Extending Persistence Using Poincaré and Lefschetz Duality

Persistent homology has proven to be a useful tool in a vari-ety 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 per-sistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic for-mulation 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 Poincare ́ duality but generalizes to non-manifold spaces. We prove stability for general triangu-lated spaces and duality as well as symmetry for triangulated manifolds