1-Dimensional intrinsic persistence of geodesic spaces

Abstract

Given a compact geodesic space [Formula: see text], we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of [Formula: see text] to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point [Formula: see text] corresponds to an isometrically embedded circle of length [Formula: see text], that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor [Formula: see text]. The theory describes geometric properties of the underlying space encoded and extractable from persistence.