Exotic R4

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


In mathematics, an exotic [math]\displaystyle{ \R^4 }[/math] is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space [math]\displaystyle{ \R^4. }[/math] 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.[1][2] There is a continuum of non-diffeomorphic differentiable structures of [math]\displaystyle{ \R^4, }[/math] as was shown first by Clifford Taubes.[3]

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 2023). For any positive integer n other than 4, there are no exotic smooth structures on [math]\displaystyle{ \R^n; }[/math] in other words, if n ≠ 4 then any smooth manifold homeomorphic to [math]\displaystyle{ \R^n }[/math] is diffeomorphic to [math]\displaystyle{ \R^n. }[/math][4]

Small exotic R4s

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

Small exotic [math]\displaystyle{ \R^4 }[/math] 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 [math]\displaystyle{ \R^4 }[/math] is called large if it cannot be smoothly embedded as an open subset of the standard [math]\displaystyle{ \R^4. }[/math]

Examples of large exotic [math]\displaystyle{ \R^4 }[/math] 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 [math]\displaystyle{ \R^4, }[/math] into which all other [math]\displaystyle{ \R^4 }[/math] can be smoothly embedded as open subsets.

Related exotic structures

Casson handles are homeomorphic to [math]\displaystyle{ \mathbb{D}^2 \times \R^2 }[/math] by Freedman's theorem (where [math]\displaystyle{ \mathbb{D}^2 }[/math] is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to [math]\displaystyle{ \mathbb{D}^2 \times \R^2. }[/math] In other words, some Casson handles are exotic [math]\displaystyle{ \mathbb{D}^2 \times \R^2. }[/math]

It is not known (as of 2022) 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 [math]\displaystyle{ \R^4 }[/math]'s from classes in [math]\displaystyle{ H^3(S^3,\mathbb{R}) }[/math][5]
  • 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 [hep-th].

References