The Atiyah-Segal completion theorem calculates the -theory of the classifying space of a compact Lie group . Namely, given such a , we know that there is a universal principal -bundle , with the property that is contractible. Given a -representation , we can form the vector bundle
via the “mixing” construction. In this way, we get a functor
and thus a homomorphism from the (complex) representation ring to the -theory of ,
This is not an isomorphism; one expects the cohomology of an infinite complex (at least if certain 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 induces an isomorphism from the -adic completion , where is the augmentation ideal in . Moreover, .
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…)