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  "body": "\n## Problem\n\nFor which integers $n = 4k + 2$ is there a closed framed smooth manifold of dimension $n$ whose [[Kervaire invariant|P4YXE08ZXF]] equals $1$?\nEquivalently, in which degrees $n$ do the [[stable homotopy groups of spheres|HGZAYPQBXW]] $\\pi^s_n$ contain an element of Kervaire invariant $1$?\n\n## Remarks\n\nKervaire showed in 1960 that the invariant vanishes in dimension $10$.\nBrowder showed in 1969 that the invariant can be nonzero only in dimensions $n = 2^j - 2$.\nFramed manifolds of Kervaire invariant $1$ exist in dimensions $2$, $6$, $14$, $30$ and $62$.\nHill, Hopkins and Ravenel showed that none exist in dimensions $2^j - 2$ with $j \\ge 8$, that is, in dimension $254$ and above.\nThis left dimension $126$ as the only open case; in 2024 Weinan Lin, Guozhen Wang and Zhouli Xu announced that a manifold of Kervaire invariant $1$ exists in dimension $126$.\n\n## Explanations\n\n### L1 human\n\nThe Kervaire invariant is a yes-or-no number attached to certain framed manifolds of dimension $4k + 2$.\nThe problem asks in which dimensions it can be \"yes\".\nThe answer turned out to be a short and strange list: $2$, $6$, $14$, $30$, $62$, and, by the 2024 announcement, $126$.\nThe question was a guide for decades of work on the homotopy groups of spheres.\n",
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