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Exotic R4

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A smooth 4-manifold homeomorphic yet not diffeomorphic to euclidean space

In mathematics, an exotic R 4 {\displaystyle \mathbb {R} ^{4}} is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space R 4 . {\displaystyle \mathbb {R} ^{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 R 4 , {\displaystyle \mathbb {R} ^{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 remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures R n ; {\displaystyle \mathbb {R} ^{n};} in other words, if n ≠ 4 then any smooth manifold homeomorphic to R n {\displaystyle \mathbb {R} ^{n}} is diffeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.}

Small exotic Rs

An exotic R 4 {\displaystyle \mathbb {R} ^{4}} is called small if it can be smoothly embedded as an open subset of the standard R 4 . {\displaystyle \mathbb {R} ^{4}.}

Small exotic R 4 {\displaystyle \mathbb {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 Rs

An exotic R 4 {\displaystyle \mathbb {R} ^{4}} is called large if it cannot be smoothly embedded as an open subset of the standard R 4 . {\displaystyle \mathbb {R} ^{4}.}

Examples of large exotic R 4 {\displaystyle \mathbb {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).

Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic R 4 , {\displaystyle \mathbb {R} ^{4},} into which all other R 4 {\displaystyle \mathbb {R} ^{4}} can be smoothly embedded as open subsets.

Related exotic structures

Casson handles are homeomorphic to D 2 × R 2 {\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}} by Freedman's theorem (where D 2 {\displaystyle \mathbb {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 . {\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.} In other words, some Casson handles are exotic D 2 × R 2 . {\displaystyle \mathbb {D} ^{2}\times \mathbb {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.

See also

  • Akbulut cork - tool used to construct exotic R 4 {\displaystyle \mathbb {R} ^{4}} 's from classes in H 3 ( S 3 , R ) {\displaystyle H^{3}(S^{3},\mathbb {R} )}
  • Atlas (topology)

Notes

  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. Asselmeyer-Maluga, Torsten; Król, Jerzy (2014-08-28). "Abelian gerbes, generalized geometries and foliations of small exotic R^4". arXiv:0904.1276 .

References

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