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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $A \\subseteq X$ be a subspace, let $x_0 \\in A$, and let $n \\ge 1$.\nIn the cube $I^n = [0, 1]^n$, let $I^{n-1}$ be the face where the last coordinate is $0$.\nLet $J^{n-1}$ be the closure of $\\partial I^n \\setminus I^{n-1}$.\nThe *relative homotopy group* $\\pi_n(X, A, x_0)$ is the set of [[homotopy|9YA6S6HD9V]] classes of maps $f \\colon I^n \\to X$ with $f(\\partial I^n) \\subseteq A$ and $f(J^{n-1}) = \\set{x_0}$.\nThe homotopies must keep these two conditions at every time.\nFor $n \\ge 2$, the sum $f + g$ is defined by the formula for absolute [[homotopy groups|P976WYEFMB]], putting $f$ and $g$ side by side in the first coordinate.\n\n## Remarks\n\nFor $n \\ge 2$ the sum makes $\\pi_n(X, A, x_0)$ a group, and for $n \\ge 3$ the group is abelian.\nFor $n = 1$ there is no group structure in general, and $\\pi_1(X, A, x_0)$ is only a pointed set.\n\nExample: when $A = \\set{x_0}$, the conditions say that $f$ sends all of $\\partial I^n$ to $x_0$, so $\\pi_n(X, x_0, x_0) = \\pi_n(X, x_0)$.\n\nExample: 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 $D^n$.\n\nNon-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.\n\nEquivalently, $\\pi_n(X, A, x_0)$ is the set of homotopy classes of maps of triples $(D^n, S^{n-1}, s_0) \\to (X, A, x_0)$.\n",
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