In mathematics, an exotic or fake R⁴ is a differentiable manifold that is homeomorphic to the Euclidean space R⁴, but not diffeomorphic. The first examples were found by Robion Kirby and Michael Freedman, 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 of R⁴, as was shown first by Clifford Taubes. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This article or section is in need of attention from an expert on the subject. ... This word should not be confused with homomorphism. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Robion Kirby is a professor of mathematics at the University of California at Berkeley who works in low-dimensional topology. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Clifford Henry Taubes is a professor of mathematics at Harvard who works in gauge field theory and differential geometry. ...

For any positive integer n other than 4, there are no exotic smooth structures on Rⁿ; in other words, if n≠4 then any smooth manifold homeomorphic to Rⁿ is diffeomorphic to Rⁿ.

Small exotic R⁴s

An exotic R⁴ is called small if it can be smoothly embedded as an open subset of the standard R⁴.

Small exotic R⁴s 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. A cobordism W between M and N is an h-cobordism if the inclusion maps M → W and N → W are homotopy equivalences. ...

Large exotic R⁴s

An exotic R⁴ is called large if it cannot be smoothly embedded as an open subset of the standard R⁴.

Examples of large exotic R⁴s 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).

There is at least one maximal exotic R⁴, into which all other R⁴s can be smoothly embedded as open subsets.

Related exotic structures

Casson handles are homeomorphic to D²×R² by Freedman's theorem (where D² is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to D²×R². In other words, some Casson handles are exotic D²×R²s. In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. ...

It is not known (as of 2006) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counter example to the smooth Poincare conjecture in dimension 4. Some plausible candidates are given by Gluck twists. In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ... In mathematics, an exotic sphere is a differentiable manifold, M, that is homeomorphic to the ordinary sphere, but not diffeomorphic. ...

See also

• exotic sphere

In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ...

References

• Alexandru Scorpan, The wild world of 4-manifolds, ISBN 0-8218-3749-4