Cobordism (original) (raw)
In der Mathematik ist der Begriff des Kobordismus (auch: Bordismus) vor allem in der Topologie und ihren Anwendungen sowie in der topologischen Quantenfeldtheorie von Bedeutung. Er gilt als die bis heute „berechenbarste“ Relation unter Mannigfaltigkeiten, die geometrisch interessant ist. Als Schöpfer der Kobordismentheorie gilt René Thom (1954), wobei einige entscheidende Ideen schon von Lew Pontrjagin (1950 und davor) vorweggenommen wurden.
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| dbo:abstract | In der Mathematik ist der Begriff des Kobordismus (auch: Bordismus) vor allem in der Topologie und ihren Anwendungen sowie in der topologischen Quantenfeldtheorie von Bedeutung. Er gilt als die bis heute „berechenbarste“ Relation unter Mannigfaltigkeiten, die geometrisch interessant ist. Als Schöpfer der Kobordismentheorie gilt René Thom (1954), wobei einige entscheidende Ideen schon von Lew Pontrjagin (1950 und davor) vorweggenommen wurden. (de) In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher. The boundary of an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions forpiecewise linear and topological manifolds. A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, . Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories. (en) En topologie différentielle, le cobordisme est une relation d'équivalence entre variétés différentielles compactes. Deux variétés compactes M et N sont dites cobordantes ou en cobordisme si leur réunion disjointe peut être réalisée comme le bord d'une variété à bord compacte L. On dit alors que cette variété L est un cobordisme entre M et N, ou bien que L réalise un cobordisme entre M et N. L'existence d'un tel cobordisme implique que M et N soient de même dimension. À proprement parler, le cobordisme n'est pas une relation d'équivalence car la classe des variétés différentielles d'une dimension donnée n n'est pas un ensemble. Toutefois, le fait que deux variétés M et N soient cobordantes dépend uniquement de la classe de difféomorphismes de ces variétés. Le cobordisme définit une relation d'équivalence sur l'ensemble des variétés différentielles de dimension n identifiées à difféomorphisme près. Par convention, une variété est supposée dénombrable à l'infini. Chaque compact peut être recouvert par un nombre fini de domaines de cartes locales, et chaque domaine s'identifie à un ouvert de Rn. Une variété différentielle a donc la puissance du continu. La classe des variétés différentielles de dimension n identifiées à difféomorphisme près s'obtient comme un quotient de l'ensemble des structures de variétés différentielles de dimension n sur l'ensemble R. Il existe une relation plus fine que le cobordisme pour les variétés différentielles orientées. Une orientation sur une variété à bord induit une orientation sur le bord. Pour une variété différentielle orientable connexe M, il existe exactement deux orientations distinctes. Si une de ces orientations est spécifiée, M est dite par abus de langage orientée. On note alors la variété M munie de la seconde orientation. Deux variétés compactes orientées M et N sont dites cobordantes lorsqu'il existe une variété à bord compacte et orientée W dont le bord est la réunion disjointe de et de N. On dit que W est un cobordisme orienté entre M et N. Il existe aussi d'autres notions de cobordisme abordées plus en avant dans l'article. (fr) 数学において、コボルディズムとは、コンパクト多様体の同値類であり、多様体の境界（フランス語で境界はコボルディズムと呼ぶ）を使って構成される。同じ次元の2つの多様体が、それらの非交和が1次元高いコンパクト多様体の境界となる。 (ja) 미분위상수학에서 보충 경계(補充境界, 영어: cobordism 코보디즘[*])는 두 개의 다양체 사이를 잇는, 이들을 경계로 하는 다양체이다. (ko) In matematica, un cobordismo è una tripla di oggetti , dove è unavarietà differenziabile, il cui bordo è l'unione disgiunta delle due varietà e . In altre parole èuna varietà il cui bordo è diviso in due parti. (it) Bordyzm – relacja równoważności w zbiorze zwartych rozmaitości różniczkowych. Na zbiorze klas abstrakcji tej relacji można zdefiniować działania w taki sposób, aby miał on strukturę pierścienia. Badanie relacji bordyzmu jest jednym z głównych nurtów w topologii algebraicznej. Dwie n-wymiarowe rozmaitości zwarte nazywamy bordycznymi, jeśli istnieje (n + 1)-wymiarowa rozmaitość różniczkowa z brzegiem której brzeg jest dyfeomorficzny z sumą rozłączną Fakt ten oznaczamy to przez Bordyzm jest relacją równoważności między rozmaitościami i . Zbiór klas abstrakcji tej relacji oznaczamy Zbiór jest grupą abelową względem dodawania zdefiniowanego następująco: gdzie jest sumą rozłączną rozmaitości i . W sumie prostej możemy zdefiniować strukturę pierścienia. Dla dowolnych klas definiujemy mnożenie jako iloczyn kartezjański przestrzeni topologicznych: które można rozszerzyć na cały zbiór Mnożenie to jest łączne i rozdzielne względem dodawania. Jednością jest klasa bordyzmów jednego punktu. Grupy określają gradację pierścienia . (pl) Бордизм, также бордантность — термин топологии, употребляющийся самостоятельно или в составе стандартныхсловосочетаний в нескольких родственных смыслах, почти во всех из них вместо бордизм раньше говорили о кобордизмах, старая терминология тоже сохранилась. (ru) Em matemática, o cobordismo é uma relação de equivalência fundamental na classe de variedades compactas da mesma dimensão, configurada usando o conceito de fronteira (bord francês, dando cobordismo) de uma variedade. Duas variedades da mesma dimensão são cobordantes se a união disjunta for o limite de uma variedade compacta uma dimensão mais alta. O limite de uma (n + 1)-variedade tridimensional W é uma variedade n-dimensional ∂W fechada, ou seja, com limite vazio. Em geral, uma variedade fechada não precisa ser um limite: a teoria do cobordismo é o estudo da diferença entre todas as variedades fechadas e aquelas que são limites. A teoria foi desenvolvida originalmente por René Thom para variedades suaves (ou seja, diferenciáveis), mas agora também existem versões para variedades topológicas e . Um cobordismo entre as variedades M e N é uma variedade compacta W cujo limite é a união disjunta de M e N, . Os cobordismos são estudados tanto pela relação de equivalência que eles geram quanto como objetos por si só. O cobordismo é uma relação de equivalência muito mais grossa que o difeomorfismo ou o homeomorfismo das variedades, e é significativamente mais fácil de estudar e calcular. Não é possível classificar variedades até difeomorfismo ou homeomorfismo em dimensões ≥ 4 - porque a palavra problema para grupos não pode ser resolvida - mas é possível classificar variedades até cobordismo. Cobordismos são objetos centrais de estudo em topologia geométrica e topologia algébrica. Na topologia geométrica, os cobordismos estão intimamente ligados à , e os são fundamentais no estudo de variedades de alta dimensão, a saber, a . Na topologia algébrica, as teorias de cobordismo são teorias de cohomologias extraordinárias fundamentais, e as categorias de cobordismos são os domínios das . (pt) Бордизм, також бордантність — термін топології, що використовується самостійно або ж у складі стандартних словосполучень в кількох споріднених сенсах. Майже у всіх з них замість бордизма раніше вживали термін кобордизм, попередня термінологія також збереглася. (uk) 在数学中，配边（英文：cobordism 来自法文的 bord）是紧流形的等价关系。它使用边界的拓扑概念。若两个流形M和N的不交并是另一个流形W的边界，那么M和N这两个流形是配边的。此外M和N的配边是W： . 配边缩写为 。M的配边类（cobordism class）是与M配边的所有流形的集合。 (zh) |
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| dbo:wikiPageExternalLink | https://archive.org/details/historyofalgebra0000dieu_g9a3/page/290 https://archive.today/20120529222452/http:/www.map.him.uni-bonn.de/B-Bordism https://web.archive.org/web/20110719102848/http:/www.map.him.uni-bonn.de/Bordism https://archive.org/details/historyofalgebra0000dieu_g9a3 |
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| dbp:author | Yuli B. Rudyak (en) |
| dbp:authorLink | Dmitri Anosov (en) |
| dbp:first | Dmitri V. (en) M. I. (en) |
| dbp:id | C/c022780 (en) b/b017030 (en) |
| dbp:last | Voitsekhovskii (en) Anosov (en) |
| dbp:title | Cobordism (en) bordism (en) |
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| rdfs:comment | In der Mathematik ist der Begriff des Kobordismus (auch: Bordismus) vor allem in der Topologie und ihren Anwendungen sowie in der topologischen Quantenfeldtheorie von Bedeutung. Er gilt als die bis heute „berechenbarste“ Relation unter Mannigfaltigkeiten, die geometrisch interessant ist. Als Schöpfer der Kobordismentheorie gilt René Thom (1954), wobei einige entscheidende Ideen schon von Lew Pontrjagin (1950 und davor) vorweggenommen wurden. (de) 数学において、コボルディズムとは、コンパクト多様体の同値類であり、多様体の境界（フランス語で境界はコボルディズムと呼ぶ）を使って構成される。同じ次元の2つの多様体が、それらの非交和が1次元高いコンパクト多様体の境界となる。 (ja) 미분위상수학에서 보충 경계(補充境界, 영어: cobordism 코보디즘[*])는 두 개의 다양체 사이를 잇는, 이들을 경계로 하는 다양체이다. (ko) In matematica, un cobordismo è una tripla di oggetti , dove è unavarietà differenziabile, il cui bordo è l'unione disgiunta delle due varietà e . In altre parole èuna varietà il cui bordo è diviso in due parti. (it) Бордизм, также бордантность — термин топологии, употребляющийся самостоятельно или в составе стандартныхсловосочетаний в нескольких родственных смыслах, почти во всех из них вместо бордизм раньше говорили о кобордизмах, старая терминология тоже сохранилась. (ru) Бордизм, також бордантність — термін топології, що використовується самостійно або ж у складі стандартних словосполучень в кількох споріднених сенсах. Майже у всіх з них замість бордизма раніше вживали термін кобордизм, попередня термінологія також збереглася. (uk) 在数学中，配边（英文：cobordism 来自法文的 bord）是紧流形的等价关系。它使用边界的拓扑概念。若两个流形M和N的不交并是另一个流形W的边界，那么M和N这两个流形是配边的。此外M和N的配边是W： . 配边缩写为 。M的配边类（cobordism class）是与M配边的所有流形的集合。 (zh) In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher. A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, . (en) En topologie différentielle, le cobordisme est une relation d'équivalence entre variétés différentielles compactes. Deux variétés compactes M et N sont dites cobordantes ou en cobordisme si leur réunion disjointe peut être réalisée comme le bord d'une variété à bord compacte L. On dit alors que cette variété L est un cobordisme entre M et N, ou bien que L réalise un cobordisme entre M et N. L'existence d'un tel cobordisme implique que M et N soient de même dimension. Il existe aussi d'autres notions de cobordisme abordées plus en avant dans l'article. (fr) Bordyzm – relacja równoważności w zbiorze zwartych rozmaitości różniczkowych. Na zbiorze klas abstrakcji tej relacji można zdefiniować działania w taki sposób, aby miał on strukturę pierścienia. Badanie relacji bordyzmu jest jednym z głównych nurtów w topologii algebraicznej. gdzie jest sumą rozłączną rozmaitości i . W sumie prostej możemy zdefiniować strukturę pierścienia. Dla dowolnych klas definiujemy mnożenie jako iloczyn kartezjański przestrzeni topologicznych: (pl) Em matemática, o cobordismo é uma relação de equivalência fundamental na classe de variedades compactas da mesma dimensão, configurada usando o conceito de fronteira (bord francês, dando cobordismo) de uma variedade. Duas variedades da mesma dimensão são cobordantes se a união disjunta for o limite de uma variedade compacta uma dimensão mais alta. Um cobordismo entre as variedades M e N é uma variedade compacta W cujo limite é a união disjunta de M e N, . (pt) |
| rdfs:label | Cobordism (en) Kobordismus (de) Cobordismo (it) Cobordisme (fr) 보충 경계 (ko) コボルディズム (ja) Bordyzm (pl) Cobordismo (pt) Бордизм (ru) Бордизм (uk) 配边 (zh) |
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