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Extending Persistence Using Poincaré and Lefschetz Duality

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An Erratum to this article was published on 08 January 2009

Abstract

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 extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds.

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Author information

Authors and Affiliations

  1. INRIA, 2004 Route des Lucioles, BP93, Sophia-Antipolis, France

    David Cohen-Steiner

  2. Department of Computer Science, Duke University, Durham, USA

    Herbert Edelsbrunner

  3. Geomagic, Research Triangle Park, NC, USA

    Herbert Edelsbrunner

  4. Department of Mathematics and Center for Computational Science, Engineering and Medicine, Duke University, Durham, NC, USA

    John Harer

Authors
  1. David Cohen-Steiner
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  2. Herbert Edelsbrunner
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  3. John Harer
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Corresponding author

Correspondence to Herbert Edelsbrunner.

Additional information

Communicated by Konstantin Mischaikow.

Research by all three authors is partially supported by DARPA under grant HR0011-05-1-0007. Research by the second author is also partially supported by NSF under grant CCR-00-86013.

An erratum to this article can be found at http://dx.doi.org/10.1007/s10208-008-9038-9

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Cohen-Steiner, D., Edelsbrunner, H. & Harer, J. Extending Persistence Using Poincaré and Lefschetz Duality. Found Comput Math 9, 79–103 (2009). https://doi.org/10.1007/s10208-008-9027-z

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  • DOI: https://doi.org/10.1007/s10208-008-9027-z

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