Rectification of interleavings and a persistent Whitehead theorem
arXiv:2010.05378 · doi:10.2140/agt.2023.23.803
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.
25 pages. v2: improved constant of Thm A and minor exposition improvements ; v3: imporved constant of Thm A and shortened exposition, final version to appear in Algebraic & Geometric Topology