In
mathematics, an exotic
is a
differentiable manifold that is
homeomorphic (i.e. shape preserving) but not
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
(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 Euclidean ...
The first examples were found in 1982 by
Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American 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 4-dimensional gene ...
and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
's theorems about smooth 4-manifolds. There is a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
of non-diffeomorphic
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
s 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 Taube ...
.
Prior to this construction, non-diffeomorphic
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.
Definition
A smooth structure on a manifold M is ...
s on spheres
exotic sphereswere already known to exist, although the question of the existence of such structures for the particular case of the
4-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
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 dim ...
(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 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 invol ...
- tool used to construct exotic
's from classes in
*
Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
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