Geometry & Topology 3 (1999) 369–396

DOI: 10.2140/gt.1999.3.369

arXiv: dg-ga/9706012

Abstract

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].

Keywords

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

Mathematical Subject Classification

Primary: 57R70

Secondary: 53C07, 57R19, 58F09

References

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Publication

Received: 28 June 1999
Accepted: 21 October 1999
Published: 25 October 1999
Proposed: Ralph Cohen
Seconded: Robion Kirby, Steve Ferry

Authors