Persistent Cup-Length

TL;DR

This paper introduces the persistent cup-length function, a new invariant in persistent cohomology that captures ring-theoretic information across filtrations, computable efficiently and stable under interleaving distances.

Contribution

It extends the cup-length concept to persistent cohomology, providing a computable and stable invariant that encodes ring structure evolution in filtrations.

Findings

The persistent cup-length function can be computed from representative cocycles.

A polynomial time algorithm for computing the persistent cup-length function is devised.

The invariant is proven to be stable under interleaving-type distances.

Abstract

Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.