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.

Notes

1. Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. Vol. 20. Providence, RI: American Mathematical Society. p. 357. doi:10.1090/gsm/020. ISBN   0-8218-0994-6. MR   1707327.
2. A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. ISBN   0-8218-3749-4
3. Akbulut, Selman (1991). "A fake compact contractible 4-manifold". Journal of Differential Geometry . 33 (2): 335–356. doi: 10.4310/jdg/1214446320 . MR   1094459.
4. Matveyev, Rostislav (1996). "A decomposition of smooth simply-connected h-cobordant 4-manifolds". Journal of Differential Geometry . 44 (3): 571–582. arXiv: dg-ga/9505001 . doi:10.4310/jdg/1214459222. MR   1431006. S2CID   15994704.
5. Curtis, Cynthia L.; Freedman, Michael H.; Hsiang, Wu Chung; Stong, Richard (1996). "A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds". Inventiones Mathematicae . 123 (2): 343–348. doi:10.1007/s002220050031. MR   1374205. S2CID   189819783.
6. Akbulut, Selman; Matveyev, Rostislav (1998). "A convex decomposition theorem for 4-manifolds". International Mathematics Research Notices . 1998 (7): 371–381. doi:10.1155/S1073792898000245. MR   1623402.
7. Akbulut, Selman; Yasui, Kouichi (2008). "Corks, plugs and exotic structures" (PDF). Journal of Gökova Geometry Topology. 2: 40–82. arXiv: 0806.3010 . MR   2466001.
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