  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.
