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| In [[differential topology]], a branch of mathematics, a '''Mazur manifold''' is a contractible, [[compact space|compact]], smooth 4-dimensional [[manifold]] (with boundary) which is not [[diffeomorphism|diffeomorphic]] to the standard [[4-ball]]. The boundary of a Mazur manifold is necessarily a [[homology sphere|homology 3-sphere]].
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| 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 <math>S^1 \times D^3</math> union a 2-handle. An observation of Mazur's shows that the [[Double (manifold)|double]] of such manifolds is [[Diffeomorphism|diffeomorphic]] to <math>S^4</math> with the standard smooth structure.
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| ==Some properties==
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| In general the [[double (manifold)|double]] of a Mazur manifold is a [[homotopy sphere|homotopy 4-sphere]], thus such manifolds are a source of possible counter-examples to the smooth [[Generalized Poincaré Conjecture|Poincaré conjecture in dimension 4]].
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| == History ==
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| [[Barry Mazur]] <ref>Mazur, Barry A note on some contractible $4$-manifolds. Ann. of Math. (2) 73 1961 221--228.</ref> and Valentin Poenaru<ref>Valentin Poenaru, Les decompositions de l'hypercube en produit topologique, Bull. Soc. Math. France 88 (1960), 113-129.</ref> discovered these manifolds simultaneously. Akbulut and Kirby showed that the [[homology sphere|Brieskorn homology spheres]] <math>\Sigma(2,5,7) </math>, <math> \Sigma(3,4,5)</math> and <math>\Sigma(2,3,13)</math> are boundaries of Mazur manifolds.<ref>S.Akbulut, R.Kirby, "Mazur manifolds," Michigan Math. J. 26 (1979), 259--284.</ref> This results were later generalized to other contractible manifolds by Casson, Harer and Stern.<ref>A.Casson, J.Harer, "Some homology lens spaces which bound rational homology balls." Pacific. J. Math. Vol 96, No 1, (1981) 23–36.</ref><ref>H.Fickle, "Knots, Z-Homology 3-spheres and contractible 4-manifolds," pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).</ref><ref>R.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.</ref> One of the Mazur manifolds is also an example of an Akbulut cork <ref>http://en.wikipedia.org/wiki/Akbulut_cork</ref> which can be used to construct exotic 4-manifolds.<ref>S.Akbulut, "A Fake compact contractible 4-manifold" , Journ. of Diff. Geom. 33, (1991), 335-356</ref>
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| Mazur manifolds have been used by Fintushel and Stern <ref>Fintushel, Ronald; Stern, Ronald J. An exotic free involution on $S^{4}$. Ann. of Math. (2) 113 (1981), no. 2, 357--365.</ref> to construct exotic actions of a group of order 2 on the [[n-sphere|4-sphere]].
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| Mazur's discovery was surprising for several reasons:
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| :* Every smooth homology sphere in dimension <math>n \geq 5</math> is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire <ref>Kervaire, Michel A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 1969 67--72.</ref> and the [[h-cobordism]] theorem. Every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). Moreover, not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the [[Homology sphere|Poincaré homology sphere]] does not bound such a 4-manifold because the [[Rokhlin invariant|Rochlin invariant]] provides an obstruction.
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| :* The [[h-cobordism|H-cobordism Theorem]] implies that, at least in dimensions <math>n \geq 6</math> there is a unique contractible <math>n</math>-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball <math>D^n</math>. It's an open problem as to whether or not <math>D^5</math> 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 <math>S^4</math>. Whether or not <math>S^4</math> admits an exotic smooth structure is equivalent to another open problem, the smooth [[Generalized Poincaré conjecture|Poincaré conjecture in dimension four]]. Whether or not <math>D^4</math> admits an exotic smooth structure is another open problem, closely linked to the [[Schoenflies problem]] in dimension four.
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| == Mazur's Observation ==
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| Let <math>M</math> be the Mazur manifold, constructed as <math>S^1 \times D^3</math> union a 2-handle. Here is a sketch of Mazur's argument that the [[Double (manifold)|double]] of such a Mazur manifold is <math>S^4</math>. <math>M \times [0,1]</math> is a contractible 5-manifold constructed as <math>S^1 \times D^4</math> union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold <math>S^1 \times S^3</math>. So <math>S^1 \times D^4</math> union the 2-handle is diffeomorphic to <math>D^5</math>. The boundary of <math>D^5</math> is <math>S^4</math>. But the boundary of <math>M \times [0,1]</math> is the [[Double (manifold)|double]] of <math>M</math>.
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| == References ==
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| <references/>
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| [[Category:Differential topology]]
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| [[Category:Manifolds]]
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Hi, everybody!
I'm Arabic male :).
I really love Two and a Half Men!
xunjie ワールドトレードプラザジュエリー祭りのシーンで表示するには、
茶色の紙袋の波をオフに設定していていますか？それとも、
時間：2010年12月6日から12月25日まで。 [http://alpha-printing.com/templates/shop/chloe.php SK-2 ��������`��] つの統合エンタープライズグループとして貿易中核産業として衣類の研究、
アベニューの場所でのマディソン（マディソン）コーチニューヨークの旗艦店は、
これらの輸出業者が突然夜に目を覚ましようにしかし、 [http://www.jaincentreleicester.com/XML/hot/list/tiffany.php �����ϩ`�� �ԥ���] 着ていたを持っていることが適切である。
子どもたちは服はカタルシス感覚は、
中国服ファッションネットワーク:2007-10から16ヒットを発表ますます従来のステアリングライニングライニングと区別、[http://amorexigente.org.br/dosyalar/shop/tomford.php �ȥ�ե��`�� ���饹] 不用意に逃亡貴族スタイルの質の高さを反映するために消費者団体、
小さな女の子隣雰囲気。
ジャケットとショートティーやその他の設計仕様とのセットの外に見ることができるに加えて、
すぐに駆けつけた他の主要な主流メディアの注目は、 [http://www.jaincentreleicester.com/assets/about/nike.html �ʥ�
����ޥå���90]
My web blog - ボッテガヴェネタ 長財布