On the Topology and Analysis

We consider systems (M,ω, g) withM a closed smooth manifold, ω a real valued closed one form and g a Riemannian metric, so that (ω, g) is a Morse-Smale pair, Definition 2. We introduce a numerical invariant ρ(ω, g) ∈ [0,∞] and using the theory of Dirichlet series and extensions of Witten-Helffer-Sjöstrand results improve Morse-Novikov theory for (M,ω, g) provided ρ(ω, g) < ∞. For example we show that the Novikov complex of (M,ω, g) is an extension of a complex of free modules over a ring Λ′ [ω],ρ (subring of the Novikov field Λ[ω]) of holomorphic functions in {s ∈ C | Re(s) > ρ} and the Novikov incidence numbers Iq(x, y, α̂), cf. Introduction, can be entirely recovered from the spectral geometry of (M,ω, g). Theorem 2 part (3) and Theorem 4 are the main new results of this paper with Theorem 1 a key ingredient in their proof.