About: Homotopy groups of spheres

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In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n:

Attributes	Values
rdf:type	yago:WikicatSpheres yago:Abstraction100002137 yago:Attribute100024264 yago:Environment113934596 yago:Situation113927383 yago:Sphere114514039 yago:State100024720
rdfs:label	Grupos de homotopía de esferas (es) Homotopy groups of spheres (en) Groupes d'homotopie des sphères (fr) Гомотопические группы сфер (ru)
rdfs:comment	En mathématiques, et plus spécifiquement en topologie algébrique, les groupes d'homotopie des sphères sont des invariants qui décrivent, en termes algébriques, comment des sphères de dimensions et égales ou différentes peuvent s'enrouler l'une sur l'autre. La notion, définie au départ pour des sphères de dimension 1 (cercles) et de dimension 2, se généralise à des sphères de toutes dimensions (les -sphères). (fr) En el campo matemático de topología algebraica, los grupos de homotopía de esferas describen cómo esferas de variadas dimensiones pueden envolverse unas a otras. Son ejemplos de invariantes topológicas, las cuales reflejan, en términos algebraicos, la estructura de esferas vistas como espacios topológicos, olvidando detalles geométricos precisos. A diferencia de los grupos de homología, los cuales son también invariantes topológicas, los grupos de homotopía son sorprendentemente complejos y difíciles de calcular. * Datos: Q3118090 (es) In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n: (en) Гомотопические группы сфер — один из основных объектов изучения теории гомотопий, области алгебраической топологии. Гомотопические группы сфер классифицируют отображения между многомерными сферами с точностью до непрерывной деформации. Гомотопические группы сфер являются дискретными алгебраическими объектами, а именно конечнопорождёнными абелевыми группами. Несмотря на то, что классификация конечнопорождённых абелевых групп очень проста, точная структура гомотопических групп сфер до конца неизвестна. (ru)
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dcterms:subject	Spheres Homotopy theory
Wikipage page ID	1104697 (xsd:integer)
Wikipage revision ID	1114224136 (xsd:integer)
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