Surgery theory (original) (raw)
Chirurgie ist eine Methode in der Topologie von Mannigfaltigkeiten. Sie wurde von Milnor und Kervaire zur Klassifikation entwickelt und dann in Arbeiten von Browder, Nowikow, Sullivan und Wall zur Klassifikation höher-dimensionaler Mannigfaltigkeiten ausgebaut. Die Grundidee der Chirurgie an einer differenzierbaren Mannigfaltigkeit ist, aus einer -dimensionalen Mannigfaltigkeit mit Einbettung die Untermenge zu entfernen und an der Stelle mit zu ersetzen. Dadurch entsteht eine neue -dimensionale Mannigfaltigkeit wobei die Sphäre und die Kugel bezeichnet.
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| dbo:abstract | Chirurgie ist eine Methode in der Topologie von Mannigfaltigkeiten. Sie wurde von Milnor und Kervaire zur Klassifikation entwickelt und dann in Arbeiten von Browder, Nowikow, Sullivan und Wall zur Klassifikation höher-dimensionaler Mannigfaltigkeiten ausgebaut. Die Grundidee der Chirurgie an einer differenzierbaren Mannigfaltigkeit ist, aus einer -dimensionalen Mannigfaltigkeit mit Einbettung die Untermenge zu entfernen und an der Stelle mit zu ersetzen. Dadurch entsteht eine neue -dimensionale Mannigfaltigkeit wobei die Sphäre und die Kugel bezeichnet. (de) En mathématiques, et particulièrement en topologie géométrique, la chirurgie est une technique, introduite en 1961 par John Milnor, permettant de construire une variété à partir d'une autre de manière « contrôlée ». On parle de chirurgie parce que cela consiste à « couper » une partie de la première variété et à la remplacer par une partie d'une autre variété, en identifiant les frontières ; ces transformations sont étroitement liées à la notion de décomposition en anses. La chirurgie est un outil essentiel dans l'étude et la classification des variétés de dimension supérieure à 4. Plus précisément, l'idée est de partir d'une variété qu'on connaît bien, et d'opérer chirurgicalement sur elle pour construire une variété ayant les propriétés que l'on souhaite, de telle sorte que les effets de ces opérations sur les groupes d'homologie, d'homotopie, ou sur d'autres invariants de la variété soient calculables. La classification des sphères exotiques par Kervaire et Milnor en 1963 amena à l'émergence de la chirurgie comme un outil majeur de la topologie en grande dimension. (fr) In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor. Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to a same cobordism class. The classification of exotic spheres by Michel Kervaire and Milnor led to the emergence of surgery theory as a major tool in high-dimensional topology. (en) 미분위상수학에서 수술(手術, 영어: surgery 서저리[*])은 다양체 속의 원기둥을 도려내고 그 자리에 다른 모양의 원기둥을 붙여 전체의 위상을 바꾸는 연산이다. 수술은 고차원 (5차원 이상) 다양체의 연구에 매우 중요한 역할을 하며, 그 이론을 수술 이론(手術理論, 영어: surgery theory)이라고 한다. (ko) Хирургия или перестройка Морса — преобразование гладких многообразий, которому подвергается многообразие уровня гладкой функции при переходе через невырожденную критическую точку; важнейшая конструкция в дифференциальной топологии. Важная роль хирургии в топологии многообразий объясняется тем, что они позволяют «деликатно» (не нарушая тех или иных свойств многообразия) уничтожать «лишние» гомотопические группы (обычно используемая с этой целью в теории гомотопий операция «приклеивания клетки» мгновенно выводит из класса многообразий).Практически все теоремы классификации структур на многообразиях основываются на изучении вопроса, когда для отображения замкнутого многообразия в клеточное пространство существуют такой бордизм и такое отображение , что , а является гомотопической эквивалентностью.Естественный путь решения этой задачи состоит в том, чтобы последовательностью хирургий уничтожить ядра гомоморфизмов (где — гомотопические группы).Если это удаётся, то результирующее отображение будет гомотопической эквивалентностью.Изучение соответствующих препятствий (лежащих в т. н. ) явилось одним из главнейших стимулов в развитии алгебраической . (ru) 在數學中，尤其是拓撲學，割補理論(surgery theory)是一種用於從另一流形對象產生一個有限維流形、並在「控制」之下的理論方法。其最初是用於處理光滑流形，之後陸續被應用於以及拓撲流形等等。 (zh) |
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| dbo:wikiPageExternalLink | http://www.map.mpim-bonn.mpg.de http://www.maths.ed.ac.uk/~aar/books/scm.pdf http://www.math.harvard.edu/~lurie/287x.html http://www.maths.ed.ac.uk/~aar/ http://www.maths.ed.ac.uk/~aar/books/wall1.pdf http://www.maths.ed.ac.uk/~aar/books/wall2.pdf http://www.maths.ed.ac.uk/~aar/fig3.png http://www.maths.ed.ac.uk/~aar/papers/lurie2011.pdf http://www.patrickorson.com/surgerygroup/ https://sites.psu.edu/surgeryforamateurs/files/2017/12/surgerybook2017-2gfid7m.pdf http://www.map.mpim-bonn.mpg.de/Category:Oberwolfach_Surgery_Seminar_2012 http://www.map.mpim-bonn.mpg.de/Category:Regensburg_Surgery_Blockseminar_2012 http://plms.oxfordjournals.org/cgi/reprint/s3-40/1/87.pdf http://plms.oxfordjournals.org/cgi/reprint/s3-40/2/193.pdf http://www.math.uchicago.edu/~shmuel/ http://www.maths.ed.ac.uk/~aar/books/index.htm |
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| dbp:author1Link | Michel Kervaire (en) Mikhail Postnikov (en) |
| dbp:author2Link | John Milnor (en) |
| dbp:authorlink | John Milnor (en) |
| dbp:first | John (en) Michel (en) Yuli B. (en) Mikail M. (en) |
| dbp:id | m/m065000 (en) |
| dbp:last | Postnikov (en) Milnor (en) Rudyak (en) Kervaire (en) |
| dbp:title | Morse surgery (en) |
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| rdfs:comment | Chirurgie ist eine Methode in der Topologie von Mannigfaltigkeiten. Sie wurde von Milnor und Kervaire zur Klassifikation entwickelt und dann in Arbeiten von Browder, Nowikow, Sullivan und Wall zur Klassifikation höher-dimensionaler Mannigfaltigkeiten ausgebaut. Die Grundidee der Chirurgie an einer differenzierbaren Mannigfaltigkeit ist, aus einer -dimensionalen Mannigfaltigkeit mit Einbettung die Untermenge zu entfernen und an der Stelle mit zu ersetzen. Dadurch entsteht eine neue -dimensionale Mannigfaltigkeit wobei die Sphäre und die Kugel bezeichnet. (de) 미분위상수학에서 수술(手術, 영어: surgery 서저리[*])은 다양체 속의 원기둥을 도려내고 그 자리에 다른 모양의 원기둥을 붙여 전체의 위상을 바꾸는 연산이다. 수술은 고차원 (5차원 이상) 다양체의 연구에 매우 중요한 역할을 하며, 그 이론을 수술 이론(手術理論, 영어: surgery theory)이라고 한다. (ko) 在數學中，尤其是拓撲學，割補理論(surgery theory)是一種用於從另一流形對象產生一個有限維流形、並在「控制」之下的理論方法。其最初是用於處理光滑流形，之後陸續被應用於以及拓撲流形等等。 (zh) En mathématiques, et particulièrement en topologie géométrique, la chirurgie est une technique, introduite en 1961 par John Milnor, permettant de construire une variété à partir d'une autre de manière « contrôlée ». On parle de chirurgie parce que cela consiste à « couper » une partie de la première variété et à la remplacer par une partie d'une autre variété, en identifiant les frontières ; ces transformations sont étroitement liées à la notion de décomposition en anses. La chirurgie est un outil essentiel dans l'étude et la classification des variétés de dimension supérieure à 4. (fr) In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor. Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. (en) Хирургия или перестройка Морса — преобразование гладких многообразий, которому подвергается многообразие уровня гладкой функции при переходе через невырожденную критическую точку; важнейшая конструкция в дифференциальной топологии. (ru) |
| rdfs:label | Surgery theory (en) Chirurgie (Mathematik) (de) Chirurgie (topologie) (fr) 수술 (수학) (ko) Перестройка Морса (ru) 割補理論 (zh) |
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