Higher homotopy group by based cubes

Let $I=[0,1]$, $n\ge 1$, and $x_0\in X$. Write $\mathrm{\partial }{I}^n$ for the points with at least one coordinate 0 or 1. The set ${\pi }_n(X,x_0)$ consists of continuous $a:I^n\to X$ with $a(\mathrm{\partial }{I}^n)=\{x_0\}$, modulo homotopies fixed on that entire boundary. This is a quotient by an equivalence relation by Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations.

The proposed product traverses $a$ first in coordinate 1: $(a\ast b)(s,u)=\{\begin{array}{ll}a(2s,u)& 0\le s\le 1\mathrm{/}2,\\ b(2s-1,u)& 1\mathrm{/}2\le s\le 1.\end{array}$ Here $u\in I^{n-1}$, absent when $n=1$. The constant map is denoted $e$ and reversal is $a^{-}(s,u)=a(1-s,u)$. The following lemmas establish well-definedness and the group laws. For $n=1$ this is the convention of Based loops and the fundamental group. Separately, ${\pi }_0(X,x_0)$ is the set of path components pointed by the component of $x_0$; no group law on it is asserted.