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… 
Persistence modules: Algebra and algorithms
TLDR
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.
Alexander Duality for Parametrized Homology
This paper extends Alexander duality to the setting of parametrized homology. Let X ⊂ Rn×R with n ≥ 2 be a compact set satisfying certain conditions, let Y = (Rn×R)\X, and let p be the projection
Zigzag persistent homology and real-valued functions
TLDR
The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory.
Death and extended persistence in computational algebraic topology
The main aim of this paper is to explore the ideas of persistent homology and extended persistent homology, and their stability theorems, using ideas from [Bubenik and Scott, 2014; Cohen-Steiner,
Parametrized homology via zigzag persistence
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological
Height Persistence and Homology Generators in R 3 Efficiently
TLDR
It is shown that the persistence for height functions on general simplicial complexes K linearly embedded in R, hence called height persistence, can be computed in O(n log n) time, which improves significantly the current best bound of O( n), ω being the exponent of matrix multiplication.
Algebraic Stability of Zigzag Persistence Modules
TLDR
This paper functorially extends each zigzag persistence module to a two-dimensional persistence module, and establishes an algebraic stability theorem for these extensions, which yields a stability result for free two- dimensional persistence modules.
Sketches of a platypus: persistent homology and its algebraic foundations
TLDR
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.
Tracking a Generator by Persistence
TLDR
This paper considers the problem of tracking generating cycles with temporal coherence, and builds upon the matrix framework proposed by Cohen-Steiner et al. to swap two consecutive simplices, so that the algorithm can process a reordering directly.
Algorithms and hardness results in computational homology
TLDR
The definition and algorithm of a canonical basis of the homology group, namely, the optimal homology basis is provided, and various issues concerning characterizing topological features in computation are addressed.
...
1
2
3
4
5
...