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Algebraic stability of zigzag persistence modules

Magnus Bakke Botnan and Michael Lesnick

Algebraic & Geometric Topology 18 (2018) 3133–3204

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 –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.

topological data analysis, persistent homology, interleavings
Mathematical Subject Classification 2010
Primary: 55N35
Secondary: 55U99
Received: 16 April 2017
Revised: 28 January 2018
Accepted: 11 March 2018
Published: 18 October 2018
Magnus Bakke Botnan
Zentrum Mathematik
Technische Universität München
Garching bei München
Michael Lesnick
Princeton Neuroscience Institute
Princeton University
Princeton, NJ
United States