Localized Homology

In this paper, we introduce localized homology, a theory for finding local geometric descriptions for topological attributes. Given a space and a cover of subspaces, we construct the blowup complex, a derived space that contains both local and global information. The persistent homology of the blowup complex localizes the topological attributes of the space. Our theory is general and applies in all dimensions. After an informal description, we formalize our approach for general spaces, adapt it for simplicial complexes, and develop a simple algorithm that works directly on the input. In each stage, we prove the theoretical equivalence of the methods. We also implement our algorithm and give preliminary results to validate our methods in practice.

• Type: Conference
• Year: 2007
• Where: SMI
• Authors: Afra Zomorodian, Gunnar Carlsson
• Keywords: Blowup Complex, Local Geometric Descriptions, SMI 2007, Topological Attributes