Rectification of interleavings and a persistent Whitehead theorem

TL;DR

This paper explores the relationships between different homotopy-invariant distances in persistent topology and proves a version of the Whitehead theorem, establishing connections between morphisms and interleavings in persistent homotopy groups.

Contribution

It demonstrates that various natural distances differ only by a multiplicative constant and proves a persistent Whitehead theorem relating morphisms and interleavings.

Findings

Different natural distances are within a constant factor of the homotopy interleaving distance.

Established a version of the persistent Whitehead theorem.

Connected morphisms inducing interleavings to stronger homotopy-invariant notions.

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.