Geometry & Topology

Circle-valued Morse theory and Reidemeister torsion

Michael Hutchings and Yi-Jen Lee

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Let X be a closed manifold with χ(X)=0, and let f:XS1 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

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

Received: 28 June 1999
Accepted: 21 October 1999
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R70: Critical points and critical submanifolds
Secondary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20] 57R19: Algebraic topology on manifolds 58F09

Morse–Novikov complex Reidemeister torsion Seiberg–Witten invariants


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.

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