Twisted Alexander polynomials detect fibered 3-manifolds

Details Twisted Alexander polynomials detect fibered 3-manifolds Twisted Alexander polynomials detect fibered 3-manifolds

2008-05-08

arxiv.org/abs/0805.1234v3

Mathematics

Geometric Topology

This is the final version of the paper. 53 pages

Scientific paper

A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S^1 x N^3 admits a symplectic structure, then N fibers over S^1. In fact we will completely determine the symplectic cone of S^1 x N in terms of the fibered faces of the Thurston norm ball of N.

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