Twisted Alexander polynomials detect fibered 3-manifolds

TL;DR

This paper proves that twisted Alexander polynomials precisely characterize when a 3-manifold is fibered, extending classical results and linking to symplectic structures on certain 4-manifolds.

Contribution

It establishes that conditions on twisted Alexander polynomials are both necessary and sufficient for a 3-manifold to be fibered, generalizing previous knot theory results.

Findings

Twisted Alexander polynomials fully characterize fibered 3-manifolds.

Necessary and sufficient conditions for fiberedness are identified.

Implications for symplectic structures on S^1 x N are derived.

Abstract

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.