A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

TL;DR

This paper proves that for any 3-manifold and non-fibered class, there exists a representation making the twisted Alexander polynomial vanish, enabling classification of certain symplectic 4-manifolds and their symplectic cones.

Contribution

It extends previous work to show the existence of representations with zero twisted Alexander polynomials for non-fibered classes, aiding in classifying symplectic 4-manifolds.

Findings

Existence of representations with zero twisted Alexander polynomial for non-fibered classes

Complete classification of symplectic 4-manifolds with free circle actions

Determination of symplectic cones for these manifolds

Abstract

In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.