Geometry & Topology

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

Thomas E Mark

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We prove a conjecture of Hutchings and Lee relating the Seiberg–Witten invariants of a closed 3–manifold X with b11 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.

Article information

Geom. Topol., Volume 6, Number 1 (2002), 27-58.

Received: 16 October 2001
Accepted: 25 January 2002
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R56: Topological quantum field theories

Seiberg–Witten invariant torsion topological quantum field theory


Mark, Thomas E. Torsion, TQFT, and Seiberg–Witten invariants of $3$–manifolds. Geom. Topol. 6 (2002), no. 1, 27--58. doi:10.2140/gt.2002.6.27.

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