In mathematics, an exotic 4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space 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 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 2018). For any positive integer n other than 4, there are no exotic smooth structures on n; in other words, if n ≠ 4 then any smooth manifold homeomorphic to n is diffeomorphic to n.
Small exotic R4s
An exotic 4 is called small if it can be smoothly embedded as an open subset of the standard 4.
Small exotic 4s 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 4 is called large if it cannot be smoothly embedded as an open subset of the standard 4.
Examples of large exotic 4s 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 4, into which all other 4s can be smoothly embedded as open subsets.
Related exotic structures
Casson handles are homeomorphic to D2×2 by Freedman's theorem (where D2 is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to D2×2. In other words, some Casson handles are exotic D2×2s.
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.
- Kirby (1989), p. 95
- Freedman and Quinn (1990), p. 122
- Taubes (1987), Theorem 1.1
- Stallings (1962), in particular Corollary 5.2
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