is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space
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
[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 remains open as of 2024).
For any positive integer n other than 4, there are no exotic smooth structures
in other words, if n ≠ 4 then any smooth manifold homeomorphic to
is called small if it can be smoothly embedded as an open subset of the standard
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.
is called large if it cannot be smoothly embedded as an open subset of the standard
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
can be smoothly embedded as open subsets.
is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to
In other words, some Casson handles are exotic
Some plausible candidates are given by Gluck twists.