Akbulut cork

In topology an Akbulut cork is a structure that is frequently used to show that in four dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut.^[1][2]

A compact contractible Stein 4-manifold 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.^{[3][4][5][6][7]}

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 diffeomorphic.^[8]

To illustrate this (without proof), consider a smooth h-cobordism W⁵ between two 4-manifolds M and N. Then within W there is a sub-cobordism K⁵ between A⁴ ⊂ M and B⁴ ⊂ N and there is a diffeomorphism

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

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.

Notes

1. R. E. Gompf and A.I. Stipsicz, 4-manifolds and Kirby calculus (p.357), AMS Pub. GSM v.20 ISBN 0-8218-0994-6
2. A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. ISBN 0-8218-3749-4
3. S. Akbulut, A Fake compact contractible 4-manifold, Journ. of Diff. Geom. 33, (1991), 335-356
4. R. Matveyev, A decomposition of smooth simply-connected h-cobordant 4-manifolds, J. Differential Geom. 44 (1996), no. 3, 571–582
5. C. L. Curtis, M. H. Freedman, W. C. Hsiang, and R. Stong, A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds, Invent. Math. 123 (1996), no. 2, 343–348
6. S. Akbulut and R. Matveyev, A convex decomposition theorem for 4-manifolds, Internat. Math. Res. Notices 1998, no. 7, 371–381
7. S. Akbulut and K. Yasui, Corks, Plugs and exotic structures, Jour. of GGT, vol 2 (2008) 40-82
8. Asselmeyer-Maluga and Brans, 2007, Exotic Smoothness and Physics
9. Scorpan, A., 2005 The Wild World of 4-Manifolds

References

• Scorpan, Alexandru (2005), The Wild World of 4-Manifolds, Providence, Rhode Island: American Mathematical Society
• Asselmeyer-Maluga, Torsten; Brans, Carl H (2007), Exotic Smoothness and Physics: Differential Topology and Spacetime Models, New Jersey, London: World Scientific