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In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , 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) KSp-theory, 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. ==Context and significance== The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated). The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished. The stable theory was still hard to compute with, in practice. What Bott periodicity offered was an insight into some highly non-trivial spaces, with central status in topology because of the connection of their cohomology with characteristic classes, for which all the (''unstable'') homotopy groups could be calculated. These spaces are the (infinite, or ''stable'') unitary, orthogonal and symplectic groups ''U'', ''O'' and Sp. In this context, ''stable'' refers to taking the union ''U'' (also known as the direct limit) of the sequence of inclusions : and similarly for ''O'' and Sp. Bott's (now somewhat awkward) use of the word ''stable'' in the title of his seminal paper refers to these stable classical groups and not to stable homotopy groups. The important connection of Bott periodicity with the stable homotopy groups of spheres comes via the so-called stable ''J''-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups . Originally described by George W. Whitehead, it became the subject of the famous Adams conjecture (1963) which was finally resolved in the affirmative by Daniel Quillen (1971). Bott's original results may be succinctly summarized in: Corollary: The (unstable) homotopy groups of the (infinite) classical groups are periodic: : : : Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results: : : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bott periodicity theorem」の詳細全文を読む スポンサード リンク
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