Localized Homology

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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.
Afra Zomorodian, Gunnar Carlsson
Added 04 Jun 2010
Updated 04 Jun 2010
Type Conference
Year 2007
Where SMI
Authors Afra Zomorodian, Gunnar Carlsson
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