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Rectification of interleavings and a persistent Whitehead theorem

Edoardo Lanari and Luis Scoccola

Algebraic & Geometric Topology 23 (2023) 803–832
Abstract

The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations of close-by real-valued functions are close-by. There are other ways of constructing homotopy-invariant distances, but not much is known about the relationships between these choices. We show that other natural distances differ from the homotopy interleaving distance in at most a multiplicative constant, and prove versions of the persistent Whitehead theorem, a conjecture of Blumberg and Lesnick that relates morphisms that induce interleavings in persistent homotopy groups to stronger homotopy-invariant notions of interleaving.

Keywords
homotopy interleaving, rectification, Whitehead theorem
Mathematical Subject Classification
Primary: 55N31, 62R40
Secondary: 18N40, 18N50, 55U10, 55U35
References
Publication
Received: 8 February 2021
Revised: 6 July 2021
Accepted: 30 October 2021
Published: 9 May 2023
Authors
Edoardo Lanari
Institute of Mathematics
Czech Academy of Sciences
Prague
Czech Republic
http://edolana.github.io
Luis Scoccola
Department of Mathematics
Northeastern University
Boston, MA
United States
http://luisscoccola.github.io

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