Parametrized Homology via Zigzag Persistence

TL;DR

This paper introduces a framework for parametrized homology using zigzag persistence, classifying features via interval measures and providing a detailed understanding of their lifespan in topological spaces.

Contribution

It extends homology to 1-parameter families with a novel classification of barcode intervals using finite rectangle measures.

Findings

Classifies homological features into four types based on interval measures.

Provides a new perspective on how features perish at interval endpoints.

Connects parametrized homology with persistence diagrams and zigzag persistence.

Abstract

This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently, as a persistence diagram. By defining persistence in terms of finite rectangle measures, we classify barcode intervals into four classes. Each of these conveys how the homological features perish at both ends of the interval over which they are defined.