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| | The writer's name is Andera and she believes it seems quite great. The favorite hobby for him and his kids is to perform lacross and he would never give it up. Mississippi is where his house is. Invoicing is what I do for a living but I've always needed my own company.<br><br>My site :: real psychics - [http://www.octionx.sinfauganda.co.ug/node/22469 www.octionx.sinfauganda.co.ug] - |
| In [[mathematics]], '''equivariant cohomology''' is a theory from [[algebraic topology]] which applies to spaces with a ''[[group (mathematics)|group]] [[group action|action]]''. It can be viewed as a common generalization of [[group cohomology]] and an ordinary [[cohomology theory]].
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| Specifically, given a group <math>G</math> (discrete or not), a [[topological space]] <math>X</math> and an action
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| :<math>G\times X\rightarrow X,</math>
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| equivariant cohomology determines a [[graded ring]]
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| :<math>H^*_GX,</math>
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| the ''equivariant cohomology ring''. If <math>G</math> is the [[trivial group]], this is just the ordinary [[cohomology ring]] of <math>X</math>, whereas if <math>X</math> is [[contractible]], it reduces to the group cohomology of <math>G</math>.
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| == Outline construction ==
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| Equivariant cohomology can be constructed as the ordinary cohomology of a suitable space determined by <math>X</math> and <math>G</math>, called the ''homotopy [[orbit space]]''
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| :<math>X_{hG}</math> of <math>G</math>
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| on <math>X</math>. (The 'h' distinguishes it from the ordinary [[orbit space]] <math>X_G</math>.)
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| If <math>G</math> is the trivial group this space <math>X_{hG}</math> will turn out to be just <math>X</math> itself, whereas if <math>X</math> is contractible the space will be a [[classifying space]] for <math>G</math>.
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| === Properties of the homotopy orbit space ===
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| * If <math>G\times X\rightarrow X</math> is a free action then <math>X_{hG}\sim X_G.\ </math>
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| * If <math>G\times X\rightarrow X</math> is a trivial action then <math>X_{hG}\sim X\times BG.</math>
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| * In particular (as a special case of either of the above) if <math>G</math> is trivial then <math>X_{hG}\sim X.\ </math>
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| === Borel construction of the homotopy orbit space ===
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| The '''homotopy quotient''', also called '''homotopy orbit space''' or '''Borel construction''', is a “homotopically correct” version of the [[orbit space]] (the quotient of <math>X</math> by its <math>G</math>-action) in which <math>X</math> is first replaced by a larger but [[homotopy equivalent]] space so that the action is guaranteed to be [[group action|free]].
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| To this end, construct the [[universal bundle]] <math>EG\rightarrow BG</math> for <math>G</math> and recall that <math>EG</math> has a free <math>G</math>-action. Then the product <math>X\times EG</math>—which is homotopy equivalent to <math>X</math> since <math>EG</math> is contractible—has a “diagonal” <math>G</math>-action defined by taking the <math>G</math>-action on each factor: moreover, this action is free since it is free on <math>EG</math>. So we define the homotopy orbit space to be the orbit space of this <math>G</math>-action.
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| This construction is denoted by
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| :<math>X_{hG} = X\times_G EG.</math>
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| == See also ==
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| *[[Equivariant differential form]]
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| ==References==
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| *{{Springer|id=e/e036090|title=Equivariant cohomology}}
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| *{{Cite journal
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| | last = Tu
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| | first = Loring W.
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| | title = What Is . . . Equivariant Cohomology?
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| | journal = AMS Notices
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| | volume = 58
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| | issue = 03
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| | pages = 423–426
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| | date = March 2011
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| | url = http://www.ams.org/notices/201103/rtx110300423p.pdf }}
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| ==Further reading==
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| *Michel Brion, Equivariant cohomology and equivariant intersection theory [http://www-fourier.ujf-grenoble.fr/~mbrion/notesmontreal.pdf]
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| *V. W. Guillemin and S. Sternberg. Supersymmetry and equivariant de Rham theory . Springer-V erlag, Berlin, 1999
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| *CM Vergne, [http://www.math.jussieu.fr/~vergne/publications2/publications/clermontu.pdf Cohomologie équivariante et théorème de Stokes]
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Symplectic topology]]
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| [[Category:Group actions]]
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The writer's name is Andera and she believes it seems quite great. The favorite hobby for him and his kids is to perform lacross and he would never give it up. Mississippi is where his house is. Invoicing is what I do for a living but I've always needed my own company.
My site :: real psychics - www.octionx.sinfauganda.co.ug -