#### Volume 3, issue 1 (1999)

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Circle-valued Morse theory and Reidemeister torsion

### Michael Hutchings and Yi-Jen Lee

Geometry & Topology 3 (1999) 369–396
 arXiv: dg-ga/9706012
##### Abstract

Let $X$ be a closed manifold with $\chi \left(X\right)=0$, and let $f:X\to {S}^{1}$ 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 ${b}_{1}\left(X\right)>0$, the invariant $I$ equals a counting invariant ${I}_{3}\left(X\right)$ 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 and refines the theorem of Meng and Taubes [Math. Res. Lett 3 (1996) 661–674].

##### Keywords
Morse–Novikov complex, Reidemeister torsion, Seiberg–Witten invariants
##### Mathematical Subject Classification
Primary: 57R70
Secondary: 53C07, 57R19, 58F09
##### Publication
Received: 28 June 1999
Accepted: 21 October 1999
Published: 25 October 1999
Proposed: Ralph Cohen
Seconded: Robion Kirby, Steve Ferry
##### Authors
 Michael Hutchings Department of Mathematics Stanford University Stanford California 94305 USA Yi-Jen Lee Department of Mathematics Princeton University Princeton New Jersey 08544 USA