cha, Trans. Am. Math. Soc. 355, No.10, 4187–4200 (2003; Zbl 1028.57004)], [H. Goda, T. Kitano, T. Morifuji, Comment. Math. Helv. 80, No. 1, 51–61 (2005; Zbl 1066.57008)], [S. Friedl, T. Kim, Topology 45, No. 6, 929–953 (2006; Zbl 1105.57009)] and [T. Kitayama, “Normalization of twisted Alexander invariants”, preprint 2007 (arXiv:0705.2371)].
In general, the constraint of monicness and degree for the ordinary Alexander polynomial falls short from characterizing fibered 3-manifolds. The main result of present paper shows that that the collection of all twisted Alexander polynomials does detect fiberedness; equivalently, it proves that twisted Alexander polynomials detect whether \((N, \phi)\) fibers under the assumption that the Thurston norm of \(\phi\) is known.
Moreover, by making use of some of their previous works (see in particular [S. Friedl and S. Vidussi, Am. J. Math. 130, No. 2, 455–484 (2008; Zbl 1154.57021)]), the authors show that, if a manifold of the form \(S^1 \times N^3\) admits a symplectic structure, then \(N\) fibers over \(S^1\).

Reviewer: Maria Rita Casali (Modena)

MSC:

57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
57N10	Topology of general \(3\)-manifolds (MSC2010)
57R17	Symplectic and contact topology in high or arbitrary dimension

Keywords:

3-manifold; twisted Alexander polynomial; fibered 3-manifold; symplectic 4-manifold; Thurston norm; Seiberg-Witten invariants; finite solvable groups

Citations:

Zbl 1009.57021; Zbl 1028.57004; Zbl 1066.57008; Zbl 1105.57009; Zbl 1154.57021

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