TheInfoList

In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, an exotic $\R^4$ is a differentiable manifold that is homeomorphic but not diffeomorphic to the
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensio ...
$\R^4.$ 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 cardinality of the continuum, continuum of non-diffeomorphic differentiable structures of $\R^4,$ as was shown first by Clifford Taubes. Prior to this construction, non-diffeomorphic smooth structures on spheresexotic sphereswere 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 2021). For any positive integer ''n'' other than 4, there are no exotic smooth structures on $\R^n;$ 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, ''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 $\mathbb^2 \times \R^2$ by Freedman's theorem (where $\mathbb^2$ is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to $\mathbb^2 \times \R^2.$ In other words, some Casson handles are exotic $\mathbb^2 \times \R^2.$ 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 Exotic sphere#4-dimensional exotic spheres and Gluck twists, Gluck twists.

*Akbulut cork - tool used to construct exotic $\R^4$'s from classes in $H^3\left(S^3,\mathbb\right)$ *Atlas (topology)