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The '''Atiyah–Segal completion theorem''' is a [[theorem]] in [[mathematics]] about [[equivariant]] [[K-theory]] in [[homotopy theory]]. Let ''G'' be a [[compact group|compact]] [[Lie group]] and let ''X'' be a ''G''-[[CW-complex]]. The theorem then states that the projection map
== variously ==


:<math>\pi\colon X\times EG\to X</math>  
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induces an isomorphism of [[profinite group|prorings]]
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:<math>\pi^*\colon K_G^*(X)\hat{_I}\to K_G^*(X\times EG)</math>.   
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Here, the induced map has as [[Domain of a function|domain]] the [[completion (ring theory)|completion]] of the ''G''-equivariant K-theory of ''X'' with respect to ''I'', where ''I'' denotes the [[augmentation ideal]] of the [[representation ring]] of ''G''.
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In the special case of ''X'' a point, the theorem specializes to give an isomorphism <math>K^*(BG)\cong R(G)\hat{_I}</math> between the K-theory of the [[classifying space]] of ''G'' and the completion of the representation ring.
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The theorem can be interpreted as giving a comparison between the geometrical process of completing a ''G''-space by making the [[group action|action]] free and the algebraic process of completing with respect to an ideal.
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<ref name=Greenlees>{{cite conference
| author = Greenlees, J.P.C.
| year = 1996
| title = An introduction to equivariant K-theory.
| conference = Equivariant homotopy and cohomology theory
| booktitle = CBMS Regional Conference Series
| volume = 91
| pages = 143–152
| publisher = Published for the Conference Board of the Mathematical Sciences,
    Washington, DC
| url =
| conferenceurl =
}}</ref>


The theorem was first proved for [[finite groups]] by [[Michael Atiyah]] in 1961,
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<ref name=Atiyah1961>{{cite journal
 
| last = Atiyah|first= M.F.|authorlink=Michael Atiyah
  <li>[http://www.xiaobinlab.com/bbs/forum.php?mod=viewthread&tid=337503&fromuid=23411 http://www.xiaobinlab.com/bbs/forum.php?mod=viewthread&tid=337503&fromuid=23411]</li>
| year = 1961
 
| title = Characters and cohomology of finite groups
  <li>[http://www.150800.net/forum.php?mod=viewthread&tid=2813174 http://www.150800.net/forum.php?mod=viewthread&tid=2813174]</li>
| volume = 9
 
| issue = 1
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
| pages = 23–64
 
| url = http://www.springerlink.com/index/3228267279648852.pdf
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24711217 http://enseignement-lsf.com/spip.php?article64#forum24711217]</li>
| accessdate = 2008-06-19
 
}}</ref>
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and a proof of the general case was published by Atiyah together with [[Graeme Segal]] in 1969.
 
<ref name=Atiyah1969>{{cite journal
</ul>
| last1 = Atiyah|first1= M.F.|author1-link=Michael Atiyah
| last2 = Segal|first2= G.B.|author2-link=Graeme Segal
| year = 1969
| title = Equivariant K-theory and completion
| journal = [[Journal of Differential Geometry]]
| volume = 3
| pages = 1–18
| url = http://intlpress.com/JDG/archive/pdf/1969/3-1&2-1.pdf
| accessdate = 2008-06-19
}}</ref>
Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups.
<ref name=Jackowski1985>{{cite journal
| last = Jackowski | first= S.
| year = 1985
| title = Families of subgroups and completion
| journal = J. Pure Appl. Algebra
| volume = 37
| issue = 2
| pages = 167–179
| doi = 10.1016/0022-4049(85)90094-5
}}</ref>
<ref name=Adams1988>{{cite journal
| last1 = Adams | first1= J.F.
| last2 = Haeberly|first2= J.P.|last3= Jackowski|first3= S.; |last4=May|first4= J.P.|author4-link=J. Peter May
| year = 1988
| title = A generalization of the Atiyah-Segal Completion Theorem
| journal = Topology
| volume = 27
| issue = 1
| pages = 1–6
| doi = 10.1016/0040-9383(88)90002-X
}}</ref>


==See also==
== Louis Vuitton Replica Handbags Nz  A study by the NRMP ==
*[[Segal conjecture]]


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{{DEFAULTSORT:Atiyah-Segal completion theorem}}
 
[[Category:K-theory]]
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