Twisted Alexander polynomials and fibered 3-manifolds

TL;DR

This paper reviews key ideas behind how twisted Alexander polynomials can identify fibered 3-manifolds and discusses related conjectures about symplectic 4-manifolds with free S^1-actions.

Contribution

It summarizes previous proofs that twisted Alexander polynomials detect fibered 3-manifolds and provides new evidence for a conjecture relating symplectic 4-manifolds and fibered orbit spaces.

Findings

Twisted Alexander polynomials detect fibered 3-manifolds.

A closed 3-manifold is fibered iff its product with S^1 is symplectic.

Supporting evidence for the conjecture on symplectic 4-manifolds with free S^1-actions.

Abstract

In a series of papers the authors proved that twisted Alexander polynomials detect fibered 3-manifolds, and they showed that this implies that a closed 3-manifold N is fibered if and only if S^1 x N is symplectic. In this note we summarize some of the key ideas of the proofs. We also give new evidence to the conjecture that if $ is a symplectic 4-manifold with a free S^1-action, then the orbit space is fibered.