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  "body": "\n## Description\n\nAn *exotic $\\R^4$* is a [[smooth manifold|F45VQ47GPV]] that is [[homeomorphic|0QT00PB5G2]] to the [[Euclidean space|0XPR1E85W0]] $\\R^4$ but not [[diffeomorphic|0KZZ70ZHXP]] to $\\R^4$ with its standard smooth structure.\nExotic $\\R^4$'s exist.\nThey were found in the early 1980s as a consequence of [[Freedman's classification|HSD6FKCYRC]] of topological $4$-manifolds and [[Donaldson's theorem|34B7K77P31]] on smooth ones.\n\n## Remarks\n\nFor $n \\ne 4$, every smooth manifold homeomorphic to $\\R^n$ is diffeomorphic to $\\R^n$, so dimension $4$ is the only exception.\nClifford Taubes showed that there are uncountably many pairwise non-diffeomorphic exotic $\\R^4$'s.\nSome of them, the small exotic $\\R^4$'s, embed smoothly as open subsets of the standard $\\R^4$; others, the large ones, do not.\nThe constructions use handle decompositions and Casson handles rather than explicit coordinates.\n\n## Explanations\n\n### L1 human\n\nFour-dimensional space can be given smooth structures, ways of doing calculus, that are different from the usual one.\nEach of them is the same as ordinary $\\R^4$ for continuous deformation, but no smooth change of coordinates turns it into ordinary $\\R^4$.\nThis happens in no other dimension, and it is one of the strangest facts of geometry.\n",
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