topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth

h-cobordism In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M ...

theorem fails. It was named after

Turkish Turkish may refer to: *a Turkic language spoken by the Turks * of or about Turkey ** Turkish language *** Turkish alphabet ** Turkish people, a Turkic ethnic group and nation *** Turkish citizen, a citizen of Turkey *** Turkish communities and mi ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Selman Akbulut Selman Akbulut (born 1949) is a Turkish mathematician, specializing in research in topology, and geometry. He was a professor at Michigan State University until February 2020. Career In 1975 he earned his Ph.D. from the University of California ...

. A

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...

Stein Stein is a German, Yiddish and Norwegian word meaning "stone" and "pip" or "kernel". It stems from the same Germanic root as the English word stone. It may refer to: Places In Austria * Stein, a neighbourhood of Krems an der Donau, Lower Austr ...

4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. T ...

C

with involution

F

on its boundary is called an Akbulut cork, if

F

extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork

(C,F)

is called a cork of a smooth 4-manifold

X

, if removing

C

from

X

and re-gluing it via

F

changes the smooth structure of

X

(this operation is called "cork twisting"). Any exotic copy

X'

of a closed simply connected 4-manifold

X

differs from

X

by a single cork twist. The basic idea of the Akbulut cork is that when attempting to use the h-corbodism theorem in four dimensions, the cork is the sub-cobordism that contains all the exotic properties of the spaces connected with the cobordism, and when removed the two spaces become trivially h-cobordant and smooth. This shows that in four dimensions, although the theorem does not tell us that two

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s are

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

(only

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 isomorphi ...

), they are "not far" from being diffeomorphic.Asselmeyer-Maluga and Brans, 2007, ''Exotic Smoothness and Physics'' To illustrate this (without proof), consider a smooth h-cobordism

W^5

between two 4-manifolds

M

and

N

. Then within

W

there is a sub-cobordism

K^5

between

A^4 \subset M

and

B^4 \subset N

and there is a diffeomorphism :

W \setminus \operatorname\, K \cong \left(M \setminus \operatorname\, A \right) \times \left,1\right

which is the content of the h-cobordism theorem for ''n'' ≥ 5 (here int ''X'' refers to the interior of a manifold ''X''). In addition, ''A'' and ''B'' are diffeomorphic with a diffeomorphism that is an

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

on the boundary ∂''A'' = ∂''B''.Scorpan, A., 2005 ''The Wild World of 4-Manifolds'' Therefore, it can be seen that the h-corbordism ''K'' connects ''A'' with its "inverted" image ''B''. This submanifold ''A'' is the Akbulut cork.

Notes

References

* * Topology Differential topology {{topology-stub