Reconstruction properties of selective Rips complexes

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.