Finite reconstruction with selective Rips complexes
Boštjan Lemež, Žiga Virk · Topology and its Applications · 2026
Selective Rips complexes corresponding to a sequence of parameters are a generalization of Rips complexes utilizing the idea of thin simplices. In this paper we prove they can be used to reconstruct the homotopy type of a closed Riemannian manifold X using a finite sample Y of X . In particular, for any sequence of parameters with positive limit and any closed Riemannian manifold X we prove the following: there exists a proximity parameter δ , such that for each metric space Y that is at Gromov-Hausdorff distance less than δ to X , the selective Rips complex of Y attains the homotopy type of X . This result is a generalization of Latchev's reconstruction result from Rips complexes to selective Rips complexes. When restricted to Rips complexes, our approach yields a novel proof for the Latschev's theorem. We also present a functorial setting, which is new even in the case of Rips complexes. The latter provides an interval of constant persistent homology, where homology is isomorphic to that of X .