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 with involution on its boundary is called an Akbulut cork, if extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside. A cork is called a cork of a smooth 4-manifold, if removing from and re-gluing it via changes the smooth structure of . Any exotic copy of a closed simply connected 4-manifold differs from 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 manifolds are diffeomorphic, they are "not far" from being diffeomorphic.
To illustrate this, consider a smooth h-cobordism between two 4-manifolds and. Then within there is a sub-cobordism between and and there is a diffeomorphism
which is the content of the h-cobordism theorem for n ≥ 5. In addition, A and B are diffeomorphic with a diffeomorphism that is an involution on the boundary ∂A = ∂B. Therefore, it can be seen that the h-corbordism K connects A with its "inverted" image B. This submanifold A is the Akbulut cork.