Akbulut cork

In mathematics , Akbulut corks appear in the theory of 4-dimensional manifolds. In particular, they are used in the construction of exotic 4-dimensional spaces ( 's). ${\displaystyle \mathbb {R} ^{4}}$

While the h-cobordism theorem holds for topological 4-manifolds , this is not the case for differentiable 4-manifolds. Instead, the following sentence by Cynthia L. Curtis , Michael Freedman , Wu-Chung Hsiang , Robert Stong applies :

In every 5-dimensional h-cobordism between 4-dimensional manifolds and there are compact , contractible 4-dimensional submanifolds with a border and an h-cobordism contained as a submanifold with border (compact and contractible) between and , so that outside of a trivial Cobordism is, therefore, a diffeomorphism ${\displaystyle W}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle A\subset M,B\subset N}$ ${\displaystyle W}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle W}$ ${\displaystyle K}$

${\displaystyle W\setminus Int(K)\simeq (M\setminus Int(A))\times \left[0,1\right]}$

gives. can be chosen so that it is diffeomorphic to the full sphere , that it is simply connected and that there is a diffeomorphism whose limitation to the boundary is an involution . ${\displaystyle K}$ ${\displaystyle {\mathbf {D} }^{5}}$ ${\displaystyle W\setminus K}$ ${\displaystyle A\to B}$

Two h-co-ordinate 4-manifolds do not have to be diffeomorphic, but one can gain from by cutting out a compact, contractible submanifold and gluing it back in by means of an involution of . ${\displaystyle M,N}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle \partial A}$

The 4-manifold is homeomorphic, but not diffeomorphic to the full sphere, and is called an Akbulut cork . It is named after Selman Akbulut . ${\displaystyle A}$ ${\displaystyle {\mathbf {D} }^{4}}$

Any Akbulut cork can be embedded in an exotic one. More precisely, one can find in the above sentence a containing and outside of trivial, open, h-cobordism that is homeomorphic to . ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle U\subset W}$ ${\displaystyle \mathbb {R} ^{4}\times \left[0,1\right]}$

literature

• A. Scorpan: The wild world of 4-manifolds , Amer. Math. Soc. 2005, ISBN 978-0-8218-3749-8
• Robert Gompf, Andras Stipsicz: 4-manifolds and Kirby calculus , American Mathematical Society 1999

Individual evidence

1. ↑ Curtis, Freedman, Hsiang, Stong: A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds , Inventiones Mathematicae, Volume 126, 1996, pp. 343-348
2. ↑ S. Akbulut, A Fake compact contractible 4-manifold, Journal of Differential Geometry, Volume 33, 1991, pp. 335-356
3. ↑ S. Akbulut, An exotic 4-manifold, Journal of Differential Geometry, Volume 33, 1991, pp. 357-361, Project Euclid