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

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

2002-04-16

arxiv.org/abs/math/0204211v1

Pacific J. Math. 208 (2003)

Mathematics

Geometric Topology

18 pages, 3 figures. To appear in Pacific J. Math

Scientific paper

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.

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