1-Dimensional Intrinsic Persistence of Geodesic Spaces

TL;DR

This paper develops a theoretical framework for understanding the intrinsic 1-dimensional persistence of geodesic spaces using Rips and Čech filtrations, linking critical points to geometric features like holes and circles.

Contribution

It introduces a comprehensive theory connecting persistence features to geometric properties of geodesic spaces, including critical points, hole sizes, and relationships between Rips and Čech persistences.

Findings

A Rips critical point corresponds to an embedded circle of length 3c.

Homology persistence encodes the lengths of the smallest bases in locally contractible spaces.

Rips and Čech persistences are isomorphic up to a factor of 3/4.

Abstract

Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or \v{C}ech filtration of $X$ to obtain what we call a persistence. 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 close, Rips and \v{C}ech induced persistences. Amongst other results we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, 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 \v{C}ech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.