The Atiyah-Segal completion theorem

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.

 

Equivariant K-theory III

Next, I would like to describe an alternative description of relative K-theory which is sometimes convenient (e.g. when describing the Thom isomorphism in K-theory). Let be a compact Lie group, as always. Let be a pair of compact -spaces (with ); then we have defined therelative -theory via

Here is equipped with a distinguished basepoint (corresponding to ), and as a result this makes sense. As usual, we can use this definition to make into a cohomology theory on compact -spaces.

To describe without use of the space , we can proceed as follows.

Definition 1 We let be the category of complexes of -vector bundles

on , which when restricted to are exact. A morphism in is a map of chain complexes.

The idea is that we are going to assign to every element of an element of the relative K-theory , by effectively taking the alternating sum . In order to do this, we will start by modifying the complex by adding acyclic complexes. Namely, we start by adding complexes of the form

where is a -vector bundle on , to make all but the first term of trivial (i.e. coming from an -representation). With this change made, we can assume that all but the first term of is stably trivial. Then the first term of is stably trivial when restricted to by exactness of . Consequently, we can quotient all the terms by and get a complex of -vector bundles ; this is exact at the basepoint of . Now, taking the alternating sum as desired, we get a map

This map does not see stable equivalence; that is, if we add to a complex a complex of the form , the image in does not change. Moreover, it is homotopy invariant.

In fact, one can give a presentation of the group in this way. We start with the set of all such complexes in as above. We identify complexes which are chain homotopic to each other. Then, we mod out by the relation of (geometric) homotopy: if one has complexes which can be obtained by restriction to the end faces of a complex in , then they should both be identified. Given these identifications, one gets precisely the group .

I don’t really want to prove these things in detail, partially because I don’t want to get too bogged down with this project.

 

Equivariant K-theory II

Last time, I described the construction which assigns to every compact -space (for a compact Lie group) the equivariant K-group . We saw that this was a functor from the (equivariant) homotopy category to commutative rings, using more or less the same arguments as in ordinary homotopy theory, only with small alterations.

The purpose of this post is to describe more of Segal’s paper. Actually, I won’t be covering any legitimate K-theory in this post; that’ll have to wait for a third. I’ll mostly be describing various classical constructions for vector bundles in the equivariant setting.

In the classical theory of (ordinary) vector bundles on compact spaces, a basic result is the Serre-Swan theorem, which identifies the category of (complex) vector bundles with the category of projective modules over the ring of complex-valued continuous functions on . This is essentially a reflection of the fact that any vector bundle on , say , can be obtained as a retract of some trivial bundle . Taking retracts corresponds to choosing idempotents in the ring of -by- matrices in , and this description via idempotents applies as well to projective modules over (or, in fact, any commutative ring).

The crucial statement here, that any vector bundle is a retract of a trivial one, fails in the equivariant case, simply because a vector bundle on which acts nontrivially can’t be a retract of a vector bundle with trivial action. But we have something reasonably close to it.

 

Definition 1 Given a -representation and a -space , we can form a vector bundle , which his naturally -equivariant.

This bundle is, equivalently, formed by taking the equivariant map . -vector bundles on are identified with -representations, so we just have to pull back.

Anyway, the claim is:

Theorem 2 (Equivariant Serre-Swan (Segal)) Any -vector bundle is a direct summand of a bundle for some -representation .

 

Equivariant K-theory

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.