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  "body": "\n## Description\n\nAn *exotic sphere* is a [[smooth manifold|F45VQ47GPV]] $\\Sigma$ that is [[homeomorphic|0QT00PB5G2]] to the standard sphere $S^n$ but not [[diffeomorphic|0KZZ70ZHXP]] to it.\nEquivalently, it is $S^n$ with a [[smooth structure|R2TQ82PCNB]] that is not diffeomorphic to the standard one.\n\nMilnor found the first examples in dimension $7$.\nHe took total spaces of bundles over $S^4$ with fibre $S^3$.\nFor suitable bundles he built a Morse function with exactly two critical points, so the total space is homeomorphic to $S^7$.\nHe then defined an invariant of such a manifold from a compact $8$-manifold that it bounds, using the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].\nThis invariant is zero for the standard $S^7$ and nonzero for some of the bundles, so these are not diffeomorphic to $S^7$.\n\n## Remarks\n\nKervaire and Milnor showed that for $n \\ge 5$ the oriented diffeomorphism classes of smooth homotopy $n$-spheres form a finite abelian group $\\Theta_n$ under [[connected sum|Y0K0K5NGWW]].\nFor $n = 7$ this group is cyclic of order $28$.\nFor $n = 1, 2, 3, 5, 6$ there are no exotic spheres.\nIn dimension $4$ it is unknown whether exotic spheres exist; this is the [[smooth four-dimensional Poincaré conjecture|P70VGYNFS5]].\n\n## Explanations\n\n### L1 human\n\nAn exotic sphere is a space that is a sphere for a topologist who allows bending and stretching, but not for one who also demands smoothness.\nThere is no smooth way to straighten it into the round sphere.\nThe first one appears in dimension seven.\nUp to orientation-preserving diffeomorphism, there are $28$ different smooth versions of the $7$-sphere.\n",
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