Exotic R4

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an exotic 
 
 
 
 
 
 R
 
 
 4
 
 
 
 
 {\displaystyle \mathbb {R} ^{4}}
 
 is a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> that is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> (i.e. shape preserving) but not <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphic</a> (i.e. non smooth) to the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
 
 
 
 
 
 R
 
 
 4
 
 
 .
 
 
 {\displaystyle \mathbb {R} ^{4}.}
 
 The first examples were found in 1982 by <a href="/facts/Michael_Freedman/FxESCuA7">Michael Freedman</a> and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and <a href="/facts/Simon_Donaldson/zkQ5xq5u">Simon Donaldson</a>'s theorems about smooth 4-manifolds. There is a <a href="/facts/Cardinality_of_the_continuum/ob578Q9l">continuum</a> of non-diffeomorphic <a href="/facts/Differentiable_structure/nHHTJud2">differentiable structures</a> 
 
 
 
 
 
 R
 
 
 4
 
 
 ,
 
 
 {\displaystyle \mathbb {R} ^{4},}
 
 as was shown first by <a href="/facts/Clifford_Taubes/fB11uJb7">Clifford Taubes</a>.
Prior to this construction, non-diffeomorphic <a href="/facts/Smooth_structure/lfvfhqxr">smooth structures</a> on spheres – <a href="/facts/Exotic_sphere/3YKNwX1I">exotic spheres</a> – were already known to exist, although the question of the existence of such structures for the particular case of the <a href="/facts/4-sphere/bFu2S7E2">4-sphere</a> remained open (and remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures 
 
 
 
 
 
 R
 
 
 n
 
 
 ;
 
 
 {\displaystyle \mathbb {R} ^{n};}
 
 in other words, if n ≠ 4 then any smooth manifold homeomorphic to 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 is diffeomorphic to 
 
 
 
 
 
 R
 
 
 n
 
 
 .
 
 
 {\displaystyle \mathbb {R} ^{n}.}

Exotic R4 open-in-new

Exotic R4