Relative homotopy group

Remarks

For $n \ge 2$ the sum makes $\pi_n(X, A, x_0)$ a group, and for $n \ge 3$ the group is abelian. For $n = 1$ there is no group structure in general, and $\pi_1(X, A, x_0)$ is only a pointed set.

Example: when $A = \set{x_0}$, the conditions say that $f$ sends all of $\partial I^n$ to $x_0$, so $\pi_n(X, x_0, x_0) = \pi_n(X, x_0)$.

Example: for $n \ge 2$, $\pi_n(D^n, S^{n-1}, x_0) \cong \pi_{n-1}(S^{n-1}, x_0) \cong \Z$, by the long exact sequence of the pair and the contractibility of the disk $D^n$.

Non-example: a map $f$ with $f(\partial I^n) \subseteq A$ that does not send $J^{n-1}$ to $x_0$ is not a representative; the base point condition on $J^{n-1}$ is part of the definition.

Equivalently, $\pi_n(X, A, x_0)$ is the set of homotopy classes of maps of triples $(D^n, S^{n-1}, s_0) \to (X, A, x_0)$.