Counting closed orbits of gradient flows of circle-valued maps
Andrei Pajitnov · arXiv (Cornell University) · 2001
Abstract. Let M be a closed connected manifold, f be a Morse map from M to a circle, v be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex C ∗ = C∗(f, v). There is a chain homotopy equivalence between C ∗ and completed simplicial chain complex of the infinite cyclic covering of M. The first main result of the paper is the construction of a functorial chain homotopy equivalence between these two complexes. The second main result states that the torsion of this chain homotopy equivalence equals to the Lefschetz zeta function of the gradient flow for an arbitrary gradient-like vector field v satisfying the transversality condition. 1.