Death and extended persistence in computational algebraic topology

TL;DR

This paper investigates persistent and extended persistent homology, their stability, and applies these concepts to non-orientable surfaces like the Klein bottle, providing computational examples and theoretical insights.

Contribution

It extends persistent homology theory to non-orientable surfaces and explores the stability of extended persistent homology with computational illustrations.

Findings

Extended persistent homology is applicable to non-orientable surfaces.

Stability theorems for persistent homology are validated.

Computational methods for Klein bottle homology are demonstrated.

Abstract

The main aim of this paper is to explore the ideas of persistent homology and extended persistent homology, and their stability theorems, using ideas from [Bubenik and Scott, 2014; Cohen-Steiner, Edelsbrunner, and Harer, 2007; and Cohen-Steiner, Edelsbrunner, and Harer, 2009], as well as other sources. The secondary aim is to explore the homology (and cohomology) of non-orientable surfaces, using the Klein bottle as an example. We also use the Klein bottle as an example for the computation of (extended) persistent homology, referring to it throughout the paper.