Atiyah–Segal completion theorem

Remarks

The special case $X = \text{point}$ gives $K^0(BG) \cong R(G)^\wedge_I$ and $K^1(BG) = 0$. Atiyah had proved this case earlier for finite groups. The general proof embeds $G$ in a unitary group, which reduces the theorem to explicit computations for unitary groups and their maximal tori.

Example: for the circle group $G = U(1)$, $R(G) = \Z[t, t^{-1}]$ and $I = (t - 1)$. With $x = t - 1$, the completion is the power series ring $\Z[\![x]\!]$. On the other side, $BU(1) = \C P^\infty$ and $K^0(\C P^n) = \Z[x]/(x^{n+1})$ with $x = [H] - 1$, so the inverse limit is also $\Z[\![x]\!]$.

Completion is needed: $R(G)$ itself is much smaller than $K^0(BG)$; for $G = \Z/2$, $R(G) = \Z[s]/(s^2 - 1)$ is countable, while $K^0(B\Z/2) \cong \Z \oplus \Z_2$, with $\Z_2$ the uncountable group of $2$-adic integers.

The Segal conjecture is the analogous statement for stable cohomotopy and the Burnside ring.

Going deeper

For a student

The theorem is a model for many completion results relating equivariant invariants of $G$-spaces to non-equivariant invariants of Borel constructions. It is used to compute the K-theory of classifying spaces of finite and compact Lie groups. It also shows what the Borel construction loses: exactly the information in $K_G^*(X)$ that disappears under $I$-adic completion.