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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $S^n = \\set{x \\in \\R^{n+1} : \\norm{x} = 1}$ be the unit $n$-sphere with base point $s_0 = (1, 0, \\ldots, 0)$.\nFor $k \\ge 1$, the group $\\pi_k(S^n)$ is the [[homotopy group|P976WYEFMB]] $\\pi_k(S^n, s_0)$.\nThe *homotopy groups of spheres* are the groups $\\pi_k(S^n)$ for all $k, n \\ge 1$.\n\n## Remarks\n\nChanging the base point gives isomorphic groups, because $S^n$ is path-connected; this is why it is usually omitted.\nExample: $\\pi_1(S^1) \\cong \\Z$, and the class of a loop is its winding number.\nExample: $\\pi_k(S^n) = 0$ for $1 \\le k \u003c n$.\nGive $S^n$ the CW structure with one $0$-cell and one $n$-cell.\nBy cellular approximation, a based map $S^k \\to S^n$ is homotopic to a map into the $k$-skeleton, which is a point.\n\nNon-example: the groups are not the homology groups of spheres.\nThe homology $H_3(S^2)$ is zero, but $\\pi_3(S^2) \\cong \\Z$, generated by the Hopf map.\n\nFurther values: $\\pi_n(S^n) \\cong \\Z$ for all $n \\ge 1$, $\\pi_k(S^1) = 0$ for $k \\ge 2$, and $\\pi_{n+1}(S^n) \\cong \\Z/2$ for $n \\ge 3$.\n",
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