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  "body": "\n## Definition\n\nLet $r, n \\ge 1$, and let $SO(n)$ be the [[special orthogonal group|0CGYNDCWRR]].\nWrite $S^{r+n} = \\partial(D^{r+1} \\times D^n) = (S^r \\times D^n) \\cup (D^{r+1} \\times S^{n-1})$, and $S^n = D^n / \\partial D^n$ with base point the image of $\\partial D^n$.\nFor a map $f \\colon S^r \\to SO(n)$, define $J(f) \\colon S^{r+n} \\to S^n$ as follows:\n\n1. on $S^r \\times D^n$, set $J(f)(x, v) = [f(x) v]$, the class of $f(x) v \\in D^n$;\n2. on $D^{r+1} \\times S^{n-1}$, let $J(f)$ be constant at the base point.\n\nThe *J-homomorphism* is the map $J \\colon \\pi_r(SO(n)) \\to \\pi_{r+n}(S^n)$, $[f] \\mapsto [J(f)]$, on [[homotopy groups|P976WYEFMB]].\n\n## Remarks\n\nThe two clauses agree on $S^r \\times S^{n-1}$, since $f(x)$ preserves the unit sphere, so $f(x) v \\in \\partial D^n$.\nThe map $J$ is a well-defined group homomorphism.\n\nExample: for $n = 2$ and $r = 1$, let $f \\colon S^1 \\to SO(2)$ send $z$ to rotation by $z$, viewing $\\R^2$ as $\\C$.\nThen $J(f) \\colon S^3 \\to S^2$ is the Hopf map, up to sign and homotopy, and it generates $\\pi_3(S^2) \\cong \\Z$.\n\nNon-example: suppose $f(x)$ is allowed to be any invertible linear map.\nThen $f(x) v$ need not lie on $\\partial D^n$ for $v \\in S^{n-1}$, and may even leave $D^n$, so the two clauses do not define a map.\n\nThe maps $J$ commute with the stabilisation maps $SO(n) \\to SO(n+1)$ and with suspension, so they give the stable J-homomorphism $J \\colon \\pi_r(SO) \\to \\pi_r^s$.\n",
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