1-Dimensional Intrinsic Persistence of Geodesic Spaces

Abstract

Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech 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 Čech 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 Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.

35 pages, 8 figures. Definition 4.1 has been updated. Now it also holds for non semi-locally simply connected spaces. The details of this modification are in the last section "Comments on this version of the manuscript"

References in corpus (3)

• Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability
• Topological and uniform structures on universal covering spaces
• A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs

Cited by in corpus (17)

• Rips complexes as nerves and a Functorial Dowker-Nerve Diagram
• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
• Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability
• Footprints of geodesics in persistent homology
• Metric reconstruction via optimal transport
• On Homotopy Types of Vietoris-Rips Complexes of Metric Gluings
• Geometric Approaches on Persistent Homology
• Contractions in persistence and metric graphs
• Vietoris-Rips Complexes of Regular Polygons
• Metric Graph Approximations of Geodesic Spaces
• Critical edges in Rips complexes and persistence
• Persistent Homology with Selective Rips complexes detects geodesic circles
• Elements of higher homotopy groups undetectable by polyhedral approximation
• Operations on Metric Thickenings
• Rigidity of terminal simplices in persistent homology
• Geometric Bounds for Persistence
• Nonlocal loss of first homotopy in polyhedral approximations of Peano continua