Homology, Homotopy and Applications, vol. 15, pages 51-72, 2013. |

Preprint | |

DOI: | 10.4310/HHA.2013.v15.n1.a3 |

arXiv: | 1102.3389 |

Given a function f: X → R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X = R

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Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. Proceedings of the Annual Symposium on Computational Geometry, pages 247–256, 2009.

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Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. Quantifying transversality by measuring the robustness of intersections. Manuscript, Deptartment of Computer Science, Duke University, Durham, North Carolina, 2009.