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


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].

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