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Parametrized homology
via zigzag persistence
Gunnar Carlsson, Vin de Silva, Sara Kališnik and Dmitriy Morozov |
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Algebraic & Geometric Topology 19 (2019) 657–700
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Abstract |
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This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. This information is encoded as a “barcode” of real intervals, each corresponding to a homological feature supported over that interval; or, equivalently, as a persistence diagram. Points in the persistence diagram are classified algebraically into four classes; geometrically, the classes identify the distinct ways in which homological features perish at the boundaries of their interval of persistence. We study the conditions under which spaces fibered over the real line have a well-defined parametrized homology; we establish the stability of these invariants and we show how the four classes of persistence diagram correspond to the four diagrams that appear in the theory of extended persistence. |
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Keywords
persistent homology, zigzag persistence, levelset zigzag
persistence, extended persistence
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Mathematical Subject Classification 2010
Primary: 55N35, 55N99
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References |
Publication
Received: 21 January 2017
Revised: 27 June 2018
Accepted: 1 August 2018
Published: 12 March 2019
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Authors |
| Gunnar Carlsson | |
| Department of Mathematics Stanford University Stanford, CA United States |
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| Vin de Silva | |
| Department of Mathematics Pomona College Claremont, CA United States |
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| Sara Kališnik | |
| Department of Mathematics Wesleyan
University Middletown, CT United States |
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| Dmitriy Morozov | |
| Lawrence Berkeley National
Laboratory Berkeley Institute for Data Science Berkeley, CA United States |
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