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.^[1][2]

A compact contractible Stein 4-manifold ${\displaystyle C}$ with involution ${\displaystyle F}$ on its boundary is called an Akbulut cork, if ${\displaystyle F}$ 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 ${\displaystyle (C,F)}$ is called a cork of a smooth 4-manifold ${\displaystyle X}$, if removing ${\displaystyle C}$ from ${\displaystyle X}$ and re-gluing it via ${\displaystyle F}$ changes the smooth structure of ${\displaystyle X}$ (this operation is called "cork twisting"). Any exotic copy ${\displaystyle X'}$ of a closed simply connected 4-manifold ${\displaystyle X}$ differs from ${\displaystyle X}$ 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. This shows that in four dimensions, although the theorem does not tell us that two manifolds are diffeomorphic (only homeomorphic), they are "not far" from being diffeomorphic.^[8]

To illustrate this (without proof), consider a smooth h-cobordism ${\displaystyle W^{5}}$ between two 4-manifolds ${\displaystyle M}$ and ${\displaystyle N}$. Then within ${\displaystyle W}$ there is a sub-cobordism ${\displaystyle K^{5}}$ between ${\displaystyle A^{4}\subset M}$ and ${\displaystyle B^{4}\subset N}$ and there is a diffeomorphism

${\displaystyle W\setminus \operatorname {int} \,K\cong \left(M\setminus \operatorname {int} \,A\right)\times \left[0,1\right],}$

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 on the boundary ∂A = ∂B.^[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.