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  "body": "\n## Summary\n\nThe homotopy groups of spheres are the groups $\\pi_k(S^n)$ of homotopy classes of based maps from the $k$-sphere to the $n$-sphere.\nSome are simple: $\\pi_k(S^n) = 0$ for $k \u003c n$, and $\\pi_n(S^n) \\cong \\Z$, detected by the degree.\nFor $k \u003e n$ they are mostly finite but highly irregular, and no complete formula is known.\nTheir computation has driven much of algebraic topology, through fibrations, spectral sequences and stable homotopy theory.\n\n## Explanations\n\n### L1 human\n\nThese groups ask in how many essentially different ways one sphere can be wrapped around another.\nWrapping a circle around a circle is counted by the winding number, an integer.\nSurprisingly, a $3$-sphere can be wrapped around a $2$-sphere in infinitely many different ways, by the Hopf map and its multiples, even though the $3$-sphere has higher dimension.\nSmall changes in the dimensions give very different answers, which is why these groups are famous.\n",
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