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).
