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

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

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 ${\displaystyle K^{*}(BG)\cong R(G)_{\widehat {I\,}}}$ 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 taking the homotopy quotient of a G-space, by making the action free before passing to the quotient, 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 Alexander Merkurjev, holding in the case that the group is algebraic over the complex numbers.