A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)

TL;DR

This paper introduces refined configurations of Betti numbers and homology for angle-valued maps on compact ANRs, enhancing topological invariants with robustness and computational applications, extending previous real-valued map results.

Contribution

It develops a new refinement of Novikov-Betti numbers and homology for angle-valued maps, including configurations of points and submodules, with properties like robustness and duality.

Findings

Configurations are robust under perturbations.

Supports of configurations relate to Novikov homology.

Applications include computational topology methods.

Abstract

For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral cohomology class represented by the map. For each degree the first refinement is a configuration of points with multiplicity located in the punctured complex plane whose total cardinality is the Novikov-Betti number of the pair. The second refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and which has the same support as the first configuration. When the field is a the field of complex numbers the second configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of the ANR defined by the angle-valued map. One discusses the properties of these configurations namely, robustness with respect to continuous perturbation of the angle-valued map and the Poincar\'e Duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov-Betti numbers replacing standard homology and standard Betti numbers.