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Let X be a closed manifold with zero Euler characteristic, and let f: X -->S^1 be a circle-valued Morse function. We define an invariant I which countsclosed orbits of the gradient of f, together with flow lines between thecritical points. We show that our invariant equals a form of topologicalReidemeister 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 andflow lines according to their homology classes. (Previously we only consideredtheir intersection numbers with a fixed level set.) The proof here isindependent of the previous proof and also simpler.  Aside from its Morse-theoretic interest, this work is motivated by the factthat when X is three-dimensional and b_1(X)>0, the invariant I equals acounting invariant I_3(X) which was conjectured in our previous paper to equalthe Seiberg-Witten invariant of X. Our result, together with this conjecture,implies that the Seiberg-Witten invariant equals the Turaev torsion. This wasconjectured by Turaev [Math. Res. Lett. 4 (1997) 679-695] and refines thetheorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].

Circle-valued Morse theory and Reidemeister torsion