Algebraic stability of zigzag persistence modules

M.B. Botnan, Michael Lesnick

Research output: Contribution to JournalArticleAcademicpeer-review


The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of ℝ–valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. We establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al.

Original languageEnglish
Pages (from-to)3133-3204
Number of pages72
JournalAlgebraic and Geometric Topology
Issue number6
Publication statusPublished - 18 Oct 2018
Externally publishedYes


Dive into the research topics of 'Algebraic stability of zigzag persistence modules'. Together they form a unique fingerprint.

Cite this