On the Conical Novikov Homology

TL;DR

This paper refines Novikov homology for Morse forms on manifolds, introducing a subring that enhances the algebraic structure and relates to the homology of certain chain complexes, with specific focus on cases where the covering group is isomorphic to 0^2.

Contribution

It introduces a new subring 0_\u03b3 of the Novikov completion, over which the Novikov complex is defined, and extends the algebraic and geometric understanding of Novikov homology for specific group cases.

Findings

The Novikov complex is defined over a refined subring 0_0, improving algebraic structure.

For G0^2 and irrationality degree 2, 0_0 is isomorphic to series in two variables with integer coefficients.

The work generalizes classical approximation algorithms and extends circle-valued Morse theory results.

Abstract

Let $\omega$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([\omega])=0$. Let $\xi:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat \Lambda}_\xi$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $\omega$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat \Lambda}_\xi$ by the set of zeroes of $\omega$. In this paper we introduce a refinement of this construction. We define a subring $\hat\Lambda_\Gamma$ of ${\hat \Lambda}_\xi$ and show that the Novikov complex $\mathcal{N}_*$ is defined actually over $\hat\Lambda_\Gamma$ and computes the homology of the chain complex $C_*(\hat M)\underset{\Lambda}{\otimes}\hat\Lambda_\Gamma $. When $G\approx\mathbf{Z}^2$, and the irrationality degree of $\xi$ equals 2, the ring $\hat\Lambda_\Gamma$ is isomorphic to the ring of series in $2$ variables $x, y$ of the form $\sum_{r\in\mathbf{N}} a_r x^{n_r}y^{m_r}$ where $a_r, n_r, m_r\in\mathbf{Z}$ and both $n_r, \ m_r$ converge to $\infty$ when $r\to \infty$. The algebraic part of the proof is based on a suitable generalization of the classical algorithm of approximating irrational numbers by rationals. The geometric part is a straightforward generalization of the author's proof of the particular case of this theorem concerning the circle-valued Morse maps. In Appendix 1 we give an overview of E. Pitcher's work on circle-valued Morse theory (1939). We show that Pitcher's lower bounds for the number of critical points of a circle-valued Morse map coincide with the torsion-free part of the Novikov inequalities. In Appendix 2 we construct a circle-valued Morse map and its gradient such that its unique Novikov incidence coefficient is a power series in one variable with an arbitrarily small convergence radius.