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  "body": "\n## Definition\n\nLet $SO = \\operatorname{colim}_n SO(n)$ be the infinite [[special orthogonal group|0CGYNDCWRR]], and let $r \\ge 0$.\nLet $J \\colon \\pi_r(SO) \\to \\pi_r^s$ be the stable [[J-homomorphism|YYJPRD1HYW]] into the [[stable homotopy group of spheres|HGZAYPQBXW]] $\\pi_r^s$.\nThe *image of J* in degree $r$ is the subgroup $\\im J \\subseteq \\pi_r^s$.\n\n## Remarks\n\nBy Bott periodicity, for $r \\ge 1$ the group $\\pi_r(SO)$ is $\\Z/2$ if $r \\equiv 0, 1 \\pmod 8$, $\\Z$ if $r \\equiv 3, 7 \\pmod 8$, and $0$ otherwise.\nFor $r = 4k - 1$, the image is cyclic of order the denominator of $B_{2k}/4k$, where $B_{2k}$ is the [[Bernoulli number|0KC00K7KY4]].\nFor $r \\equiv 0, 1 \\pmod 8$ with $r \\ge 1$, $J$ is injective and the image is $\\Z/2$.\n\nExample: in degree $3$, $B_2 = 1/6$ gives $B_2 / 4 = 1/24$, and the image of J is all of $\\pi_3^s \\cong \\Z/24$.\n\nNon-example: in degree $2$, $\\pi_2(SO) = 0$, so the image of J is $0$; the nonzero element $\\eta^2$ of $\\pi_2^s \\cong \\Z/2$ is not in the image of J.\n\nJ. F. Adams determined the image, with the upper bound depending on the Adams conjecture, which was later proved by Quillen and by Sullivan.\n",
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