Exotic R4

In mathematics, an exotic $\R^4$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space $\R^4.$ 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 $\R^4,$ as was shown first by Clifford Taubes.

Prior to this construction, non-diffeomorphic smooth structures on spheres exotic sphereswere 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 $\R^n;$ in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to $\R^n$ is diffeomorphic to $\R^n.$

Small exotic R⁴s

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

Small exotic $\R^4$ 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 $\R^4$ is called large if it cannot be smoothly embedded as an open subset of the standard $\R^4.$

Examples of large exotic $\R^4$ 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).

showed that there is a maximal exotic $\R^4,$ into which all other $\R^4$ can be smoothly embedded as open subsets.

Related exotic structures

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

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 $\R^4$'s from classes in $H^3(S^3,\mathbb)$
*Atlas (topology)

Notes

References

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* {{cite journal, last = Taubes , first = Clifford Henry , author-link = Clifford Henry Taubes , title = Gauge theory on asymptotically periodic 4-manifolds , url = http://projecteuclid.org/euclid.jdg/1214440981 , journal = Journal of Differential Geometry , volume = 25 , year = 1987 , issue = 3 , pages = 363–430 , doi = 10.4310/jdg/1214440981 , mr = 882829 , id = {{Project Euclid, 1214440981, doi-access = free

4-manifolds
Differential structures