Atiyah–Segal completion theorem

The Atiyah-Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let "G" be a compact Lie group and let "X" be a "G"-CW-complex. The theorem then states, that the projection map

:$picolon\; X\; imes\; EG\; o\; X$

induces an isomorphism of prorings

:$pi^*colon\; K\_G^*(X)hat\{\_I\}\; o\; K\_G^*(X\; imes\; EG)$.

Here, the induced map has as domain the 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".

In the special case of "X" a point, the theorem specializes to give an isomorphism $K^*(BG)cong\; R(G)hat\{\_I\}$ between the K-theory of the classifying space of "G" and the completion of the representation ring.

The theorem can be interpreted as giving a comparison between the geometrical process of completing a "G"-space by making the action free and the algebraic process of completing with respect to an ideal.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 = ]

The theorem was first proved for finite groups by Michael Atiyah in 1961,cite journal
author = Atiyah, M.F.
year = 1961
title = Characters and cohomology of finite groups
volume = 9
issue = 1
pages = 23-64
url = http://www.springerlink.com/index/3228267279648852.pdf
accessdate = 2008-06-19] and a proof of the general case was published by Atiyah together with Graeme Segal in 1969.cite journal
author = Atiyah, M.F.
coauthors = Segal, G.B.
year = 1969
title = Equivariant K-theory and completion
journal = J. Differential Geometry
volume = 3
pages = 1-18
url = http://intlpress.com/JDG/archive/pdf/1969/3-1&2-1.pdf
accessdate = 2008-06-19] Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups.cite journal
author = Jackowski, S.
year = 1985
title = Families of subgroups and completion
journal = J. Pure Appl. Algebra
volume = 37
issue = 2
pages = 167-179] cite journal
author = Adams, J.F.
coauthors = Haeberly, J.P.; Jackowski, S.; May, J.P.
year = 1988
title = A generalization of the Atiyah-Segal Completion Theorem
journal = Topology
volume = 27
issue = 1
pages = 1-6]

References