Geometry & Topology

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

Thomas E Mark

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.

Article information

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

Dates
Accepted: 25 January 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882788

Digital Object Identifier
doi:10.2140/gt.2002.6.27

Mathematical Reviews number (MathSciNet)
MR1885588

Zentralblatt MATH identifier
1021.57007

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

Citation

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. https://projecteuclid.org/euclid.gt/1513882788

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