Parametrized homology via zigzag persistence

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

doi:10.2140/agt.2019.19.657

Parametrized homology via zigzag persistence

Journal Article · Tue Mar 12 00:00:00 EDT 2019 · Algebraic & Geometric Topology

DOI:https://doi.org/10.2140/agt.2019.19.657· OSTI ID:1604670

Carlsson, Gunnar ^[1]; de Silva, Vin ^[2]; Kališnik, Sara ^[3]; Morozov, Dmitriy ^[4]

Stanford Univ., CA (United States)
Pomona College, Claremont, CA (United States)
Wesleyan Univ., Middletown, CT (United States)
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

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; here, 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.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); Gordon and Betty Moore Foundation; Alfred P. Sloan Foundation

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1604670

Journal Information:: Algebraic & Geometric Topology, Journal Name: Algebraic & Geometric Topology Journal Issue: 2 Vol. 19; ISSN 1472-2747

Publisher:: Mathematical Sciences PublishersCopyright Statement

Country of Publication:: United States

Language:: English

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Cited By (2)

Decomposition of Exact pfd Persistence Bimodules Cochoy, Jérémy; Oudot, Steve Discrete & Computational Geometry, Vol. 63, Issue 2 https://doi.org/10.1007/s00454-019-00165-z	journal	December 2019
Decomposition of exact pfd persistence bimodules Cochoy, Jérémy; Oudot, Steve arXiv https://doi.org/10.48550/arxiv.1605.09726	preprint	January 2016

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