REVIEW of: "Friedl Stefan - Vidussi, Stefano, Twisted Alexander polynomials detect fibered 3-manifolds, Ann. Math. (2) 173, No. 3, 1587-1643 (2011)." [DE05960690X]

It is well known that, if a knot K S3 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, ) (where N is a compact, connected, oriented 3- manifolds with empty or toroidal boundary and 2 H1(N;Z)) to fiber: see [Ann. Sci. Ecol. Norm. Super. (4) 35, No. 2, 153-171 (2002; Zbl 1009.57021)], [Trans. Am. Math. Soc. 355, No.10, 4187-4200 (2003; Zbl 1028.57004)], [Comment. Math. Helv. 80, No. 1, 51-61 (2005; Zbl 1066.57008)], [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, ) fibers under the assumption that the Thurston norm of is known. Moreover, by making use of some of their previous works (see in particular [Am. J. Math. 130, No. 2, 455- 484 (2008; Zbl 1154.57021)]), the authors show that, if a manifold of the form S1 × N3 admits a symplectic structure, then N fibers over S1.