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…)

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 ${G}$ be a compact Lie group, as always. Let ${(X, A)}$ be a pair of compact ${G}$-spaces (with ${ A \subset X}$); then we have defined therelative ${K_G}$-theory via

$\displaystyle K_G(X, A) \equiv \widetilde{K}_G(X/A).$

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

To describe ${K_G(X, A)}$ without use of the space ${X/A}$, we can proceed as follows.

Definition 1 We let ${C_G(X, A)}$ be the category of complexes of ${G}$-vector bundles

$\displaystyle 0 \rightarrow E_0 \rightarrow \dots \rightarrow E_n \rightarrow 0$

on ${X}$, which when restricted to ${A}$ are exact. A morphism in ${C_G(X, A)}$ is a map of chain complexes.

The idea is that we are going to assign to every element ${E_\bullet}$ of ${C_G(X,A)}$ an element of the relative K-theory ${K_G(X, A) = \widetilde{K}_G(X/A)}$, by effectively taking the alternating sum ${\sum (-1)^i [E_i]}$. In order to do this, we will start by modifying the complex ${E_\bullet}$ by adding acyclic complexes. Namely, we start by adding complexes of the form

$\displaystyle 0 \rightarrow 0 \rightarrow \dots \rightarrow 0 \rightarrow F \rightarrow F \rightarrow 0 \rightarrow 0 \rightarrow \dots$

where ${F}$ is a ${G}$-vector bundle on ${A}$, to make all but the first term of ${E_\bullet}$ trivial (i.e. coming from an ${R(G)}$-representation). With this change made, we can assume that all but the first term of ${E_\bullet}$ is stably trivial. Then the first term of ${E_\bullet}$ is stably trivial when restricted to ${A}$ by exactness of ${E_\bullet|_A}$. Consequently, we can quotient all the terms by ${A}$ and get a complex of ${G}$-vector bundles ${E_\bullet|_{X/A}}$; this is exact at the basepoint of ${X/A}$. Now, taking the alternating sum as desired, we get a map

$\displaystyle C_G(X, A) \rightarrow K_G(X, A).$

This map does not see stable equivalence; that is, if we add to a complex ${E_\bullet}$ a complex of the form ${0 \rightarrow F \rightarrow F \rightarrow 0}$, the image in ${K_G(X, A)}$ does not change. Moreover, it is homotopy invariant.

In fact, one can give a presentation of the group ${K_G(X, A)}$ in this way. We start with the set of all such complexes in ${C_G(X, A)}$ 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 ${E_\bullet, F_\bullet \in C_G(X, A)}$ which can be obtained by restriction to the end faces of a complex in ${C_G(X \times [0, 1], A \times [0, 1])}$, then they should both be identified. Given these identifications, one gets precisely the group ${K_G(X, A)}$.

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. (more…)

Last time, I described the construction which assigns to every compact ${G}$-space ${X}$ (for ${G}$ a compact Lie group) the equivariant K-group ${K_G(X)}$. 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 ${\mathrm{Vect}(X)}$ of (complex) vector bundles with the category of projective modules over the ring ${C(X; \mathbb{C})}$ of complex-valued continuous functions on ${X}$. This is essentially a reflection of the fact that any vector bundle on ${X}$, say ${E \rightarrow X}$, can be obtained as a retract of some trivial bundle ${\mathbb{C}^n \times X \rightarrow X}$. Taking retracts corresponds to choosing idempotents in the ring of ${n}$-by-${n}$ matrices in ${C(X; \mathbb{C})}$, and this description via idempotents applies as well to projective modules over ${C(X; \mathbb{C})}$ (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 ${G}$ 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 ${G}$-representation ${V}$ and a ${G}$-space ${X}$, we can form a vector bundle ${V \times X \rightarrow X}$, which his naturally ${G}$-equivariant.

This bundle is, equivalently, formed by taking the equivariant map ${X \rightarrow \ast}$. ${G}$-vector bundles on ${\ast}$ are identified with ${G}$-representations, so we just have to pull back.

Anyway, the claim is:

Theorem 2 (Equivariant Serre-Swan (Segal)) Any ${G}$-vector bundle ${E \rightarrow X}$ is a direct summand of a bundle ${V \times X \rightarrow X}$ for some ${G}$-representation ${V}$. (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…)