Relative homotopy classes and groups

Let $x_0\in A\subseteq X$, with $A$ carrying its subspace topology, and $n\ge 1$. Set $F=I^{n-1}\times \{0\}$ and let $J$ be the union of the other faces of $I^n$. A relative representative is a continuous map $a:I^n\to X$ with $a(F)\subseteq A$ and $a(J)=\{x_0\}$. Two representatives are equivalent if connected by a continuous homotopy satisfying these same conditions at every time. Reversal of time and pasting in time give the equivalence relation, with constant homotopies giving reflexivity. Its classes form ${\pi }_n(X,A,x_0)$, pointed by the constant map.

For $n=1$, $F=\{0\}$ and $J=\{1\}$: representatives are paths from a variable point of $A$ to $x_0$. This is a pointed set, with no group operation asserted. Relative ${\pi }_0$ is not defined. For $n\ge 2$, concatenation uses coordinate 1 as in Higher homotopy group by based cubes, leaving the distinguished last coordinate untouched. The group law is established below. If $A=\{x_0\}$ this is the absolute cubical definition.