#### Geometry & Topology, Vol. 3 (1999)
Paper no. 15, pages 369-396.

## Circle-valued Morse theory and Reidemeister torsion

### Michael Hutchings, Yi-Jen Lee

**Abstract**.
Let X be a closed manifold with zero Euler characteristic, and let f: X --> S^1 be a circle-valued Morse function. We define an invariant I which counts closed orbits of the gradient of f, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679-695].

We proved a similar result in our previous paper [Topology, 38 (1999) 861-888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a fixed level set.) The proof here is independent of the previous proof and also simpler.

Aside from its Morse-theoretic interest, this work is motivated by the fact that when X is three-dimensional and b_1(X)>0, the invariant I equals a counting invariant I_3(X) which was conjectured in our previous paper to equal the Seiberg-Witten invariant of X. Our result, together with this conjecture, implies that the Seiberg-Witten invariant equals the Turaev torsion. This was conjectured by Turaev [Math. Res. Lett. 4 (1997) 679-695] and refines the theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].
**Keywords**.
Morse-Novikov complex, Reidemeister torsion, Seiberg-Witten invariants

**AMS subject classification**.
Primary: 57R70.
Secondary: 53C07, 57R19, 58F09.

**DOI:** 10.2140/gt.1999.3.369

**E-print:** `arXiv:dg-ga/9706012`

Submitted to GT on 28 June 1999.
Paper accepted 21 October 1999.
Paper published 26 October 1999.

Notes on file formats
Michael Hutchings, Yi-Jen Lee

Dept of Math, Stanford University

Stanford, CA 94305, USA

Dept of Math, Princeton University

Princeton, NJ 08544, USA

Email: hutching@math.stanford.edu, ylee@math.princeton.edu

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**