# 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

${\displaystyle \pi \colon X\times EG\to X}$

induces an isomorphism of prorings

${\displaystyle \pi ^{*}\colon K_{G}^{*}(X){\hat {_{I}}}\to K_{G}^{*}(X\times EG)}$.

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 ${\displaystyle K^{*}(BG)\cong R(G){\hat {_{I}}}}$ 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. [1]

The theorem was first proved for finite groups by Michael Atiyah in 1961, [2] and a proof of the general case was published by Atiyah together with Graeme Segal in 1969. [3] Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups. [4] [5] The corresponding statement for algebraic K-theory was proven by Merkujev, holding in the case that the group is algebraic over the complex numbers.

## References

1. ^ Greenlees, J.P.C. (1996). "An introduction to equivariant K-theory.". CBMS Regional Conference Series. Equivariant homotopy and cohomology theory. Published for the Conference Board of the Mathematical Sciences, Washington, DC. pp. 143–152.
2. ^ Atiyah, M.F. (1961). "Characters and cohomology of finite groups" (PDF) 9 (1): 23–64. Retrieved 2008-06-19.
3. ^ Atiyah, M.F.; Segal, G.B. (1969). "Equivariant K-theory and completion" (PDF). Journal of Differential Geometry 3: 1–18. Retrieved 2008-06-19.
4. ^ Jackowski, S. (1985). "Families of subgroups and completion". J. Pure Appl. Algebra 37 (2): 167–179. doi:10.1016/0022-4049(85)90094-5.
5. ^ Adams, J.F.; Haeberly, J.P.; Jackowski, S.;; May, J.P. (1988). "A generalization of the Atiyah-Segal Completion Theorem". Topology 27 (1): 1–6. doi:10.1016/0040-9383(88)90002-X.