Barcodes for closed one form - an alternative to Novikov theory

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.