Finite reconstruction with selective Rips complexes

TL;DR

This paper extends the theory of topological reconstruction of metric spaces, showing that selective Rips complexes can recover the homotopy type of a Riemannian manifold close to the space, generalizing previous results.

Contribution

It introduces selective Rips complexes and proves they can reconstruct the homotopy type of manifolds, extending Latchev's theorem and providing a new functorial framework.

Findings

Selective Rips complexes recover manifold homotopy types

Generalization of Latchev's reconstruction theorem

New functorial setting for complex construction

Abstract

Selective Rips complexes corresponding to a sequence of parameters are a generalization of Vietoris-Rips complexes utilizing the idea of thin simplices. We prove that if a metric space $Y$ is close (in Gromov-Hausdorff distance) to a closed Riemannian manifold $X$, then selective Rips complexes of $Y$ for certain parameters attain the homotopy type of $X$. This result is a generalization of Latchev's reconstruction result from Vietoris-Rips complexes to selective Rips complexes. In particular, we present a novel proof for the Latschev's theorem as a special case. We also present a functorial setting, which is new even in the case of Vietoris-Rips complexes.