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

Abstract

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.