About: Bott periodicity theorem

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In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.

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  • In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, (real) KO-theory and (quaternionic) , associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres. (en)
  • 博特周期性定理描述了酉群的同伦群和正交群同伦群的周期性。简单的讲： 注意第2和第3个等式蕴涵了正交群的同伦群具有周期8。拉乌尔·博特开始是用莫尔斯理论证明的，后来又出现了K理论的证明。 (zh)
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  • Raoul Bott (en)
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  • Raoul (en)
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  • Bott (en)
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  • 1957 (xsd:integer)
  • 1959 (xsd:integer)
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  • 博特周期性定理描述了酉群的同伦群和正交群同伦群的周期性。简单的讲： 注意第2和第3个等式蕴涵了正交群的同伦群具有周期8。拉乌尔·博特开始是用莫尔斯理论证明的，后来又出现了K理论的证明。 (zh)
  • In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. (en)
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  • Bott periodicity theorem (en)
  • 博特周期性定理 (zh)
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