Persistent Cohomology Operations

Abstract

The work presented in this dissertation includes the study of cohomology and cohomological operations within the framework of Persistence. Although Persistence was originally defined for homology, recent research has developed persistent approaches to other algebraic topology invariants. The work in this document extends the field of persistence to include cohomology classes, cohomology operations and characteristic classes. By starting with presenting a combinatorial formula to compute the Stiefel-Whitney homology class, we set up the groundwork for Persistent Characteristic Classes. To discuss persistence for the more general cohomology classes, we construct an algorithm that allows us to find the Poincar'{e} Dual to a homology class. Then, we develop two algorithms that compute persistent cohomology, the general case and one for a specific cohomology class. We follow this with defining and composing an algorithm for extended persistent cohomology. In addition, we construct an algorithm for determining when a cohomology class is decomposible and compose it in the context of persistence. Lastly, we provide a proof for a concise formula for the first Steenrod Square of a given cohomology class and then develop an algorithm to determine when a cohomology class is a Steenrod Square of a lower dimensional cohomology class.