Bifiltrations and persistence paths for 2–Morse functions
Ryan Budney, Tomasz Kaczyński · Algebraic & Geometric Topology · 2023
We study the homotopy type of bifiltrations of compact manifolds induced as the preimage of filtrations of R 2 for generic smooth functions f W M !R 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.