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Parametrized homology via zigzag persistence

Gunnar Carlsson, Vin de Silva, Sara Kališnik and Dmitriy Morozov

Algebraic & Geometric Topology 19 (2019) 657–700
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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Keywords
persistent homology, zigzag persistence, levelset zigzag persistence, extended persistence
Mathematical Subject Classification 2010
Primary: 55N35, 55N99
References
Publication
Received: 21 January 2017
Revised: 27 June 2018
Accepted: 1 August 2018
Published: 12 March 2019
Authors
Gunnar Carlsson
Department of Mathematics
Stanford University
Stanford, CA
United States
Vin de Silva
Department of Mathematics
Pomona College
Claremont, CA
United States
Sara Kališnik
Department of Mathematics Wesleyan University
Middletown, CT
United States
Dmitriy Morozov
Lawrence Berkeley National Laboratory
Berkeley Institute for Data Science
Berkeley, CA
United States