Circle-valued Morse theory and Reidemeister torsion

Let X be a closed manifold with chi(X) = 0, and let f : X -> 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 [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].