Reeb complexes and topological persistence

TL;DR

This paper introduces Reeb complexes to analyze how homology generators evolve along functions, establishing their relation to persistent homology and levelset zigzags through spectral sequences.

Contribution

It formalizes Reeb complexes and connects them to existing topological data analysis methods like persistent homology and zigzags.

Findings

Reeb complexes capture homology flow along functions.

Reeb complexes relate closely to levelset zigzags.

Spectral sequences reveal the connection between these methods.

Abstract

We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data analysis such as levelset zigzags and persistent homology. We make this relation precise and in particular explain how Reeb complexes and levelset zigzags can be extracted from the first pages of respective spectral sequences with the same termination.