Refinement of Novikov - Betti numbers and of Novikov homology provided by an angle valued map

TL;DR

This paper introduces a refined framework for Novikov - Betti numbers and Novikov homology using angle-valued maps, providing a more detailed algebraic and topological understanding of the underlying spaces.

Contribution

It develops a novel configuration-based approach that refines classical invariants like Novikov - Betti numbers and homology through complex eigenvalue configurations and module structures.

Findings

Refined configurations of complex numbers and modules for (X,f)

Induced orthogonal Hilbert submodules in L2-homology

Connection with von Neumann algebra structures

Abstract

To a pair (X,f), X compact ANR and f a continuous angle valued map defined on X, a fixed field and a nonnegative integer one assigns a finite configuration of complex numbers with multiplicities located in the punctured complex plane and a finite configuration of free modules over the ring of Laurent polynomials (with coefficients in the fixed field) indexed by the same complex numbers. This is done in analogy with the configuration of eigenvalues and of generalized eigenspaces of an invertible linear operator in a finite dimensional complex vector space. The configuration of complex numbers refines the Novikov - Betti number and the configuration of free modules refines the Novikov homology associated with the cohomology class defined by f, in the same way the collection of eigenvalues and of generalized eigen-spaces refine the dimension of the vector space and the vector space on which the operator acts. In the case the field is the field of complex numbers the configuration of free modules induces by "von-Neumann completion" a configuration of mutually orthogonal closed Hilbert submodules of the L 2--homology of the infinite cyclic cover of X determined by the map f, which is an Hilbert module over the von-Neumann algebra of complex L-infinity functions on the unit circle in the complex plane.