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  "body": "\n## Definition\n\nLet $k \\ge 0$ be an integer.\nFor $n \\ge 1$, let $E \\colon \\pi_{n+k}(S^n) \\to \\pi_{n+k+1}(S^{n+1})$ be the suspension homomorphism, which sends the class of $f$ to the class of its [[suspension|Q7RFNJH76Q]] $\\Sigma f$.\nThe *$k$-th stable homotopy group of spheres* is the colimit\n$$\\pi_k^s = \\operatorname{colim}_{n} \\pi_{n+k}(S^n)$$\nof the sequence of [[homotopy groups of spheres|YD6VAXXSBM]] $\\pi_{k+1}(S^1) \\to \\pi_{k+2}(S^2) \\to \\pi_{k+3}(S^3) \\to \\cdots$ along the maps $E$.\n\n## Remarks\n\nBy the [[Freudenthal suspension theorem|7DM2KSS2DB]], $E \\colon \\pi_{n+k}(S^n) \\to \\pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n \\ge k + 2$.\nSo $\\pi_k^s \\cong \\pi_{n+k}(S^n)$ for every $n \\ge k + 2$.\n\nExample: $\\pi_0^s \\cong \\Z$, because $\\pi_n(S^n) \\cong \\Z$ for all $n$ and $E$ preserves the degree.\n\nExample: $\\pi_1^s \\cong \\Z/2$, generated by the suspensions of the Hopf map $S^3 \\to S^2$; the group $\\pi_3(S^2) \\cong \\Z$ itself is not yet in the stable range, since $n = 2 \u003c k + 2 = 3$.\n\nThe next values are $\\pi_2^s \\cong \\Z/2$, $\\pi_3^s \\cong \\Z/24$, $\\pi_4^s = \\pi_5^s = 0$, $\\pi_6^s \\cong \\Z/2$ and $\\pi_7^s \\cong \\Z/240$.\nComposition of maps makes $\\pi_*^s = \\bigoplus_k \\pi_k^s$ a graded commutative ring.\nFor $k \\ge 1$, the group $\\pi_k^s$ is finite.\n",
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