Exotic R4
# Exotic R4

In mathematics, an exotic $\mathbb {R} ^{4}$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space $\mathbb {R} ^{4}.$ 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. There is a continuum of non-diffeomorphic differentiable structures of $\mathbb {R} ^{4},$ as was shown first by Clifford Taubes.

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 2023). For any positive integer n other than 4, there are no exotic smooth structures on $\mathbb {R} ^{n};$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to $\mathbb {R} ^{n}$ is diffeomorphic to $\mathbb {R} ^{n}.$

## Small exotic $R^4s$

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

Small exotic $\mathbb {R} ^{4}$ 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 $R^4s$

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

Examples of large exotic $\mathbb {R} ^{4}$ 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 $\mathbb {R} ^{4},$ into which all other $\mathbb {R} ^{4}$ can be smoothly embedded as open subsets.

## Related exotic structures

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

It is not known (as of 2022) 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.

