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 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.
Highlights
Given a closed manifold M and a cohomology class a ∈ Hd1R(M ), one can define the so-called Novikov homology HN(a), introduced by Novikov [11, 12]
HN(a) is defined by picking a Morse representative α ∈ a and a cover on which α pulls back to an exact form df, and mimicking the definition of Morse homology using fas the underlying Morse function
The Novikov Morse Homology Theorem says that HN(a) is isomorphic to the twisted singular homology H (M, Nov(a))
Summary
Given a closed manifold M and a cohomology class a ∈ Hd1R(M ), one can define the so-called Novikov homology HN(a), introduced by Novikov [11, 12].
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