Exotic sphere (original) (raw)

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En mathématiques, et plus précisément en topologie différentielle, une sphère exotique est une variété différentielle M qui est homéomorphe, mais non difféomorphe, à la n-sphère euclidienne standard. Autrement dit, M est une sphère du point de vue de ses propriétés topologiques, mais sa structure différentielle (qui définit, par exemple, la notion de vecteur tangent) n'est pas la structure usuelle, d'où l'adjectif « exotique ».

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dbo:abstract In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). The first exotic spheres were constructed by John Milnor in dimension as -bundles over . He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and Milnor showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. (en) En mathématiques, et plus précisément en topologie différentielle, une sphère exotique est une variété différentielle M qui est homéomorphe, mais non difféomorphe, à la n-sphère euclidienne standard. Autrement dit, M est une sphère du point de vue de ses propriétés topologiques, mais sa structure différentielle (qui définit, par exemple, la notion de vecteur tangent) n'est pas la structure usuelle, d'où l'adjectif « exotique ». (fr) 미분위상수학에서 이국적 초구 모노이드(異國的超球monoid, 영어: monoid of exotic spheres)는 어떤 주어진 차원에서, 매끄러운 초구와 위상 동형이지만 미분 동형이 아닐 수 있는 매끄러운 유향 다양체들의 집합이다. 이 집합은 연결합을 통해 가환 모노이드를 이룬다. 4차원이 아닌 다른 모든 차원에서 이국적 초구 모노이드는 유한 아벨 군이지만, 4차원의 경우는 이 모노이드가 심지어 유한 집합인지 여부도 알려지지 않았다. (ko) In de differentiaaltopologie, een deelgebied van de wiskunde, is een exotische sfeer een differentieerbare variëteit die homeomorf is met de standaard euclidische -sfeer, maar niet daarmee diffeomorf. Dat betekent dat een dergelijke variëteit vanuit topologisch oogpunt weliswaar een sfeer is, maar vanuit het oogpunt van haar differentieerbare structuur juist geen sfeer is. Dus als dimensie heeft, bestaat er een homeomorfisme , maar is geen enkele een diffeomorfisme. De eerste exotische sferen werden in 1956 door John Milnor geconstrueerd in dimensie als -bundels over . Hij toonde aan dat er ten minste 7 differentieerbare structuren op de 7-sfeer bestaan. In 1959 liet Milnor in elke dimensie zien dat de diffeomorfisme klassen van georiënteerde exotische sferen niet-triviale elementen van een abelse monoïde onder verbonden som vormen, wat een eindige abelse groep is, wanneer de dimensie niet gelijk is aan 4. De classificatie van exotische sferen door Michel Kervaire en John Milnor (1963) heeft laten zien dat de georiënteerde exotische 7-sferen de niet-triviale elementen van een cyclische groep van orde 28 onder de operatie van de verbonden som zijn. (nl) Екзотична сфера — гладкий многовид М, що гомеоморфний, але не дифеоморфний стандартній n-сфері. (uk) Экзотическая сфера — гладкое многообразие М, которое гомеоморфно, но не диффеоморфно стандартной n-сфере. (ru)
dbo:wikiPageExternalLink http://www.nilesjohnson.net/seven-manifolds.html http://www.map.mpim-bonn.mpg.de/Gluck_construction%23References http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.math.sunysb.edu/~jack/PREPRINTS/pcity-lec.pdf http://www.uni-math.gwdg.de/schick/publ/Groups%20of%20homotopy%20spheres%20I.pdf http://www.nilesjohnson.net/ https://eprints.ucm.es/17187/1/On%20twins%20in%20the%20four%20sphere.pdf https://web.archive.org/web/20110401030522/http:/www.manifoldatlas.him.uni-bonn.de/Exotic_spheres https://www.ams.org/notices/201106/rtx110600804p.pdf http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/
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dbp:align right (en)
dbp:authorLink Egbert Brieskorn (en)
dbp:authorlink Cameron Gordon (en) Michael Freedman (en) William Browder (en) Robert Gompf (en) John Milnor (en) Selman Akbulut (en) Julius Shaneson (en) Michel Kervaire (en) Sylvain Cappell (en)
dbp:first John (en) Michael (en) Robert (en) Scott (en) William (en) Cameron (en) Julius (en) Kevin (en) Michel (en) Selman (en) Egbert (en) Sylvain (en) Yuli B. (en)
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dbp:last Gordon (en) Morrison (en) Walker (en) Freedman (en) Akbulut (en) Browder (en) Brieskorn (en) Gompf (en) Milnor (en) Rudyak (en) Cappell (en) Kervaire (en) Shaneson (en)
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dbp:title Milnor sphere (en)
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rdfs:comment En mathématiques, et plus précisément en topologie différentielle, une sphère exotique est une variété différentielle M qui est homéomorphe, mais non difféomorphe, à la n-sphère euclidienne standard. Autrement dit, M est une sphère du point de vue de ses propriétés topologiques, mais sa structure différentielle (qui définit, par exemple, la notion de vecteur tangent) n'est pas la structure usuelle, d'où l'adjectif « exotique ». (fr) 미분위상수학에서 이국적 초구 모노이드(異國的超球monoid, 영어: monoid of exotic spheres)는 어떤 주어진 차원에서, 매끄러운 초구와 위상 동형이지만 미분 동형이 아닐 수 있는 매끄러운 유향 다양체들의 집합이다. 이 집합은 연결합을 통해 가환 모노이드를 이룬다. 4차원이 아닌 다른 모든 차원에서 이국적 초구 모노이드는 유한 아벨 군이지만, 4차원의 경우는 이 모노이드가 심지어 유한 집합인지 여부도 알려지지 않았다. (ko) Екзотична сфера — гладкий многовид М, що гомеоморфний, але не дифеоморфний стандартній n-сфері. (uk) Экзотическая сфера — гладкое многообразие М, которое гомеоморфно, но не диффеоморфно стандартной n-сфере. (ru) In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). (en) In de differentiaaltopologie, een deelgebied van de wiskunde, is een exotische sfeer een differentieerbare variëteit die homeomorf is met de standaard euclidische -sfeer, maar niet daarmee diffeomorf. Dat betekent dat een dergelijke variëteit vanuit topologisch oogpunt weliswaar een sfeer is, maar vanuit het oogpunt van haar differentieerbare structuur juist geen sfeer is. Dus als dimensie heeft, bestaat er een homeomorfisme , maar is geen enkele een diffeomorfisme. (nl)
rdfs:label Exotic sphere (en) Sphère exotique (fr) 이국적 초구 모노이드 (ko) Exotische sfeer (nl) Экзотическая сфера (ru) Екзотична сфера (uk)
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