Simple homotopy type of the Novikov complex and Lefschetz $\zeta$-function of the gradient flow

TL;DR

This paper establishes a homotopy equivalence between the Novikov complex and the chain complex of the cyclic covering for generic gradients, linking torsion to the Lefschetz zeta-function, and provides a survey of Morse-Novikov theory.

Contribution

It proves a homotopy equivalence for the Novikov complex with the chain complex of the cyclic covering, relating torsion to the Lefschetz zeta-function for generic gradients.

Findings

Homotopy equivalence between Novikov complex and cyclic covering chains

Torsion of the equivalence equals the Lefschetz zeta-function

Novikov complex is rational over the ring of rational functions

Abstract

Let f be a Morse map from a closed manifold to a circle. S.P.Novikov constructed an analog of the Morse complex for f. The Novikov complex is a chain complex defined over the ring of Laurent power series with integral coefficients and finite negative part. This complex depends on the choice of a gradient-like vector field. The homotopy type of the Novikov complex is the same as the homotopy type of the completed complex of the simplicial chains of the cyclic covering associated to f. In the present paper we prove that for every C^0-generic f-gradient there is a homotopy equivalence between these two chain complexes, such that its torsion equals to the Lefschetz zeta-function of the gradient flow. For these gradients the Novikov complex is defined over the ring of rational functions and the Lefschetz zeta-function is also rational. The paper contains also a survey of Morse-Novikov theory and of the previous results of the author on the C^0-generic properties of the Novikov complex.