Rigidity of terminal simplices in persistent homology

TL;DR

This paper investigates the rigidity of terminal simplices in persistent homology, providing conditions under which these simplices remain unchanged under small perturbations of the filtration function, thus addressing stability in discrete settings.

Contribution

It introduces a sufficient condition for the rigidity of terminal simplices in persistent homology, linking stability to barcodes in adjacent dimensions.

Findings

Provides a criterion for terminal simplex rigidity

Links stability conditions to barcodes in dimensions n and n+1

Enhances understanding of persistence stability in discrete complexes

Abstract

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the critical simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on $\epsilon > 0$ implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through $\epsilon$-perturbations of filtration function. The condition for a homology class or a bar in dimension n depends on the barcodes in dimensions n and n+1.