HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an exotic \R^4 is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
that is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
\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 of \R^4, as was shown first by
Clifford Taubes Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taub ...
. Prior to this construction, non-diffeomorphic smooth structures on spheres
exotic sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of a ...
swere 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 2022). 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 In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same di ...
(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 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 "fle ...
s 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 2022) 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 the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or di ...
in dimension 4. Some plausible candidates are given by Gluck twists.


See also

*
Akbulut cork In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compact contractible Stein 4-manifold C with involut ...
- 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