Persistence and Topological Complexity

TL;DR

This paper introduces persistent versions of topological complexity and related invariants, establishing their stability and demonstrating their effectiveness in distinguishing spaces beyond traditional persistent homology.

Contribution

It develops persistent analogs of topological complexity and zero-divisor-cup-length, proves their stability, and shows they can better differentiate certain spaces.

Findings

Persistent invariants are stable under Gromov-Hausdorff distance.

Persistent topological complexity can distinguish spaces more effectively than persistent homology.

Erosion distances are bounded by twice the Gromov-Hausdorff distance.

Abstract

Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology.