# 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

$\pi\colon X\times EG\to X$

induces an isomorphism of prorings

$\pi^*\colon K_G^*(X)\hat{_I}\to K_G^*(X\times EG)$.

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 $K^*(BG)\cong R(G)\hat{_I}$ 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.