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Bifiltrations and persistence paths for $2$–Morse functions

Ryan Budney and Tomasz Kaczynski

Algebraic & Geometric Topology 23 (2023) 2895–2924
Abstract

We study the homotopy type of bifiltrations of compact manifolds induced as the preimage of filtrations of 2 for generic smooth functions f : M 2. The primary goal of the paper is to allow for a simple description of the multigraded persistent homology associated to such filtrations. Our main result is a description of the evolution of the bifiltration of f in terms of cellular attachments. Analogs of the Morse–Conley equation and Morse inequalities along so-called persistence paths are derived, and a scheme for computing pathwise barcodes is proposed.

Keywords
persistent homology, Morse theory, bifiltrations
Mathematical Subject Classification
Primary: 57R35
Secondary: 55M99, 55N31
References
Publication
Received: 25 October 2021
Revised: 9 March 2022
Accepted: 1 May 2022
Published: 7 September 2023
Authors
Ryan Budney
Mathematics and Statistics
University of Victoria
Victoria, BC
Canada
Tomasz Kaczynski
Département de Mathématiques
Université de Sherbrooke
Sherbrooke, QC
Canada

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