On the Conical Novikov Homology

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.