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

Thomas E Mark

Geometry & Topology 6 (2002) 27–58

arXiv: math.DG/9912147


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.

Seiberg–Witten invariant, torsion, topological quantum field theory
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57R56
Forward citations
Received: 16 October 2001
Accepted: 25 January 2002
Published: 29 January 2002
Proposed: Robion Kirby
Seconded: Ronald Stern, Ronald Fintushel
Thomas E Mark
Department of Mathematics
University of California
California 94720-3840