Norms on the cohomology of a 3-manifold and SW theory

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 3manifold N -the Alexander and the Thurston norm. We will start by giving a "new" proof, applying SW theory, of McMullen's inequality between these two norms, and then use these norms to study two problems related to symplectic 4-manifolds of the form S 1 N . First we will prove that -as long as N is irreducible -the unit balls of the Thurston and Alexander norms are related in a way that is similar to the case of fibered 3-manifolds, supporting the conjecture that N has to be fibered over S 1 . Second, we will provide the first example of a 2-cohomology class on a symplectic manifold (of the form S 1 N ) that lies in the positive cone and satisfies Taubes' "more constraints", but cannot be represented by a symplectic form, disproving a conjecture of Li and Liu (Li-Liu, 2001, Section 4.1).