Circle-valued Morse theory and Reidemeister torsion
Michael L. Hutchings, Yi‐Jen Lee · Geometry & Topology · 1999
Let X be a closed manifold with χ(X) = 0, and let f : X → S 1 be a circlevalued 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 [28].We proved a similar result in our previous paper [7], 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 proof in [7], 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 [7] 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 [28] and refines the theorem of Meng and Taubes [14].