Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

Let X be a closed oriented Riemannian manifold with chi(X) = 0 and h(1)(X) > 0, and let phi:X --> S-1 be a circle-valued Morse function. Under some mi ld assumptions on phi, we prove a formula relating 1. the number of closed orbits of the gradient How of phi in different homo logy classes; 2. the torsion of the Novikov complex, which counts gradient Row lines betw een critical points of phi: and 3. a kind of Reidemeister torsion of X determined by the homotopy class of phi. When dim(X) = 3, we state a conjecture related to Taubes's "SW = Gromov" th eorem, and we use it to deduce (for closed manifolds, module signs) the Men g-Taubes relation between the Seiberg-Witten invariants and the "Milnor tor sion" of X. (C) 1999 Elsevier Science Ltd. All rights reserved.