Английская Википедия: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>C</math> with involution <math>F</math> on its boundary is called an Akbulut cork, if <math>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>(C,F)</math> is called a cork of a smooth 4-manifold <math>X</math>, if removing <math>C</math> from <math>X</math> and re-gluing it via <math>F</math> changes the smooth structure of <math>X</math> (this operation is called "cork twisting"). Any exotic copy <math>X'</math> of a closed simply connected 4-manifold <math>X</math> differs from <math>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>W^5</math> between two 4-manifolds <math>M</math> and <math>N</math>. Then within <math>W</math> there is a sub-cobordism <math>K^5</math> between <math>A^4 \subset M</math> and <math>B^4 \subset N</math> and there is a diffeomorphism
- <math>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
References
- ↑ Шаблон:Cite book
- ↑ A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. Шаблон:Isbn
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Asselmeyer-Maluga and Brans, 2007, Exotic Smoothness and Physics
- ↑ Scorpan, A., 2005 The Wild World of 4-Manifolds
