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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 \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 ...
\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 \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 \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, ''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 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 \R^4's from classes in H^3(S^3,\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