Polytope Novikov homology

Abstract

Let M be a closed manifold and $${\mathcal {A}} \subseteq H^1_{\mathrm {dR}}(M)$$ A ⊆ H dR 1 ( M ) a polytope. For each $$a \in {\mathcal {A}}$$ a ∈ A , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $${\mathcal {A}}$$ A . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.