Geometry & Topology
- Geom. Topol.
- Volume 3, Number 1 (1999), 369-396.
Circle-valued Morse theory and Reidemeister torsion
Let be a closed manifold with , and let be a circle-valued Morse function. We define an invariant which counts closed orbits of the gradient of , 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 is three-dimensional and , the invariant equals a counting invariant which was conjectured in our previous paper to equal the Seiberg–Witten invariant of . 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].
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
Permanent link to this document
Digital Object Identifier
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
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