
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 be a closed manifold and a polytope. For each we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope . 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.
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