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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mazur manifold ： ウィキペディア英語版 Mazur manifold In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold (with boundary) which is not diffeomorphic to the standard 4-ball. The boundary of a Mazur manifold is necessarily a homology 3-sphere. Frequently the term Mazur manifold is restricted to a special class of the above definition: 4-manifolds that have a handle decomposition containing exactly three handles: a single 0-handle, a single 1-handle and single 2-handle. This is equivalent to saying the manifold must be of the form $S^1\; \backslash times\; D^3$ union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to $S^4$ with the standard smooth structure. == History == Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres $\backslash Sigma(2,5,7)$, $\backslash Sigma(3,4,5)$ and $\backslash Sigma(2,3,13)$ are boundaries of Mazur manifolds. This results were later generalized to other contractible manifolds by Casson, Harer and Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds. Mazur manifolds have been used by Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere. Mazur's discovery was surprising for several reasons: : * Every smooth homology sphere in dimension $n\; \backslash geq\; 5$ is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire and the h-cobordism theorem. Slightly more strongly, every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the Poincaré homology sphere does not bound such a 4-manifold because the Rochlin invariant provides an obstruction. : * The h-cobordism Theorem implies that, at least in dimensions $n\; \backslash geq\; 6$ there is a unique contractible $n$-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball $D^n$. It's an open problem as to whether or not $D^5$ admits an exotic smooth structure, but by the h-cobordism theorem, such an exotic smooth structure, if it exists, must restrict to an exotic smooth structure on $S^4$. Whether or not $S^4$ admits an exotic smooth structure is equivalent to another open problem, the smooth Poincaré conjecture in dimension four. Whether or not $D^4$ admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four. 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「Mazur manifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.