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 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 S¹ × 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 S¹. Second, we will provide the first example of a 2-cohomology class on a symplectic manifold (of the form S¹ × 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).

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