Английская Википедия:Exotic R4
Шаблон:Technical Шаблон:Short description
In mathematics, an exotic <math>\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>\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>\R^4,</math> as was shown first by Clifford Taubes.[3]
Prior to this construction, non-diffeomorphic smooth structures on spheresШаблон:Sndexotic spheresШаблон:Sndwere 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>\R^n;</math> in other words, if n ≠ 4 then any smooth manifold homeomorphic to <math>\R^n</math> is diffeomorphic to <math>\R^n.</math>[4]
Small exotic R4s
An exotic <math>\R^4</math> is called small if it can be smoothly embedded as an open subset of the standard <math>\R^4.</math>
Small exotic <math>\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>\R^4</math> is called large if it cannot be smoothly embedded as an open subset of the standard <math>\R^4.</math>
Examples of large exotic <math>\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).
Шаблон:Harvs showed that there is a maximal exotic <math>\R^4,</math> into which all other <math>\R^4</math> can be smoothly embedded as open subsets.
Related exotic structures
Casson handles are homeomorphic to <math>\mathbb{D}^2 \times \R^2</math> by Freedman's theorem (where <math>\mathbb{D}^2</math> is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to <math>\mathbb{D}^2 \times \R^2.</math> In other words, some Casson handles are exotic <math>\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>\R^4</math>'s from classes in <math>H^3(S^3,\mathbb{R})</math>[5]
- Atlas (topology)
Notes
References
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal Шаблон:MathSciNet
- Шаблон:Cite book
- Шаблон:Cite journal
