Persistent Homology of Filtered Covers

TL;DR

This paper extends the Nerve Lemma to filtered covers, establishing isomorphism of persistent homology groups, which enhances the theoretical foundation of persistent homology in topological data analysis.

Contribution

It introduces a homological algebra-based extension of the Nerve Lemma that applies to filtered covers, ensuring isomorphism of persistent homology groups.

Findings

Proves an isomorphism between persistent homology groups of filtered covers and their nerves.

Uses homological algebra to handle non-commuting maps in chain complexes.

Provides a theoretical foundation for persistent homology in topological data analysis.

Abstract

We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis, the usual Nerve Lemma does not provide isomorphism of persistent homology groups. Our argument involves some homological algebra: the key point being that although the maps produced in the standard proof of the Nerve Lemma do not commute as maps of chain complexes, the maps they induce on homology do.