Homotopy similarity of maps. Strong similarity

Abstract

Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge 0$, we define a relation $\stackrel{r}{\approx }$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\stackrel{r}{\sim }$.

Treating basepoints revised. Notation changed. Many minor changes