Interleaving Mayer–Vietoris spectral sequences

Torras-Casas, Álvaro; Pennig, Ulrich

doi:10.2140/agt.2024.24.4265

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Interleaving Mayer–Vietoris spectral sequences

Álvaro Torras-Casas and Ulrich Pennig

Algebraic & Geometric Topology 24 (2024) 4265–4306

DOI: 10.2140/agt.2024.24.4265

Abstract

We discuss the Mayer–Vietoris spectral sequence as an invariant in the context of persistent homology. In particular, we introduce the notion of $𝜀$ –acyclic carriers and $𝜀$ –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 $𝜀$ –interleavings exist between the spectral sequences associated to two different covers.

Keywords

Mayer–Vietoris, acyclic carrier, interleaving, spectral sequence, regular morphism

Mathematical Subject Classification

Primary: 55N31, 55T99

References

Publication

Received: 28 May 2021

Revised: 31 January 2023

Accepted: 11 April 2023

Published: 17 December 2024

Authors

Álvaro Torras-Casas
School of Mathematics Cardiff University Cardiff United Kingdom

Ulrich Pennig
School of Mathematics Cardiff University Cardiff United Kingdom

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Volume 24, issue 8 (2024)