Defining and Computing Topological Persistence for 1-cocycles

TL;DR

This paper introduces a novel concept of topological persistence for 1-cocycles, extending the idea to a broader class of maps, with practical applications in data ranking and discrete vector fields.

Contribution

It defines and computes topological persistence for 1-cocycles, a generalization not previously explored, using level persistence techniques.

Findings

Defined topological persistence for 1-cocycles

Developed methods to compute relevant persistence numbers

Applied the concept to practical scenarios like data ranking

Abstract

The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various directions. However, no attempt has been made so far to extend the concept of topological persistence to a generalization of `maps' such as cocycles which are discrete analogs of closed differential forms, a well known concept in differential geometry. We define a notion of topological persistence for 1-cocycles in this paper and show how to compute its relevant numbers. It turns out that, instead of the standard persistence, one of its variants which we call level persistence can be leveraged for this purpose. It is worth mentioning that 1-cocyles appear in practice such as in data ranking or in discrete vector fields.