On the conical Novikov homology

Abstract

Let ω be a Morse form on a closed connected manifold M . Let p :x M Ñ M be a regular covering with structure group G, such that p ˚prωsq " 0. The period homomorphism π1pM q Ñ R corresponding to ω factors through a homomorphism ξ : G Ñ R. The rank of Im ξ is called the irrationality degree of ξ. Denote by Λ the group ring ZG and let p 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 N˚freely generated over p Λ ξ by the set of zeroes of ω. In this paper we introduce a refinement of this construction. We define a subring p Λ Γ of p Λ ξ (depending on an auxiliary parameter Γ which is a certain cone in the vector space H 1 pG, Rq) and show that the Novikov complex N˚is defined actually over p Λ Γ and computes the homology of the chain complex C˚p x M q b Λ p Λ Γ . In the particular case when G « Z 2 , and the irrationality degree of ξ equals 2, the ring p Λ Γ is isomorphic to the ring of series in 2 variables x, y of the form ř rPN arx nr y mr where ar, nr, mr P Z and both nr, mr converge to 8 when r Ñ 8.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 [15]. As a byproduct we obtain a simple proof of the properties of the Novikov complex for the case of Morse forms of irrationality degree ą 1.The paper contains two appendices. In Appendix 1 we give an overview of the 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 (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.

Polytope Novikov homology

“…The proof of the first part is literally the same as in Theorems 2.14 and 2.19. To see that the inclusion defines a chain map, it suffices to observe that both boundary operators in (17) are identical upon restricting to the smaller complex CN • (ϑ a , A).…”

“…We conclude the present subsection by explaining how to recover [17, Theorem 5.1] from the polytope Novikov principle. For the reader's convenience, we briefly recall Pajitnov's setting, keeping the notation as close as 26 One can also deduce from category theory by observing that the functor F : mod possible to [17]. Fix a Morse-Smale pair (ω, g) on M and let p : M → M be a regular cover, such that p * [ω] = 0.…”

“…Strictly speaking, we could also use perturbations coming from Corollary 2.17 at the expanse of working with small sections. For the sake of exposition, we refrain from stating this explicitly 17. The addition and multiplication of Nov(G) are the obvious ones: ng t g + mg t g = (ng + mg) t g and (ng t g ) • (n h t h ) := ng • n h t g+h .…”

“…The rank of a cohomology class a ∈ H 1 dR (M ) is defined as rank Z (im Φa).28 The definition in[17] is slightly more general, as they define the count with respect to any transverse ω-gradient.…”

Polytope Novikov Homology

“…for all b ∈ A, in order to get the desired continuation chain map to conclude (18). For this purpose we define…”