Twisted Alexander polynomials, symplectic 4-manifolds and surfaces of minimal complexity

TL;DR

This paper explores how twisted Alexander polynomials can determine the existence of symplectic structures and minimal complexity surfaces in 4-manifolds with free circle actions, extending previous work and introducing new results.

Contribution

It provides new insights into minimal complexity surfaces in such 4-manifolds using twisted Alexander polynomials, complementing earlier findings on symplectic structures.

Findings

Criteria for the existence of symplectic structures in 4-manifolds with circle actions

New results on minimal complexity surfaces in these manifolds

Application of twisted Alexander polynomials to topological problems

Abstract

Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on surfaces of minimal complexity are new.