Rectification of interleavings and a persistent Whitehead theorem
Edoardo Lanari, Luis Scoccola · Algebraic & Geometric Topology · 2023
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 homotopyinvariant 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.