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1-Dimensional intrinsic persistence of geodesic spaces

Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

https://doi.org/10.1142/S1793525319500444Cited by:21 (Source: Crossref)

Abstract

Given a compact geodesic space $X$ $X$ , we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ $X$ 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 $c$ $c$ corresponds to an isometrically embedded circle of length $3 c$ $3 c$ , 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 $3 / 4$ $3 / 4$ . The theory describes geometric properties of the underlying space encoded and extractable from persistence.

Keywords:

AMSC: 57N65, 55N05, 55N35, 55Q05, 53C22, 30F99, 53B21

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