- Exotic R4
In

mathematics , an**exotic****R**^{4}is adifferentiable manifold that ishomeomorphic to theEuclidean space **R**^{4}, but not diffeomorphic. The first examples were found byRobion Kirby andMichael Freedman , by using the contrast between Freedman's theorems about topological 4-manifolds, andSimon Donaldson 's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphicdifferentiable structure s of**R**^{4}, as was shown first byClifford Taubes .Prior to this construction, non-diffeomorphic smooth structures on spheres —

exotic sphere s — were already known to exist, although the question of the existence of such structures for the particular case of the4-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****R**^{4}sAn 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}s 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****R**^{4}sAn 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}s 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

**R**^{4}, into which all other**R**^{4}s can be smoothly embedded as open subsets.**Related exotic structures**Casson handle s are homeomorphic to**D**^{2}×**R**^{2}by Freedman's theorem (where**D**^{2}is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to**D**^{2}×**R**^{2}. In other words, some Casson handles are exotic**D**^{2}×**R**^{2}s.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 byGluck twist s.**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

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