Torsion, TQFT, and Seiberg-Witten invariants of 3-manifolds

Abstract

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b1 ≥ 1 to an invariant that "counts" gradient flow lines - including closed orbits - of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg-Witten invariants of 3-manifolds by making use of a "topological quantum field theory," which makes the calculation completely explicit. We also realize a version of the Seiberg-Witten invariant of X as the intersection number of a pair of totally real submanifolds of a product of vortex moduli spaces on a Riemann surface constructed from geometric data on X. The analogy with recent work of Ozsváth and Szabó suggests a generalization of a conjecture of Salamon, who has proposed a model for the Seiberg-Witten-Floer homology of X in the case that X is a mapping torus.