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
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 ...
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
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
in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to
is diffeomorphic to
Small exotic R4s
An exotic
is called small if it can be smoothly embedded as an open subset of the standard
Small exotic
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
is called large if it cannot be smoothly embedded as an open subset of the standard
Examples of large exotic
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
into which all other
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
by Freedman's theorem (where
is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to
In other words, some Casson handles are exotic
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
's from classes in
*
Atlas (topology)
Notes
References
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* {{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