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 \cite{FV08a,FV08,FV07}. The results on surfaces of minimal 
complexity are new.

1. Université du Québec à Montréal
   Montréal, Québec, Canada
   and
   University of Warwick
   Coventry, UK
   
2. Department of Mathematics
   University of California
   Riverside, CA 92521, USA