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Glasnik matematički, Vol. 57 No. 1, 2022.

Izvorni znanstveni članak

https://doi.org/10.3336/gm.57.1.06

Reconstruction properties of selective Rips complexes

Boštjan Lemež ; Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia
Žiga Virk orcid id orcid.org/0000-0001-9016-011X ; Faculty of Computer and Information Science, University of Ljubljana, 1000 Ljubljana, Slovenia

Puni tekst: engleski pdf 202 Kb

str. 73-88

preuzimanja: 212

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Sažetak

Selective Rips complexes associated to two parameters are certain subcomplexes of Rips complexes consisting of thin simplices. They are designed to detect more closed geodesics than their Rips counterparts. In this paper we introduce a general definition of selective Rips complexes with countably many parameters and prove basic reconstruction properties associated with them. In particular, we prove that selective Rips complexes of a closed Riemannian manifold \(X\) attain the homotopy type of \(X\) at small scales.
We also completely classify the resulting persistent fundamental group and \(1\)-dimensional persistent homology.

Ključne riječi

Reconstruction results, Rips complexes, Riemannian manifolds, geodesic spaces, fundamental groups

Hrčak ID:

279801

URI

https://hrcak.srce.hr/279801

Datum izdavanja:

28.6.2022.

Posjeta: 614 *

Publication date: 30 June 2022

Volume: Vol 57

Issue: Svezak 1

Pages: 73-88

DOI: https://doi.org/10.3336/gm.57.1.06

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Reconstruction results, Rips complexes, Riemannian manifolds, geodesic spaces, fundamental groups

Reconstruction properties of selective Rips complexes

Boštjan Lemež[1]

Email: bostjan.lemez@imfm.si

vZiga Virk[2]

Email: ziga.virk@fri.uni-lj.si

 

[1] Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia

[2] Faculty of Computer and Information Science, University of Ljubljana, 1000 Ljubljana, Slovenia

Abstract

Selective Rips complexes associated to two parameters are certain subcomplexes of Rips complexes consisting of thin simplices. They are designed to detect more closed geodesics than their Rips counterparts. In this paper we introduce a general definition of selective Rips complexes with countably many parameters and prove basic reconstruction properties associated with them. In particular, we prove that selective Rips complexes of a closed Riemannian manifold \(X\) attain the homotopy type of \(X\) at small scales. We also completely classify the resulting persistent fundamental group and \(1\)-dimensional persistent homology.

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