Homotopy similarity of maps. Strong similarity

TL;DR

This paper introduces a new relation on homotopy classes of maps between pointed cellular spaces and conjectures its equivalence to an existing notion of r-similarity, aiming to deepen understanding of homotopy relations.

Contribution

It defines a novel relation on homotopy classes and proposes the conjecture that this relation always matches the r-similarity, advancing the theory of homotopy similarities.

Findings

Proposes a new relation $\, ext{ extasciitilde}^r$ on homotopy classes.

Conjectures that this relation coincides with the r-similarity $\, ext{ extasciitilde}_r$.

Provides a framework for future proof or counterexamples.

Abstract

Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge0$, we define a relation $\overset r\approx$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\overset r\sim$.