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 with involution on its boundary is called an Akbulut cork, if 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 is called a cork of a smooth 4-manifold , if removing from and re-gluing it via changes the smooth structure of (this operation is called "cork twisting"). Any exotic copy of a closed simply connected 4-manifold differs from 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 between two 4-manifolds and . Then within there is a sub-cobordism between and and there is a diffeomorphism
![{\displaystyle W\setminus \operatorname {int} \,K\cong \left(M\setminus \operatorname {int} \,A\right)\times \left[0,1\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2fdd1c23095cb2433fd5c4d5a57920af027efa)
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 edit
- ^ 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.
- ^ A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. ISBN 0-8218-3749-4
- ^ Akbulut, Selman (1991). "A fake compact contractible 4-manifold". Journal of Differential Geometry. 33 (2): 335–356. doi:10.4310/jdg/1214446320. MR 1094459.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Akbulut, Selman; Yasui, Kouichi (2008). "Corks, plugs and exotic structures" (PDF). Journal of Gökova Geometry Topology. 2: 40–82. arXiv:0806.3010. MR 2466001.
- ^ Asselmeyer-Maluga and Brans, 2007, Exotic Smoothness and Physics
- ^ Scorpan, A., 2005 The Wild World of 4-Manifolds
References edit
- 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
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