Persistent Homology, Matroids and Cobordisms

TL;DR

This paper explores how the complex combinatorial structure of homology classes in filtered simplicial complexes can be encoded using filtered matroids and rooted forests, which can be realized as cobordisms, enriching the understanding of persistent homology.

Contribution

It introduces a novel encoding of homology class combinatorics via filtered matroids and rooted forests, and demonstrates their realization as cobordisms, advancing topological data analysis methods.

Findings

Homological information can be represented by filtered matroids.

Rooted forests effectively encode homology class combinatorics.

Rooted forests can be realized as cobordisms.

Abstract

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.