Hyperplane restrictions of indecomposable $n$-dimensional persistence modules

Understanding the structure of indecomposable $n$‑dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular $(n-1)$‑dimensional persistence module with finite support is a hyperplane restriction of an indecomposable $n$‑dimensional persistence module. We extend this result to the following: If $M$ is any finitely presented $(n-1)$‑dimensional persistence module with finite support, then there exists an indecomposable ndimensional persistence module $M^\prime$ such that $M$ is the restriction of $M^\prime$ to a hyperplane. We also show that any finite zigzag persistence module is the restriction of some indecomposable $3$‑dimensional persistence module to a path.

Keywords

persistent homology

2010 Mathematics Subject Classification

13C05, 55N35

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This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1650116.

Received 12 December 2020

Received revised 22 July 2021

Accepted 23 August 2021

Published 24 August 2022