Exotic R4

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 2024) 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.

Notes

References

References

1. Kirby (1989), p. 95
2. Freedman and Quinn (1990), p. 122
3. Taubes (1987), Theorem 1.1
4. Stallings (1962), in particular Corollary 5.2
5. (2014-08-28). "Abelian gerbes, generalized geometries and foliations of small exotic R^4".