G ‐invariant persistent homology

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo( X ) of all self‐homeomorphisms of a topological space X . This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo‐distance d G associated with a group G  ⊂ Homeo( X ), we need to adapt persistent homology and consider G ‐invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G . In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G ‐invariant persistent homology with respect to the natural pseudo‐distance d G . We also show how G ‐invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self‐homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.