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

We prove a conjecture of Hutchings and Lee relating the Seiberg–Witten invariants of a closed 3–manifold $X$ with $b_{1} \geq 1$ 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.

Keywords

Seiberg–Witten invariant, torsion, topological quantum field theory

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57R56

References

Forward citations

Publication

Received: 16 October 2001

Accepted: 25 January 2002

Published: 29 January 2002

Proposed: Robion Kirby

Seconded: Ronald Stern, Ronald Fintushel

Authors

Thomas E Mark
Department of Mathematics University of California Berkeley California 94720-3840 USA

Volume 6, issue 1 (2002)

Thomas E Mark