Geometry & Topology, Vol. 6 (2002) Paper no. 2, pages 27--58.

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

Thomas Mark

Abstract. We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 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 Ozsvath and Szabo 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

AMS subject classification. Primary: 57M27. Secondary: 57R56.

DOI: 10.2140/gt.2002.6.27

E-print: arXiv:math.DG/9912147

Submitted to GT on 16 October 2001. Paper accepted 25 January 2002. Paper published 29 January 2002.

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

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