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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Topological K-theory ： ウィキペディア英語版 Topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch. == Definitions == Let be a compact Hausdorff space and . Then is the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, usually denotes complex -theory whereas real -theory is sometimes written as . The remaining discussion is focussed on complex -theory, the real case being similar. As a first example, note that the -theory of a point are the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers are the integers. There is also a reduced version of -theory, $\backslash widetilde(X)$, defined for a compact pointed space (cf. reduced homology). This reduced theory is intuitively modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles and are said to be stably isomorphic if there are trivial bundles and , so that . This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, $\backslash widetilde(X)$ can be defined as the kernel of the map induced by the inclusion of the base point into . -theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces :$\backslash widetilde(X/A)\backslash to\backslash widetilde(X)\backslash to\backslash widetilde(A)$ extends to a long exact sequence :$\backslash cdots\; \backslash to\; \backslash widetilde(SX)\; \backslash to\; \backslash widetilde(SA)\; \backslash to\; \backslash widetilde(X/A)\; \backslash to\; \backslash widetilde(X)\; \backslash to\; \backslash widetilde(A).$ Let be the -th reduced suspension of a space and then define :$\backslash widetilde^(X):=\backslash widetilde(S^nX),\; \backslash qquad\; n\backslash geq\; 0.$ Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without base points: :$K^(X)=\backslash widetilde^(X\_+).$ Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers. 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「Topological K-theory」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.