CIRCLE-VALUED MORSE THEORY, REIDEMEISTER TORSION, AND SEIBERG–WITTEN INVARIANTS OF 3-MANIFOLDS

Abstract

Let X be a closed oriented Riemannian manifold with χ( X)=0 and b 1( X)>0, and let φ : X→S 1 be a circle-valued Morse function. Under some mild assumptions on φ, we prove a formula relating 1. the number of closed orbits of the gradient flow of φ in different homology classes; 2. the torsion of the Novikov complex, which counts gradient flow lines between critical points of φ; and 3. a kind of Reidemeister torsion of X determined by the homotopy class of φ. When dim( X)=3, we state a conjecture related 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.