Exotic R4

In mathematics, an exotic $$\R^4$$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space $$\R^4.$$ 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 $$\R^4,$$ 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 remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures $$\R^n;$$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to $$\R^n$$ is diffeomorphic to $$\R^n.$$

Small exotic R4s
An exotic $$\R^4$$ is called small if it can be smoothly embedded as an open subset of the standard. $$\R^4.$$

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

Examples of large exotic $$\R^4$$ 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).

showed that there is a maximal exotic $$\R^4,$$ into which all other $$\R^4$$ can be smoothly embedded as open subsets.

Related exotic structures
Casson handles are homeomorphic to $$\mathbb{D}^2 \times \R^2$$ 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 $$\mathbb{D}^2 \times \R^2.$$ In other words, some Casson handles are exotic $$\mathbb{D}^2 \times \R^2.$$

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.