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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 Eilenberg–Ganea theorem ： ウィキペディア英語版 Eilenberg–Ganea theorem In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group ''G'' with certain conditions on its cohomological dimension (namely 3 ≤ cd(''G'') ≤ ''n''), one can construct an aspherical CW complex ''X'' of dimension ''n'' whose fundamental group is ''G''. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the ''Annals of Mathematics''.〔 *〕 ==Definitions== Group cohomology: Let ''G'' be a group and ''X'' = ''K''(''G'', 1) is the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of ''Z'' over the group ring ''Z''() (where ''Z'' is a trivial ''Z''() module). :$\backslash cdots\; \backslash xrightarrow\; C\_n(E)\backslash xrightarrow\; C\_(E)\backslash rightarrow\; \backslash cdots\; \backslash rightarrow\; C\_1(E)\backslash xrightarrow\; C\_0(E)\backslash xrightarrow\; Z\backslash rightarrow\; 0,$ where ''E'' is the universal cover of ''X'' and ''C''''k''(''E'') is the free abelian group generated by singular ''k'' chains. Group cohomology of the group ''G'' with coefficient in ''G'' module ''M'' is the cohomology of this chain complex with coefficient in ''M'' and is denoted by ''H'' *(''G'', ''M''). Cohomological dimension: ''G'' has cohomological dimension ''n'' with coefficients in ''Z'' (denoted by cd''Z''(''G'')) if :$n=\backslash sup\; \backslash M\backslash textH^(G,M)\backslash neq\; 0\backslash \}.$ Fact: If ''G'' has a projective resolution of length ≤ ''n'', i.e. ''Z'' as trivial ''Z''() module has a projective resolution of length ≤ ''n'' if and only if ''H''''i''''Z''(''G'',''M'') = 0 for all ''Z'' module ''M'' and for all ''i'' > ''n''. Therefore we have an alternative definition of cohomological dimension as follows, ''Cohomological dimension of G with coefficient in'' ''Z'' ''is the smallest n (possibly infinity) such that G has a projective resolution of length'' ''n'', ''i.e.'' ''Z'' ''has a projective resolution of length'' ''n'' ''as a trivial'' ''Z''() ''module.'' 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「Eilenberg–Ganea theorem」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.