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…)

The semester here is now over (save for final exams), which means that I hope to start posting on this blog more frequently again. One of my goals for the next couple of months is to understand the proof of the Atiyah-Singer index theorem. I’m pretty far from that point right now, so I’ll start with the foundations; this will have the additional effect of forcing me to engage more deeply with the basic stuff. (It’s all too easy for students — and I seem to be especially prone to this — to get flaky and learn mathematical terms without actually gaining understanding!) Ideally, I’m hoping to repeat a MaBloWriMo-type project.

To start with, I’ve been reading Segal’s paper “Equivariant K-theory.” This post will cover some of the basic ideas in this paper. (more…)