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  "body": "\n## Summary\n\nThe J-homomorphism $J \\colon \\pi_r(SO(n)) \\to \\pi_{r+n}(S^n)$ turns a family of rotations of $\\R^n$ parametrised by an $r$-sphere into a map of spheres.\nIt connects the homotopy of the rotation groups, known through Bott periodicity, with the homotopy groups of spheres.\nIn the stable range it gives a homomorphism $\\pi_r(SO) \\to \\pi_r^s$ into the [[stable homotopy groups of spheres|HGZAYPQBXW]], whose image is the [[image of J|GX8K2SCWVQ]].\n\n## Explanations\n\n### L1 human\n\nA loop of rotations of the plane, such as one full turn, can be used to twist a sphere around another sphere.\nThe J-homomorphism makes this precise in every dimension.\nIts value on one full turn of the plane is the Hopf map $S^3 \\to S^2$, up to sign.\nIt explains a regular part of the otherwise chaotic homotopy groups of spheres.\n",
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