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 [math]\displaystyle{ C }[/math] with involution [math]\displaystyle{ F }[/math] on its boundary is called an Akbulut cork, if [math]\displaystyle{ F }[/math] 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 [math]\displaystyle{ (C,F) }[/math] is called a cork of a smooth 4-manifold [math]\displaystyle{ X }[/math], if removing [math]\displaystyle{ C }[/math] from [math]\displaystyle{ X }[/math] and re-gluing it via [math]\displaystyle{ F }[/math] changes the smooth structure of [math]\displaystyle{ X }[/math] (this operation is called "cork twisting"). Any exotic copy [math]\displaystyle{ X' }[/math] of a closed simply connected 4-manifold [math]\displaystyle{ X }[/math] differs from [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ W^5 }[/math] between two 4-manifolds [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math]. Then within [math]\displaystyle{ W }[/math] there is a sub-cobordism [math]\displaystyle{ K^5 }[/math] between [math]\displaystyle{ A^4 \subset M }[/math] and [math]\displaystyle{ B^4 \subset N }[/math] and there is a diffeomorphism
- [math]\displaystyle{ W \setminus \operatorname{int}\, K \cong \left(M \setminus \operatorname{int}\, A \right) \times \left[0,1\right], }[/math]
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
- ↑ Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society. p. 357. doi:10.1090/gsm/020. ISBN 0-8218-0994-6.
- ↑ 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.
- ↑ Matveyev, Rostislav (1996). "A decomposition of smooth simply-connected h-cobordant 4-manifolds". Journal of Differential Geometry 44 (3): 571–582. doi:10.4310/jdg/1214459222.
- ↑ 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.
- ↑ Akbulut, Selman; Matveyev, Rostislav (1998). "A convex decomposition theorem for 4-manifolds". International Mathematics Research Notices 1998 (7): 371–381. doi:10.1155/S1073792898000245.
- ↑ Akbulut, Selman; Yasui, Kouichi (2008). "Corks, plugs and exotic structures". Journal of Gökova Geometry Topology 2: 40–82. http://gokovagt.org/journal/2008/jggt08-akbulutyasui.pdf.
- ↑ Asselmeyer-Maluga and Brans, 2007, Exotic Smoothness and Physics
- ↑ 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
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