Barcodes for closed one form - an alternative to Novikov theory

TL;DR

This paper generalizes barcodes for angle-valued maps to topological closed one forms on compact ANRs, extending Novikov theory and establishing stability and duality properties with applications in geometry, dynamics, and data analysis.

Contribution

It introduces a novel extension of Novikov complexes to a broader class of closed one forms on ANRs, with proven stability and duality properties.

Findings

Extended barcodes to closed one forms on ANRs.

Proved stability and Poincaré duality properties.

Applications demonstrated in geometry, dynamics, and data analysis.

Abstract

We extend the configurations discussed in Burghelea's book and Burghelea-Haller's paper on topology of angle-valued maps, equivalently the closed, open and closed-open bar codes from real- or angle-valued maps, to topological closed one forms on compact ANRs. As a consequence one provides an extension of the classical Novikov complex associated to a closed smooth one form and a vector field the form is Lyapunov for, to a considerably larger class of situations. We establish strong stability properties and Poincar\'e duality properties when the underlying space is a closed manifold. Applications to Geometry, Dynamics and Data Analysis are the targets of our research. A different approach towards such bar codes was proposed in Usher-Zhang's work.