Persistent Homology and Persistent Cohomology: A Review

Persistent homology has become an important tool in non-linear data reduction. Its sister theory, Persistent cohomology, has attracted less attention in the past eventhough it has many advantages. We surveyed several literatures and present a summary of the status quo in persistent (co)homology covering the theory, computations, representations in terms of cycles (for persistent homology) and cocycles (for persistent cohomology), lens (quotient) space and their equivalence. Moreover, we computed the persistent homology and persistent cohomology for the 2 - sphere both manually and computationally (Using Ripserer). In both cases, same result was obtained, particularly in the computation of their barcodes much more that persistent cohomology is not only faster in computation than persistent homology, but also uses less memory in a little time.