Exotic R4

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In mathematics, an exotic ${\displaystyle \mathbb {R} ^{4}}$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space ${\displaystyle \mathbb {R} ^{4}.}$ The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.^[1][2] There is a continuum of non-diffeomorphic differentiable structures ${\displaystyle \mathbb {R} ^{4},}$ as was shown first by Clifford Taubes.^[3]

Prior to this construction, non-diffeomorphic smooth structures on spheres – exotic spheres – were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures ${\displaystyle \mathbb {R} ^{n};}$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$^[4]