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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 of the homotopy quotient : : If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map is a homotopy equivalence and so one gets: If ''X'' is a manifold, ''G'' a compact Lie group and 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 by its -action) in which 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
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