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

Let X be a compact oriented Riemannian manifold and let \(\phi:X\to S^1\) be a circle-valued Morse function. Under some mild assumptions on \(\phi\), we prove a formula relating: (a) the number of closed orbits of the gradient flow of \(\phi\) of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of \(\phi\); and (c) a kind of Reidemeister torsion of X determined by the homotopy class of \(\phi\). When \(\dim(X)=3\) and \(b_1(X)>0\), we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X.