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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 Complex cobordism ： ウィキペディア英語版 Complex cobordism In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute. The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum. ==Spectrum of complex cobordism== The complex bordism MU *(''X'') of a space ''X'' is roughly the group of bordism classes of manifolds over ''X'' with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows. The space MU(''n'') is the Thom space of the universal ''n''-plane bundle over the classifying space BU(''n'') of the unitary group U(''n''). The natural inclusion from U(''n'') into U(''n''+1) induces a map from the double suspension S2MU(''n'') to MU(''n''+1). Together these maps give the spectrum MU; namely, it is the homotopy colimit of MU(''n''). Examples: MU(0) is the sphere spectrum. MU(1) is the desuspension $\backslash Sigma^\; \backslash mathbb\backslash mathbf^\backslash infty$ of $\backslash mathbb\backslash mathbf^\backslash infty$. The nilpotence theorem states that, for any ring spectrum ''R'', the kernel of $\backslash pi\_$ * R \to \operatorname_ *(R) consists of nilpotent elements.〔http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf〕 The theorem implies in particular that, if ''S'' is the sphere spectrum, then for any ''n'' > 0, every element of $\backslash pi\_n\; S$ is nilpotent (a theorem of Nishida). (Proof: if ''x'' is in $\backslash pi\_n\; S$, then ''x'' is a torsion but its image in $\backslash operatorname\_$ *(S) \simeq L, the Lazard ring, cannot be torsion since ''L'' is a polynomial ring. Thus, ''x'' must be in the kernel.) 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「Complex cobordism」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.