  The aim of this paper is to discuss some applications of the relation between
Seiberg-Witten theory and two natural norms defined on the first cohomology
group of a closed 3-manifold N - the Alexander and Thurston norms. We start by
giving a "new" proof of McMullen's inequality between these norms, and then use
these norms to study two problems related to symplectic 4-manifolds of the form
S^1xN. First we prove that - as long as N is irreducible - the unit balls of
these norms are related in a way similar to the case of fibered 3-manifolds,
supporting the conjecture that N is fibered. Second, we provide the first
example of a 2-cohomology class on a symplectic manifold that lies in the
positive cone and satisfies Taubes' "more constraints", but cannot be
represented by a symplectic form.
