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2019 Parametrized homology via zigzag persistence
Gunnar Carlsson, Vin de Silva, Sara Kališnik, Dmitriy Morozov
Algebr. Geom. Topol. 19(2): 657-700 (2019). DOI: 10.2140/agt.2019.19.657

Abstract

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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Gunnar Carlsson. Vin de Silva. Sara Kališnik. Dmitriy Morozov. "Parametrized homology via zigzag persistence." Algebr. Geom. Topol. 19 (2) 657 - 700, 2019. https://doi.org/10.2140/agt.2019.19.657

Information

Received: 21 January 2017; Revised: 27 June 2018; Accepted: 1 August 2018; Published: 2019
First available in Project Euclid: 19 March 2019

zbMATH: 07075112
MathSciNet: MR3924175
Digital Object Identifier: 10.2140/agt.2019.19.657

Subjects:
Primary: 55N35, 55N99

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 2 • 2019
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