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

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

1999-12-17

arxiv.org/abs/math/9912147v2

Geom.Topol. 6 (2002) 27-58

Mathematics

Differential Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper2.abs.html

Scientific paper

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.

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