Akbulut cork

In topology an Akbulut cork is a structure is frequently used to show that in four dimensions, the smooth h-cobordism theorem fails. It was named after Selman Akbulut.

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 (only homeomorphic), they are "not far" from being diffeomorphicAsselmeyer-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 ⊂ "M" and "B"⁴ ⊂ "N" and there is a diffeomorphism

:$W\; setminus\; operatorname\{int\},\; K\; cong\; left(M\; setminus\; operatorname\{int\},\; A\; ight)\; imes\; left\; [0,1\; ight]\; ,$

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"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

*
*