Persistent Cup Product Structures and Related Invariants

TL;DR

This paper introduces a new stable 2D persistence module structure based on the cohomological cup product, extending persistent invariants and defining a novel persistent LS-category invariant for topological data analysis.

Contribution

It develops the persistent cup module, studies its stability, and introduces a generalized stable persistent invariant including the persistent LS-category.

Findings

The persistent cup module is stable under interleaving distance.

Persistent $ ext{ell}$-cup modules are stable and have associated persistence diagrams.

A generalized stable persistent invariant extends rank, cup-length, and LS-category invariants.

Abstract

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g.~the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length $\ell\geq0$ and the other is the filtration parameter. This new persistence structure, called the persistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter $\ell$, we obtain a 1-dimensional persistence module, called the persistent $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant, which extends both the rank invariant (also referred to as persistent Betti number), Puuska's rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant.