Counting closed orbits of gradients of circle-valued maps

TL;DR

This paper establishes a functorial chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex for circle-valued Morse maps, linking torsion to the Lefschetz zeta function.

Contribution

It constructs a functorial chain homotopy equivalence and relates its torsion to the Lefschetz zeta function for gradient flows with hyperbolic closed orbits.

Findings

Constructed a functorial chain homotopy equivalence.

Proved the torsion equals the Lefschetz zeta function.

Applied to Morse maps with hyperbolic closed orbits.

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 corresponding 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, if $v$ has only hyperbolic closed orbits.