Glasnik Matematicki, Vol. 57, No. 1 (2022), 73-88. \( \)

RECONSTRUCTION PROPERTIES OF SELECTIVE RIPS COMPLEXES

Boštjan Lemež and Žiga Virk

Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia
e-mail:bostjan.lemez@imfm.si

Faculty of Computer and Information Science, University of Ljubljana, 1000 Ljubljana, Slovenia
e-mail:ziga.virk@fri.uni-lj.si

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.

2020 Mathematics Subject Classification.   53C22, 55N35, 55Q05, 55U10, 57N65

Key words and phrases.   Reconstruction results, Rips complexes, Riemannian manifolds, geodesic spaces, fundamental groups

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

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