Geometry & Topology

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

Thomas E Mark

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
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

Keywords
Seiberg–Witten invariant torsion topological quantum field theory

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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References

  • S B Bradlow, Vortices in Holomorphic Line Bundles over Closed Kähler Manifolds, Comm. Math. Phys. 135 (1990) 1–17
  • S K Donaldson, Monopoles, Knots, and Vortices, (1997) lecture notes transcribed by Ivan Smith
  • S K Donaldson, Topological Field Theories and Formulae of Casson and Meng-Taubes, from: “Proceedings of the Kirbyfest”, Geometry and Topology Monographs, 2 (1999)
  • R Fintushel, R Stern, Knots, Links, and 4-Manifolds, Invent. Math. 134 (1998) 363–400
  • M Hutchings, Y-J Lee, Circle-Valued Morse theory and Reidemeister Torsion, preprint (1997)
  • M Hutchings, Y-J Lee, Circle-Valued Morse Theory, Reidemeister Torsion, and Seiberg-Witten Invariants of 3-Manifolds, Topology, 38 (1999) 861–888
  • E Ionel, T H Parker, Gromov Invariants and Symplectic Maps, Math. Ann. 314 (1999) 127–158
  • A Jaffe, C H Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhäuser, Boston, Mass. (1980)
  • I G Macdonald, Symmetric Products of an Algebraic Curve, Topology 1 (1962) 319–343
  • G Meng, C H Taubes, SW = Milnor Torsion, Math. Res. Lett. 3 (1996) 661–674
  • J Milnor, A Duality Theorem for Reidemeister Torsion, Ann. Math. 76 (1962) 137–147
  • J Milnor, Lectures on the h-Cobordism Theorem, Princeton Unversity Press (1965)
  • T Mrowka, P Ozsváth, B Yu, Seiberg-Witten Monopoles on Seifert Fibered Spaces, Comm. Anal. Geom. 5 (1997) 685–791
  • P Ozsváth, Z Szabó, Holomorphic Disks and Invariants for Rational Homology Three-Spheres.
  • P Ozsváth, Z Szabó, Holomorphic Disks and Three-Manifold Invariants: Properties and Applications.
  • D Salamon, Seiberg-Witten Invariants of Mapping Tori, Symplectic Fixed Points, and Lefschetz Numbers, from: “Proceedings of 6th Gökova Geometry-Topology Conference” (1998) 117–143
  • V G Turaev, Reidemeister Torsion in Knot Theory, Russian Math. Surveys, 41 (1986) 119–182