In

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, an exotic $\backslash R^4$ is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensio ...

$\backslash R^4.$ 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 cardinality of the continuum, continuum of non-diffeomorphic differentiable structures of $\backslash R^4,$ as was shown first by Clifford Taubes.
Prior to this construction, non-diffeomorphic smooth structures on spheresexotic 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 still remains open as of 2021). For any positive integer ''n'' other than 4, there are no exotic smooth structures on $\backslash R^n;$ in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to $\backslash R^n$ is diffeomorphic to $\backslash R^n.$
Small exotic R^{4}s

Large exotic R^{4}s

Related exotic structures

Casson handles are homeomorphic to $\backslash mathbb^2\; \backslash times\; \backslash R^2$ by Freedman's theorem (where $\backslash mathbb^2$ is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to $\backslash mathbb^2\; \backslash times\; \backslash R^2.$ In other words, some Casson handles are exotic $\backslash mathbb^2\; \backslash times\; \backslash R^2.$ It is not known (as of 2017) 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 Exotic sphere#4-dimensional exotic spheres and Gluck twists, Gluck twists.See also

*Akbulut cork - tool used to construct exotic $\backslash R^4$'s from classes in $H^3(S^3,\backslash mathbb)$ *Atlas (topology)Notes

References

* * * * * * * {{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 = {{Euclid, 1214440981, doi-access = free 4-manifolds Differential structures