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  "body": "\n## Definition\n\nLet $(X, x_0)$ be a [[pointed space|T9DG8DDE75]], and let $n \\ge 1$ be an integer.\nLet $I^n = [0, 1]^n$ be the $n$-cube, with boundary $\\partial I^n$.\nThe *$n$-th homotopy group* $\\pi_n(X, x_0)$ is the set of [[homotopy|9YA6S6HD9V]] classes of continuous maps $f \\colon I^n \\to X$ with $f(\\partial I^n) = \\set{x_0}$.\nHere the homotopies $F \\colon I^n \\times [0, 1] \\to X$ must satisfy $F(s, t) = x_0$ for all $s \\in \\partial I^n$ and all $t$.\nThe group operation is $[f] + [g] = [f + g]$, where\n$$(f + g)(s_1, s_2, \\ldots, s_n) = \\begin{cases} f(2 s_1, s_2, \\ldots, s_n) \u0026 \\text{if } s_1 \\le \\tfrac12, \\\\ g(2 s_1 - 1, s_2, \\ldots, s_n) \u0026 \\text{if } s_1 \\ge \\tfrac12. \\end{cases}$$\n\n## Remarks\n\nSince $I^n / \\partial I^n$ is homeomorphic to $S^n$, the elements of $\\pi_n(X, x_0)$ are also the pointed homotopy classes of pointed maps $(S^n, s_0) \\to (X, x_0)$.\nFor $n = 1$ the definition gives the [[fundamental group|S1XK3HXHFX]].\nThe set $\\pi_0(X, x_0)$ is defined as the set of path components of $X$, with the component of $x_0$ as base point; it is not a group in general.\n\nExample: $\\pi_n(\\R^m, 0)$ is trivial for all $n \\ge 1$.\nFor $f$ as above, $F(s, t) = (1 - t) f(s)$ is a homotopy that keeps $\\partial I^n$ at $0$ and ends at the constant map.\n\nNon-example: a map $f \\colon I^n \\to X$ that does not send all of $\\partial I^n$ to $x_0$ does not define an element of $\\pi_n(X, x_0)$.\nLikewise, free homotopy classes of maps $S^1 \\to X$ form in general a different set: they correspond to conjugacy classes of $\\pi_1(X, x_0)$ when $X$ is path-connected.\n",
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