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.

 K-theory with compact supports

The “index” of an elliptic operator on a compact manifold {X} is going to turn out to be a homomorphism from {K(TX) \rightarrow \mathbb{Z}}, where {K(TX)} is the compactly supported K-theory of the tangent bundle {TX}. The idea is that an elliptic operator {D} on {X} between vector bundles {E, F} on {X} is going to have a symbol, a linearization

\displaystyle \sigma(D): \pi^* E \rightarrow \pi^* F

where {\pi: TX \rightarrow X} is the tangent bundle (technically, the cotangent bundle). So the idea is that {\sigma(D)} is a complex of complex vector bundles on {TX}, and we want it to represent something in the {K}-theory of {TX}.

So, anyway, we need a good way of thinking of {K}-theory with compact supports. Given a locally compact space {X}, we define

\displaystyle K(X) = \widetilde{K}(X^+)

where {X^+} is its one-point compactification. Then, a consequence of the previous description of relative K-theory is that one can describe {K(X)} by taking all complexes of vector bundles on {X} which are exact outside a compact set, and quotienting by the relation of homotopy and stable equivalence. In this way, {\sigma(D)}—when {D} is elliptic—is going to give an element of {K(TX)}.

The Thom isomorphism

Next, I would like to describe the Thom isomorphism in K-theory. This is the next main tool in the Atiyah-Singer papers. Given an oriented {n}-dimensional real vector bundle {V} over a topological space {X}, the classical Thom isomorphism states that the reduced cohomology of the Thom space {B(V)/S(V)} (where {B(V)} is the unit ball bundle and {S(V)} is the sphere bundle) is the cohomology of {X} shifted by {n}. The isomorphism

\displaystyle H^*(X) \simeq \widetilde{H}^*( B(V)/S(V))

is given by multiplication by the Euler class.

The Thom isomorphism in K-theory is going to be similar. To start with, we’ll need Bott periodicity. Let {X} be a compact space, and {V} a complex vector bundle over {X} with fibers {V_x}. Then there is a canonical element of {K(V)}, denoted {\lambda_V}, which is the relative Koszul complex of {V}. That is, the fiber over a vector {v \in V_x \subset V} is the complex

\displaystyle 0 \rightarrow \mathbb{C} \stackrel{v}{\rightarrow} V_x \stackrel{\wedge v}{\rightarrow} \lambda^2 V_x \rightarrow \dots.

This is exact outside the zero-section of {V}. Note that {K(V)} has been defined as the ordinary K-theory of the Thom space of {V}.

Now, there is a map

\displaystyle K(X) \rightarrow K(V)

which is given by multiplication by {\lambda_V}. Namely, if we have an element of {K(X)}, represented by a complex {E_\bullet} of vector bundles on {X}, then we use the projection {\pi: V \rightarrow X} to pull back to get a complex {\pi^* E_\bullet} of vector bundles on {V}. This is not necessarily exact outside a compact set, though, because {\pi^{-1}} of a compact subset of {X} is not compact. However, we take the tensor product {\lambda_V \otimes \pi^* E_\bullet}, which is (by e.g. the Kunneth formula) exact outside the zero section of {V}, and which is therefore a representative element of {K(V)}.

Theorem 2 (Bott periodicity) The map of multiplication by {\lambda_V} is an isomorphism.

This is a bit different from the usual statement of Bott periodicity, but it reduces to it. Namely, let’s suppose the vector bundle is trivial. (One can reduce to this locally by proving the result locally on {X}.) Then, the construction {\lambda_V} has the property that {\lambda_{V \oplus W } = \lambda_V \lambda_W}, so one reduces to the case where {V } is the trivial bundle {\mathbb{C}}. Then this is more or less the usual statement of Bott periodicity, once one unravels what the complex {\lambda_{\mathbb{C}}} is; it’s essentially the Hopf bundle minus one (up to a sign).

A key ingredient for the proof of the Atiyah-Singer index theorem is that this formulation of Bott periodicity (or the Thom isomorphism) also holds equivariantly. Namely, let’s now suppose that {V} is a {G}-vector bundle on {X}. Then {\lambda_V} represents an element of {K_G(V)}, as before, and we get a map

\displaystyle K_G(X) \rightarrow K_G(V)

by multiplication by {\lambda_V}.

Theorem 3 (Equivariant Bott periodicity) The map of multiplication by {\lambda_V} is an isomorphism {K_G(X) \rightarrow K_G(V)}.

This seems to be significantly more difficult to prove than in the non-equivariant case. When {V = \mathbb{C}} is a trivial bundle, it can be proved as before, by an analysis of clutching functions (maybe I should talk about Bott periodicity at some point). More generally, one can apply this argument whenever {V} is a sum of one-dimensional equivariant vector bundles. However, in the equivariant case, one cannot do this in general, because of the existence of irreducible representations of dimension {> 1}.