Exotic R4

In mathematics, an exotic R⁴ is a differentiable manifold that is homeomorphic to the Euclidean space R⁴, but not diffeomorphic. The first examples were found by Robion Kirby and Michael Freedman, 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 R⁴, 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 remains open. 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".

mall exotic R⁴s

An exotic R⁴ is called small if it can be smoothly embedded as an open subset of the standard R⁴.

Small exotic R⁴s 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 R⁴ is called large if it cannot be smoothly embedded as an open subset of the standard R⁴.

Examples of large exotic R⁴s 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).

There is at least one maximal exotic R⁴, into which all other R⁴s can be smoothly embedded as open subsets.

Related exotic structures

Casson handles are homeomorphic to D²×R² 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×R². In other words, some Casson handles are exotic D²×R²s.

It is not known (as of 2006) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counter example to the smooth Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.

References

*Alexandru Scorpan, "The wild world of 4-manifolds", ISBN 978-0-8218-3749-8
*Robert Gompf and Andras Stipsicz, "4-Manifolds and Kirby Calculus", Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. ISBN 0-8218-0994-6