#### 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.

Notes on file formats
Thomas Mark

Department of Mathematics, University of California

Berkeley, CA 94720-3840, USA

Email: mark@math.berkeley.edu

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