  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 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 [Math. Res. Lett. 4 (1997) 679-695] and refines the
theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].
