Circle-valued Morse theory and Reidemeister torsion

Let $X$ be a closed manifold with $χ (X) = 0$ , and let $f : X \to S^{1}$ 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 $b_{1} (X) > 0$ , the invariant $I$ equals a counting invariant $I_{3} (X)$ 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].

Keywords

Morse–Novikov complex, Reidemeister torsion, Seiberg–Witten invariants

Mathematical Subject Classification

Primary: 57R70

Secondary: 53C07, 57R19, 58F09

References

Forward citations

Publication

Received: 28 June 1999

Accepted: 21 October 1999

Published: 25 October 1999

Proposed: Ralph Cohen

Seconded: Robion Kirby, Steve Ferry

Authors

Michael Hutchings
Department of Mathematics Stanford University Stanford California 94305 USA

Yi-Jen Lee
Department of Mathematics Princeton University Princeton New Jersey 08544 USA

Volume 3, issue 1 (1999)

Michael Hutchings and Yi-Jen Lee