• Algebraic footprints of geometric features in homology
The Client : Javna agencija za raziskovalno dejavnost RS
Project type: Research projects ARRS
Project duration: 2019 - 2022
  • Description

Given a metric space, X, there are conditions on a finite sample A in X under which it is possible to recover the homology of X from the VietorisRips or the Čech complex of A. These conditions relate the local curvature with the sampling density and the scale parameter for which the complex is constructed. While the conditions can be relaxed if we substitute persistent homology for homology, the focus is still on scales that are sufficiently small to resolve the small details of the space X.

We propose to extend the reconstruction effort by considering the homology on all scales, from small to intermediate to large. This is motivated by the insight that interesting but not necessarily small-scale features of X are also reflected in the homology of the VietorisRips and Čech complexes. For example, closed geodesic loops may cause non-trivial three-dimensional homology even if they are contractible, and they can be detected this way. This means that there is a Hopf-type effect we witness: geometric features generate higher-dimensional algebraic footprints. We propose to study these effects and to develop a persistent cookbook, in which algebraic implications of geometric features are listed along with methods to reconstruct them.

Here we describe concrete methods and steps needed for this development. Two main tools are contractions and deformation contractions, which facilitate arguments of algebraic topology to deduce the existence of appropriate subspaces from the occurrence of algebraic elements. The arguments themselves contain aspects of combinatorics, of discrete geometry, and of classical topology. For example, to obtain the persistent homology we get for simple shapes like the sphere, we need to understand open covers that arise from finite configurations on the sphere. Along the way, we prove results that are of independent interest: a VietorisRips nerve theorem; new reconstructions of the homotopy type from finite samples; the classification of contractions in metric graphs; etc.