Polytope Novikov Homology

TL;DR

This paper introduces polytope Novikov homology, a generalization of Novikov homology, establishing chain homotopy equivalences within polytopes and extending classical theorems with new principles.

Contribution

It develops polytope Novikov homology, proving chain homotopy equivalences for classes in a polytope, and introduces a new polytope Novikov Principle extending previous results.

Findings

Polytope Novikov homology generalizes classical Novikov homology.

Chain homotopy equivalences hold for classes within a polytope.

A new polytope Novikov Principle is established.

Abstract

Let $M$ be a closed manifold and $\mathcal{A} \subseteq H^1_{\mathrm{dR}}(M)$ a polytope. For each $a \in \mathcal{A}$ we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $\mathcal{A}$. 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 to 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.