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In topology an Akbulut cork is a structure that is frequently used to show that in four dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut.〔R. E. Gompf and A.I. Stipsicz, 4-manifolds and Kirby calculus (p.357), AMS Pub. GSM v.20 ISBN 0-8218-0994-6〕〔A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. ISBN 0-8218-3749-4〕 A compact contractible Stein 4-manifold C with involution F on its boundary is called an Akbulut cork, if 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 (C,F) is called a cork of a smooth 4-manifold X, if removing C from X and re-gluing it via F changes the smooth structure of X (this operation is called "cork twisting"). Any exotic copy X' of a closed simply connected 4-manifold X differs from X by a single cork twist.〔S. Akbulut, A Fake compact contractible 4-manifold, Journ. of Diff. Geom. 33, (1991), 335-356〕〔R. Matveyev, A decomposition of smooth simply-connected h-cobordant 4-manifolds, J. Differential Geom. 44 (1996), no. 3, 571–582〕〔C. L. Curtis, M. H. Freedman, W. C. Hsiang, and R. Stong, A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds, Invent. Math. 123 (1996), no. 2, 343–348〕〔S. Akbulut and R. Matveyev, A convex decomposition theorem for 4-manifolds, Internat. Math. Res. Notices 1998, no. 7, 371–381〕〔S. Akbulut and K. Yasui, Corks, Plugs and exotic structures, Jour. of GGT, vol 2 (2008) 40-82〕 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.〔Asselmeyer-Maluga and Brans, 2007, ''Exotic Smoothness and Physics''〕 To illustrate this (without proof), consider a smooth h-cobordism ''W''5 between two 4-manifolds ''M'' and ''N''. Then within ''W'' there is a sub-cobordism ''K''5 between ''A''4 ⊂ ''M'' and ''B''4 ⊂ ''N'' and there is a diffeomorphism : 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''.〔Scorpan, A., 2005 ''The Wild World of 4-Manifolds''〕 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== 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Akbulut cork」の詳細全文を読む スポンサード リンク
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