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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 Equivariant cohomology ： ウィキペディア英語版 Equivariant cohomology In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space ''X'' with action of a topological group ''G'' is defined as the ordinary cohomology ring with coefficient ring $\backslash Lambda$ of the homotopy quotient $EG\; \backslash times\_G\; X$: :$H\_G^$ *(X; \Lambda) = H^ *(EG \times_G X; \Lambda). If $G$ is the trivial group, this is the ordinary cohomology ring of $X$, whereas if $X$ is contractible, it reduces to the cohomology ring of the classifying space $BG$ (that is, the group cohomology of $G$ when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map $EG\; \backslash times\_G\; X\; \backslash to\; X/G$ is a homotopy equivalence and so one gets: $H\_G^$ *(X; \Lambda) = H^ *(X/G; \Lambda). If ''X'' is a manifold, ''G'' a compact Lie group and $\backslash Lambda$ is the field of real numbers or the field of complex numbers (the most typical situation), then the above cohomology may be computed using so-called Cartan model (see equivariant differential forms.) The construction should not be confused as a more naive cohomology of invariant differential forms: if ''G'' is a compact Lie group, then, by the averaging argument, any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information. The Koszul duality is known to hold between equivariant cohomology and ordinary cohomology. == Homotopy quotient == The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of $X$ by its $G$-action) in which $X$ is first replaced by a larger but homotopy equivalent space so that the action is guaranteed to be free. To this end, construct the universal bundle ''EG'' → ''BG'' for ''G'' and recall that ''EG'' admits a free ''G''-action. Then the product ''EG'' × ''X'' —which is homotopy equivalent to ''X'' since ''EG'' is contractible—admits a “diagonal” ''G''-action defined by (''e'',''x'').''g'' = (''eg'',''g−1x''): moreover, this diagonal action is free since it is free on ''G''. So we define the homotopy quotient ''X''''G'' to be the orbit space (''EG'' × ''X'')/''G'' of this free ''G''-action. In other words, the homotopy quotient is the associated ''X''-bundle over ''BG'' obtained from the action of ''G'' on a space ''X'' and the principal bundle ''EG'' → ''BG''. This bundle ''X'' → ''X''''G'' → ''BG'' is called the Borel fibration. 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「Equivariant cohomology」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.