Incidence coefficients in the Novikov complex for Morse forms: rationality and exponential growth properties

TL;DR

This paper investigates the algebraic properties of incidence coefficients in the Novikov complex for Morse forms, proving their rationality and exponential growth behavior for generic gradients, and providing explicit examples.

Contribution

It extends the understanding of Novikov complex incidence coefficients, showing they are rational functions and belong to specific algebraic rings for generic gradients, and confirms Arnold's exponential growth conjecture.

Findings

Incidence coefficients are rational functions for generic Morse forms.

These coefficients belong to localized or completed group rings.

The exponential growth conjecture is confirmed for generic gradients.

Abstract

In this paper we continue the study of generic properties of the Novikov complex, began in the work "The incidence coefficients in the Novikov complex are generically rational functions" ( dg-ga/9603006). For a Morse map $f:M\to S^1$ there is a refined version of Novikov complex, defined over the Novikov completion of the fundamental group ring. We prove that for a $C^0$ generic $f$-gradient the corresponding incidence coefficients belong to the image in the Novikov ring of a (non commutative) localization of the fundamental group ring. The Novikov construction generalizes also to the case of Morse 1-forms. In this case the corresponding incidence coefficients belong to the suitable completion of the ring of integral Laurent polynomials of several variables. We prove that for a given Morse form $\omega$ and a $C^0$ generic $\omega$-gradient these incidence coefficients are rational functions. The incidence coefficients in the Novikov complex are obtained by counting the algebraic number of the trajectories of the gradient, joining the zeros of the Morse form. There is V.I.Arnold's version of the exponential growth conjecture, which concerns the total number of trajectories. We confirm it for any given Morse form and a $C^0$ dense set of its gradients. We also give an example of explicit computation of the Novikov complex.