## Geometry & Topology

### Circle-valued Morse theory and Reidemeister torsion

#### Abstract

Let $X$ be a closed manifold with $χ(X)=0$, and let $f:X→S1$ 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 $b1(X)>0$, the invariant $I$ equals a counting invariant $I3(X)$ 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 and refines the theorem of Meng and Taubes [Math. Res. Lett 3 (1996) 661–674].

#### Article information

Source
Geom. Topol., Volume 3, Number 1 (1999), 369-396.

Dates
Accepted: 21 October 1999
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883151

Digital Object Identifier
doi:10.2140/gt.1999.3.369

Mathematical Reviews number (MathSciNet)
MR1716272

Zentralblatt MATH identifier
0929.57019

#### Citation

Hutchings, Michael; Lee, Yi-Jen. Circle-valued Morse theory and Reidemeister torsion. Geom. Topol. 3 (1999), no. 1, 369--396. doi:10.2140/gt.1999.3.369. https://projecteuclid.org/euclid.gt/1513883151

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