Polytope Novikov homology.

Area of Science:

• Differential Topology
• Algebraic Topology
• Homological Algebra

Background:

• The Novikov homology is a significant invariant in topology, often used in studying smooth manifolds and dynamical systems.
• Existing generalizations of Novikov homology have limitations in capturing complex topological structures.
• The need for a more flexible framework to study topological invariants on manifolds with specific finiteness conditions.

Purpose of the Study:

• To introduce and define a new concept: polytope Novikov homology.
• To establish chain homotopy equivalence between polytope Novikov complexes derived from different cohomology classes.
• To explore applications of this new homology theory, including a novel approach to the Novikov Morse Homology Theorem and a generalized Novikov Principle.

Main Methods:

• Construction of a polytope Novikov chain complex incorporating a multiple finiteness condition defined by a polytope.
• Utilizing algebraic topology techniques to prove chain homotopy equivalence of the constructed complexes.
• Developing applications based on the properties of the polytope Novikov homology.

Main Results:

• The definition of polytope Novikov homology, generalizing ordinary Novikov homology.
• Proof that cohomology classes within a given polytope yield chain homotopy equivalent polytope Novikov complexes.
• A novel approach to the (twisted) Novikov Morse Homology Theorem and a new polytope Novikov Principle, extending existing results.

Conclusions:

• Polytope Novikov homology provides a powerful new tool for studying topological invariants on manifolds.
• The established chain homotopy equivalence simplifies comparisons between different topological structures.
• The presented applications demonstrate the broad utility of this generalized homology theory in advancing topological and geometric understanding.