An Excision Theorem for Persistent Homology

TL;DR

This paper proves an excision property for persistent homology groups, extending the applicability of the Mayer-Vietoris sequence to a broader class of filtrations in topological data analysis.

Contribution

It establishes an excision theorem for persistent homology and extends the Mayer-Vietoris sequence to more general filtrations, broadening theoretical foundations.

Findings

Excision property holds for a large class of filtrations.

Method extends Mayer-Vietoris sequence to new filtrations.

Applicable to a broader class of topological data analysis scenarios.

Abstract

We demonstrate that an excision property holds for persistent homology groups. This property holds for a large class of filtrations, and in fact we show that given any filtration on a larger space, we can extend it to a filtration of two subspaces which guarantees that the excision property holds for the triple. This method also applies to the Mayer-Vietoris sequence in persistent homology introduced by DiFabio and Landi in 2011, extending their results to a much larger class of filtrations.