The Atiyah-Segal completion theorem calculates the ${K}$-theory of the classifying space ${BG}$ of a compact Lie group ${G}$. Namely, given such a ${G}$, we know that there is a universal principal ${G}$-bundle ${EG \rightarrow BG}$, with the property that ${EG}$ is contractible. Given a ${G}$-representation ${V}$, we can form the vector bundle

$\displaystyle EG \times_G V \rightarrow BG$

via the “mixing” construction. In this way, we get a functor

$\displaystyle \mathrm{Rep}(G) \rightarrow \mathrm{Vect}(BG),$

and thus a homomorphism from the (complex) representation ring${R(G)}$ to the ${K}$-theory of ${BG}$,

$\displaystyle R(G) \rightarrow K^0(BG).$

This is not an isomorphism; one expects the cohomology of an infinite complex (at least if certain ${\lim^1}$ terms vanish) to have a natural structure of a complete topological group. Modulo this, however, it turns out that:

Theorem (Atiyah-Segal) The natural map ${R(G) \rightarrow K^0(BG)}$ induces an isomorphism from the ${I}$-adic completion ${R(G)_{I}^{\wedge} \simeq K^0(BG)}$, where ${I}$ is the augmentation ideal in ${R(G)}$. Moreover, ${K^1(BG) =0 }$.

The purpose of this post is to describe a proof of the Atiyah-Segal completion theorem, due to Adams, Haeberly, Jackowski, and May. This proof uses heavily the language of pro-objects, which was discussed in the previous post (or rather, the dual notion of ind-objects was discussed). Remarkably, their approach uses this formalism to eliminate almost all the actual computations, by reducing to a special case. (more…)