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

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.

These are the expanded notes of the talk given by the first author at the Postnikov memorial conference at Bedlewo, Poland in June 2007