On the conical Novikov homology

Let ω be a Morse form on a closed connected manifold M . Let p : M ^ → M be a regular covering with structure group G such that p ∗ ( [ ω ] ) = 0 . The period homomorphism π 1 ( M ) → R corresponding to ω factors through a homomorphism ξ : G → R . The rank of Im ξ is called the irrationality degree of ξ . Denote by Λ the group ring Z G and let Λ ^ ξ be its Novikov completion. Choose a transverse ω -gradient v . The classical construction of counting the flow lines of v defines the Novikov complex freely generated over Λ ^ ξ by the set of zeroes of ω . We introduce a refinement of this construction. We define a subring Λ ^ Γ of Λ ^ ξ (depending on an auxiliary parameter Γ which is a certain cone in the vector space H 1 ( G , R ) ) and show that the Novikov complex is defined actually over Λ ^ Γ and computes the homology of the chain complex C ∗ ( M ^ ) ⊗ Λ Λ ^ Γ . In the particular case when G ≈ Z 2 , and the irrationality degree of ξ equals 2, the ring Λ ^ Γ is isomorphic to the ring of series in two variables x , y of the form ∑ r ∈ N a r x n r y m r where a r , n r , m r ∈ Z and both n r , m r converge to ∞ when r → ∞ . 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 (Pazhitnov in Ann Fac Sci Toulouse Math 4(2):297–338, 1995 ). As a byproduct we obtain a simple proof of the properties of the Novikov complex for the case of Morse forms of irrationality degree > 1 . The paper contains two appendices. In Appendix 1 we give an overview of 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 (1982). 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.