Exotic R4
Encyclopedia
In mathematics
, an exotic 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
.
Prior to this construction, non-diffeomorphic smooth structure
s on spheres — exotic sphere
s — were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open. 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 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.
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).
showed that
there is a unique maximal exotic R4, into which all other R4s can be smoothly embedded as open subsets.
s 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.
It is not known (as of 2009) 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an exotic R4 is a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
that is homeomorphic to the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
R4, but not diffeomorphic
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
.
The first examples were found by Robion Kirby
Robion Kirby
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
and Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
's theorems about smooth 4-manifolds. There is a continuum
Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
.
Prior to this construction, non-diffeomorphic smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
s on spheres — exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s — were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open. 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
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...
fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.
Large exotic R4s
An exotic R4 is called large if it cannot be smoothly embedded as an open subset of the standardR4.
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).
showed that
there is a unique maximal exotic R4, into which all other R4s can be smoothly embedded as open subsets.
Related exotic structures
Casson handleCasson handle
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced...
s 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.
It is not known (as of 2009) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture
Generalized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...
in dimension 4. Some plausible candidates are given by Gluck twists.