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  "body": "\n## Summary\n\nThe Hopf invariant is an integer $H(f)$ attached to a map $f \\colon S^{2n-1} \\to S^n$.\nIt is read off from the [[cup product|JX47MS1R1H]] in the cohomology of the space obtained by attaching a $2n$-cell to $S^n$ along $f$.\nIt depends only on the homotopy class of $f$ and gives a homomorphism $\\pi_{2n-1}(S^n) \\to \\Z$.\nIt showed that $\\pi_3(S^2)$ is infinite, and the question of which $n$ admit maps of Hopf invariant one is tied to division algebras.\n\n## Explanations\n\n### L1 human\n\nThe Hopf invariant measures how much a map from a big sphere to a smaller sphere is twisted.\nFor the Hopf map $S^3 \\to S^2$, the preimages of two different points are two circles that are linked once, and the Hopf invariant is the linking number, $\\pm 1$.\nFor a constant map nothing is linked, and the invariant is $0$.\n",
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