Abstract
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].
Citation
Michael Hutchings. Yi-Jen Lee. "Circle-valued Morse theory and Reidemeister torsion." Geom. Topol. 3 (1) 369 - 396, 1999. https://doi.org/10.2140/gt.1999.3.369
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