Extending persistence using poincaré and lefschetz duality

Journal Article (Journal Article)

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 3. 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 3, 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. © 2008 SFoCM.

Full Text

Duke Authors

Cited Authors

  • Cohen-Steiner, D; Edelsbrunner, H; Harer, J

Published Date

  • January 1, 2009

Published In

Volume / Issue

  • 9 / 1

Start / End Page

  • 79 - 103

Electronic International Standard Serial Number (EISSN)

  • 1615-3383

International Standard Serial Number (ISSN)

  • 1615-3375

Digital Object Identifier (DOI)

  • 10.1007/s10208-008-9027-z

Citation Source

  • Scopus