Homology Localization by Hierarchical Blowups

Topological descriptors such as the generators of homology groups are very useful in the analysis of complex data sets. It is often desired to find the smallest such generators to help localize the interesting features. One interpretation of localization utilizes a covering of the underlying space and computes generators contained within these covers. A similar construction was later used to compute persistence homology for smaller subsets in parallel before gluing the results. In this presentation, we describe a more efficient version of this construction and discuss how it can be used to find generators within a large class of subspaces.