Bi-filtrations and persistence paths for 2-Morse functions

TL;DR

This paper introduces a framework for analyzing the homotopy types of bi-filtrations induced by 2-Morse functions, providing a cellular attachment description and methods for computing persistence barcodes along paths.

Contribution

It offers a novel description of bi-filtrations via cellular attachments and introduces a scheme for computing persistence barcodes along specific paths.

Findings

Description of bi-filtration evolution through cellular attachments

Derivation of Morse-Conley type equations for persistence paths

Proposed scheme for path-wise barcode computation

Abstract

This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description of the multi-graded persistent homology associated to such filtrations. The main result of the paper is a description of the evolution of the bi-filtration of f in terms of cellular attachments. An analogy of Morse-Conley equation and Morse inequalities along so called persistence paths are derived. A scheme for computing path-wise barcodes is proposed.