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  "body": "\n## Statement\n\nLet $n \\ge 2$.\nThere is a continuous map $f \\colon S^{2n-1} \\to S^n$ with [[Hopf invariant|TEGTSY7XMX]] $H(f) = \\pm 1$ if and only if $n \\in \\set{2, 4, 8}$.\n\n## Remarks\n\nThe maps for $n = 2, 4, 8$ are the Hopf fibrations built from the complex numbers, the quaternions and the octonions.\nFor odd $n$, every Hopf invariant is $0$, and for even $n$, the Whitehead product $[\\iota_n, \\iota_n]$ has Hopf invariant $\\pm 2$, so only the value $\\pm 1$ is in question.\nAn argument with Steenrod squares shows that $n$ must be a power of $2$.\nJ. F. Adams excluded all remaining $n \\ge 16$ with secondary cohomology operations, which is [[Adams' theorem on Hopf invariant one|ZJVKZ8D64P]].\nAdams and Atiyah later gave a short proof with Adams operations in topological K-theory.\nConsequences: $\\R^n$ carries the structure of a real division algebra, with a bilinear product and no zero divisors, only for $n = 1, 2, 4, 8$, and $S^{n-1}$ is an H-space only for $n = 1, 2, 4, 8$.\n\n## Explanations\n\n### L1 human\n\nThe Hopf map wraps the $3$-sphere around the $2$-sphere with the simplest possible twist, measured by Hopf invariant one.\nThe question is in which dimensions such a simplest twist exists.\nThe answer is only in dimensions $2$, $4$ and $8$, the dimensions of the complex numbers, the quaternions and the octonions.\n\n### L2 human\n\nThe theorem explains why the list $\\R, \\C, \\mathbb{H}, \\mathbb{O}$ of division algebras stops.\nA bilinear product without zero divisors on $\\R^n$ makes $S^{n-1}$ an H-space, and the Hopf construction turns this multiplication into a map $S^{2n-1} \\to S^n$ of Hopf invariant $\\pm 1$.\nSo the topological theorem implies the algebraic one.\nThe problem also led to the Adams spectral sequence and to the use of K-theory operations in homotopy theory.\n",
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