# Exotic R4

In mathematics, an exotic ${\displaystyle \mathbb {R} ^{4}}$ is a differentiable manifold that is homeomorphic but not diffeomorphic 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 of ${\displaystyle \mathbb {R} ^{4},}$ 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 still remains open as of 2020). For any positive integer n other than 4, there are no exotic smooth structures on ${\displaystyle \mathbb {R} ^{n};}$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$[4]

## Small exotic R4s

An exotic ${\displaystyle \mathbb {R} ^{4}}$  is called small if it can be smoothly embedded as an open subset of the standard ${\displaystyle \mathbb {R} ^{4}.}$

Small exotic ${\displaystyle \mathbb {R} ^{4}}$  can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.

## Large exotic R4s

An exotic ${\displaystyle \mathbb {R} ^{4}}$  is called large if it cannot be smoothly embedded as an open subset of the standard ${\displaystyle \mathbb {R} ^{4}.}$

Examples of large exotic ${\displaystyle \mathbb {R} ^{4}}$  can be constructed using the fact that compact 4-manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work).

Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic ${\displaystyle \mathbb {R} ^{4},}$  into which all other ${\displaystyle \mathbb {R} ^{4}}$  can be smoothly embedded as open subsets.

## Related exotic structures

Casson handles are homeomorphic to ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}}$  by Freedman's theorem (where ${\displaystyle \mathbb {D} ^{2}}$  is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.}$  In other words, some Casson handles are exotic ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.}$

It is not known (as of 2017) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.

• Akbulut cork - tool used to construct exotic ${\displaystyle \mathbb {R} ^{4}}$ 's from classes in ${\displaystyle H^{3}(S^{3},\mathbb {R} )}$ [5]