Low-dimensional topology (original) (raw)
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
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| dbo:abstract | In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory. (en) En mathématiques, la topologie en basses dimensions est la branche de la topologie qui concerne les variétés de dimension inférieure ou égale à quatre. Des sujets représentatifs en sont l'étude des variétés de dimension 3 (en) et la théorie des nœuds et des tresses. Elle fait partie de la topologie géométrique. (fr) 数学における低次元位相幾何学（ていじげんいそうきかがく、英: low-dimensional topologyは、4次元、あるいはそれ以下の次元の多様体の研究をする位相幾何学の一分野である。扱われる主題は、および4次元多様体の構造論、結び目理論および組み紐群などがある。低次元トポロジーは幾何学的位相幾何学の一部と見なすことができる。 (ja) La topologia in dimensione bassa è una branca della topologia (e quindi della geometria) che studia gli "spazi di dimensione 1, 2, 3 e 4". La topologia in dimensione bassa studia soprattutto le varietà, da molteplici punti di vista. A partire dagli anni sessanta, è emersa sempre più la peculiarità di queste dimensioni, il cui studio necessita di strumenti ad hoc, più specifici delle tecniche generali fornite dalla topologia algebrica e della topologia differenziale. Da cui la nascita negli anni 60/70 di un settore apposito, che studiasse tecniche adeguate, soprattutto alle dimensioni 3 e 4. Un esempio lampante di questo fenomeno è la dimostrazione di Stephen Smale della Congettura di Poincaré: gli argomenti usati dal matematico statunitense funzionano per tutte le dimensioni superiori a 4, ma non per le altre. La stessa congettura è stata successivamente dimostrata con tecniche complesse e molto specifiche in dimensione 4 da Michael Freedman nel 1982 e in dimensione 3 da Grigori Perelman nel 2003 (i casi 1 e 2 sono molto facili, come notò Henri Poincaré già alla fine del XIX secolo). I risultati sorprendenti ottenuti da William Thurston, Simon Donaldson, Michael Freedman, Vaughan Jones e Edward Witten nell'ambito delle varietà di dimensione 3 e 4, ottenuti tra la fine degli anni settanta, e tutti gli anni ottanta, hanno valso a tutti questi una medaglia Fields, e hanno portato il settore alla ribalta della geometria e di tutta la matematica. Grigori Perelman, anch'egli vincitore di una medaglia Fields, chiude infine nel 2003 la congettura di Poincaré, insoluta per più di un secolo. (it) De laag-dimensionale topologie is in de wiskunde een tak van de topologie die variëteiten van vier of minder dimensies bestudeert. Representatieve onderwerpen zijn de structuurtheorie van 3-variëteiten, 4-variëteiten, knopentheorie en vlechtgroepen. Het wordt als een onderdeel van de meetkundige topologie beschouwd. Een aantal nieuwe ontwikkelingen vanaf de jaren 1960 hebben lage dimensies als interessant onderwerp in de meetkundige topologie extra benadrukt. De oplossing, in 1961, door Stephen Smale, van het vermoeden van Poincaré in hogere dimensies liet de dimensies drie en vier des te moeilijker lijken; en inderdaad bleken er nieuwe methoden nodig te zijn, aangezien de extra vrijheidsgraden van hogere dimensies betekenden dat vragen daar konden worden gereduceerd tot rekenmethodes die beschikbaar waren in de . Thurstons vermeetkundigingsvermoeden, door hem in de late jaren zeventig van de twintigste eeuw geformuleerd, bood een raamwerk dat suggereerde dat meetkunde en topologie in lagere dimensies nauw vervlochten zijn, en Thurstons bewijs van de geometrisering voor Haken-variëteiten maakte gebruik van een groot aantal verschillende hulpmiddelen uit voorheen slechts zwak verbonden deelgebieden uit de wiskunde. Vaughan Jones' ontdekking van de Jones-veelterm in de vroege jaren tachtig van de 20ste eeuw stuurde niet alleen de knopentheorie in een nieuwe richting, maar bracht ook een aantal nog steeds mysterieuze verbindingen aan het licht tussen laag-dimensionale topologie en de wiskundige natuurkunde. Tot slot kondigde Grigori Perelman in 2002 een bewijs aan van het drie-dimensionale vermoeden van Poincaré. Hij maakt hierbij gebruik van Richard S. Hamiltons Ricci-stroom, een idee uit het veld van de . In zijn algemeenheid heeft deze vooruitgang tot een betere integratie van de topologie en de rest van de wiskunde geleid. (nl) A topologia de baixa dimensão, ou topologia geométrica, é a área da topologia dedicada ao estudo das variedades de dimensão inferior ou igual a 4. Inclui a teoria dos nós. O problema mais famoso desta área é a Conjectura de Poincaré. (pt) 在数学中，低维拓扑是拓扑学中研究二、三、四维流形或更广义的拓扑空间的一个分支。有代表性的研究主题包括三维流形、、扭结和辫群等的结构理论。低维拓扑是几何拓扑学的一部分。 (zh) |
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| dbp:authorlink | Emil Artin (en) William Thurston (en) Oswald Teichmüller (en) |
| dbp:first | Emil (en) William (en) Oswald (en) |
| dbp:last | Thurston (en) Artin (en) Teichmüller (en) |
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| dbp:year | 1940 (xsd:integer) 1947 (xsd:integer) 1982 (xsd:integer) |
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| rdfs:comment | In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory. (en) En mathématiques, la topologie en basses dimensions est la branche de la topologie qui concerne les variétés de dimension inférieure ou égale à quatre. Des sujets représentatifs en sont l'étude des variétés de dimension 3 (en) et la théorie des nœuds et des tresses. Elle fait partie de la topologie géométrique. (fr) 数学における低次元位相幾何学（ていじげんいそうきかがく、英: low-dimensional topologyは、4次元、あるいはそれ以下の次元の多様体の研究をする位相幾何学の一分野である。扱われる主題は、および4次元多様体の構造論、結び目理論および組み紐群などがある。低次元トポロジーは幾何学的位相幾何学の一部と見なすことができる。 (ja) A topologia de baixa dimensão, ou topologia geométrica, é a área da topologia dedicada ao estudo das variedades de dimensão inferior ou igual a 4. Inclui a teoria dos nós. O problema mais famoso desta área é a Conjectura de Poincaré. (pt) 在数学中，低维拓扑是拓扑学中研究二、三、四维流形或更广义的拓扑空间的一个分支。有代表性的研究主题包括三维流形、、扭结和辫群等的结构理论。低维拓扑是几何拓扑学的一部分。 (zh) La topologia in dimensione bassa è una branca della topologia (e quindi della geometria) che studia gli "spazi di dimensione 1, 2, 3 e 4". La topologia in dimensione bassa studia soprattutto le varietà, da molteplici punti di vista. A partire dagli anni sessanta, è emersa sempre più la peculiarità di queste dimensioni, il cui studio necessita di strumenti ad hoc, più specifici delle tecniche generali fornite dalla topologia algebrica e della topologia differenziale. Da cui la nascita negli anni 60/70 di un settore apposito, che studiasse tecniche adeguate, soprattutto alle dimensioni 3 e 4. (it) De laag-dimensionale topologie is in de wiskunde een tak van de topologie die variëteiten van vier of minder dimensies bestudeert. Representatieve onderwerpen zijn de structuurtheorie van 3-variëteiten, 4-variëteiten, knopentheorie en vlechtgroepen. Het wordt als een onderdeel van de meetkundige topologie beschouwd. (nl) |
| rdfs:label | Low-dimensional topology (en) Niedrigdimensionale Topologie (de) Topologia in dimensione bassa (it) Topologie en basses dimensions (fr) 低次元トポロジー (ja) Laag-dimensionale topologie (nl) Topologia de baixa dimensão (pt) 低維拓撲 (zh) |
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