Persistent Homology and Persistent Cohomology: A Review

Abstract

Persistent homology is an important tool in non-linear data reduction. Its sister theory, persistent cohomology, has attracted less attention in the past years eventhough it has many advantages. Several literatures dealing with theory and computations of persistent homology and cohomology were surveyed. Reasons why cohomology has been neglected over time are identified and, few possible solutions to the identified problems are made available. Furthermore, using Ripserer, the computation of persistent homology and cohomology using 2-sphere both manually and computationally are carried out. In both cases, same result was obtained, particularly in the computation of their barcodes which visibly revealed the point where the two coincides. Conclusively, it is observed that persistent cohomology is not only faster in computation than persistent homology, but also uses less memory in a little time.