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  "body": "\n## Summary\n\nThe Kervaire invariant is an invariant in $\\Z/2$ of a closed framed smooth manifold of dimension $4k + 2$.\nIt is the Arf invariant of a quadratic refinement, built from the framing, of the mod $2$ intersection form on the middle cohomology.\nMichel Kervaire used it in 1960 to give a closed topological manifold of dimension $10$ with no smooth structure.\nThrough the Pontryagin–Thom construction it becomes a homomorphism on the [[stable homotopy groups of spheres|HGZAYPQBXW]].\nThe question of where it is nonzero is the Kervaire invariant problem.\n\n## Explanations\n\n### L1 human\n\nA framing of a manifold is a consistent choice of a basis of directions at every point.\nOn a framed manifold of dimension $4k + 2$, the framing lets one assign to each middle-dimensional mod $2$ class a value $0$ or $1$.\nThese values form a quadratic form over the field with two elements, and such a form has a single invariant, its Arf invariant.\nThat invariant is the Kervaire invariant.\nFor example, the torus $S^1 \\times S^1$ with the framing that comes from its structure as a Lie group has Kervaire invariant $1$.\n",
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