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.
on its boundary is called an Akbulut cork, if
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).
is called a cork of a smooth 4-manifold
(this operation is called "cork twisting").
by a single cork twist.
[3][4][5][6][7] The basic idea of the Akbulut cork is that when attempting to use the h-cobodism 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.
[8] To illustrate this (without proof), consider a smooth h-cobordism
and there is a diffeomorphism which is the content of the h-cobordism theorem for n ≥ 5 (here int X refers to the interior of a manifold X).
[9] Therefore, it can be seen that the h-corbordism K connects A with its "inverted" image B.
This submanifold A is the Akbulut cork.