Interleaving Mayer-Vietoris spectral sequences

TL;DR

This paper explores the stability and interleaving properties of Mayer-Vietoris spectral sequences in persistent homology, introducing new concepts like ε-acyclic carriers and establishing approximate nerve theorems.

Contribution

It introduces ε-acyclic carriers and equivalences, analyzes stability conditions, and adapts Serre's work to spectral sequences in persistent homology.

Findings

Established stability conditions for spectral sequences under noise

Proved an approximate nerve theorem in the context of persistent homology

Derived conditions for ε-interleavings between spectral sequences of different covers

Abstract

We discuss the Mayer-Vietoris spectral sequence as an invariant in the context of persistent homology. In particular, we introduce the notion of $\varepsilon$-acyclic carriers and $\varepsilon$-acyclic equivalences between filtered regular CW-complexes and study stability conditions for the associated spectral sequences. We also look at the Mayer-Vietoris blowup complex and the geometric realization, finding stability properties under compatible noise; as a result we prove a version of an approximate nerve theorem. Adapting work by Serre we find conditions under which $\varepsilon$-interleavings exist between the spectral sequences associated to two different covers.