# Extending Persistence Using Poincaré and Lefschetz Duality

@article{CohenSteiner2009ExtendingPU, title={Extending Persistence Using Poincar{\'e} and Lefschetz Duality}, author={David Cohen-Steiner and Herbert Edelsbrunner and John Harer}, journal={Foundations of Computational Mathematics}, year={2009}, volume={9}, pages={79-103} }

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that…

## 58 Citations

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A number of facts about persistence modules are presented; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology.

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The various choices in use, and what they allow us to prove are examined, and the inherent differences between the choices people use are discussed, and potential directions of research are speculated on.

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