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.

Classically, topological K-theory starts by assigning to any finite CW complex ${X}$ the category of vector bundles on ${X}$. This is an additive category, where all exact sequences split; one can thus define the Grothendieck group ${K(X)}$ of this category. This is obtained by taking the free abelian group on all symbols ${[E]}$ for ${E}$ a vector bundle on ${X}$, and quotienting by the relations ${[E] = [E'] + [E'']}$ whenever there is an isomorphism ${E \simeq E' \oplus E''}$. In other words, K-theory is the “group completion” of the monoid of isomorphism classes of vector bundles on ${X}$. Moreover, ${K(X)}$ becomes a ring because one has a tensor product operation on the category of vector bundles, which clearly commutes with direct sums.

1. Definitions

The point of Segal’s paper is to generalize this to the equivariant setting. Namely, let ${G}$ be a compact Lie group. A ${G}$-space will be a space ${X}$ with a continuous ${G}$-action ${X \times G \rightarrow X}$. Usually, we will want ${X}$ to be compact. So let’s assume this for the present post.

Definition 1 A ${G}$-vector bundle on ${X}$ is a vector bundle ${p: E \rightarrow X}$ where ${E}$ has the structure of a ${G}$-space such that ${p}$ is equivariant. Moreover, we require that the maps on fibers ${E_x \rightarrow E_{gx}}$ for ${x \in X, g \in G}$ be morphisms of vector spaces (i.e. linear).

Clearly, there is a category of ${G}$-vector bundles on ${X}$ and ${G}$-equivariant maps; an equivariant map between ${G}$-vector bundles is simply a map of vector bundles which is ${G}$-equivariant.

The basic example is when ${X}$ is acted upon trivially by ${G}$. In that case, we can think of a ${G}$-vector bundle as a continuously varying family of ${G}$-representations, as all the fibers ${E_x, x \in X}$ will be equipped with an action of ${G}$. An example of how we can get such equivariant vector bundles is to start with an ordinary vector bundle ${F \rightarrow X}$, and take the tensor power ${F^{\otimes k} \rightarrow X}$. This is canonically endowed with the structure of an ${S_k}$-vector bundle if ${X}$ is given the trivial ${S_k}$-action.

Another useful example is when ${X = G/H}$ for a closed subgroup ${H \subset G}$. In this case, we claim:

Proposition 2 There is an equivalence of categories between ${G}$-vector bundles on ${X = G/H}$ and finite-dimensional ${H}$-representations.

Proof: To see the idea, note that if ${E \rightarrow G/H}$ is a ${G}$-vector bundle, then the fiber over the distinguished element ${\ast \in G/H}$ (the image of the identity in ${G}$) is acted on by ${H}$, and becomes an ${H}$-representation. So there is a functor from equivariant (${G}$-) vector bundles on ${G/H}$ to ${H}$-representations given by sending ${E \rightarrow G/H}$ to ${E_{\ast}}$. One has to check that this is an equivalence of categories, but this is because one can easily check that any vector bundle over ${G/H}$ is given of the form ${E_{\ast} \times_H G}$ (where the product over ${H}$ denotes a natural identification). I won’t spell out all the details. $\Box$

The category of equivariant vector bundles on a ${G}$-space ${X}$ naturally forms an additive category. As before, we can define:

Definition 3 The equivariant K-group ${K_G(X)}$ of ${X}$ is the Grothendieck group of equivariant vector bundles on ${X}$.

One might object that the Grothendieck group is sometimes defined in terms of short exact sequences, and sometimes in terms of split exact sequences. We should probably check that an exact sequence always splits.

Proposition 4 An exact sequence of ${G}$-vector bundles on ${X}$ given by ${0 \rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0}$ (i.e. such that the fibers are all exact) necessarily splits.

Proof: To see this, we have to show that ${E' \rightarrow E }$ is a split injection. One way to see this is to choose a complement ${F'' \subset E}$ to ${E'}$; this will then map isomorphically to ${E''}$ and we will get an isomorphism ${E \simeq E' \oplus F'' \simeq E' \oplus E''}$.

In the nonequivariant case, one would do this by choosing a continuously varying family of hermitian metrics on the fibers: that is, a hermitian metric on ${E}$. Then we could take ${F''}$ to be the orthogonal complement to ${E'}$ in ${E}$. Naturally this will not work here, because we won’t necessarily get an equivariant subbundle. But we can do this if we choose an equivariant hermitian metric. This we can do by taking any hermitian metric (which is a section of ${E \otimes \overline{E}}$), and averaging under the ${G}$-action to get an equivariant section, which will be an equivariant hermitian metric. $\Box$

The group ${K_G(X)}$ is a contravariant functor in ${X}$ under equivariant maps of spaces ${f: Y \rightarrow X}$, because for such a map there is an additive functor ${f^*}$ from equivariant bundles on ${Y}$ to equivariant bundles on ${X}$.

2. The averaging trick

So far, nothing here is non-standard from ordinary K-theory: the main clever trick that cropped up was the idea of averaging a nonequivariant section to get an equivariant one. We had to check (and I technically didn’t, but it’s pretty easy) that averaging a hermitian metric would give a hermitian one. Maybe this is worth spelling out more. Let ${E}$ be an equivariant vector bundle over the ${G}$-space ${X}$; then we let ${\Gamma(E)}$ be the vector space of all sections ${X \rightarrow E}$, and we let ${\Gamma^G(E)}$ be the subspace of equivariant sections. Note that ${\Gamma(E)}$ has a ${G}$-action, while ${\Gamma^G(E)}$ is just a vector space (or it has the trivial action).

Now, ${\Gamma(E)}$ is actually a topological vector space, even a Banach space. Given a section ${s \in \Gamma(E)}$, we can define, using the normalized Haar measure on ${G}$,

$\displaystyle s^G = \int_G s \circ g dg \in \Gamma^G(E);$

this map ${\Gamma(E) \rightarrow \Gamma^G(E)}$ is just the projection onto the subspace ${\Gamma^G(E)}$ of fixed points of the ${G}$-action. Here we genuinely use the compactness of ${G}$ to use the existence of an invariant measure on ${G}$ with total mass one.

Let’s use this argument to prove a fact which is standard in ordinary K-theory, but requires a little extra work in the equivariant case. Namely, we can prove the homotopy invariance of equivariant K-theory:

Proposition 5 Let ${f, g: X \rightarrow Y}$ be equivariantly homotopic equivariant maps of ${G}$-spaces. Then ${f^* = g^* }$ as maps ${K_G(Y) \rightarrow K_G(X)}$.

Proof: As usual, we can reduce to the case of the two imbeddings ${X \rightrightarrows X \times [0, 1]}$. In this case, we essentially have to show the following. If ${E \rightarrow X \times [0, 1]}$ is a vector bundle on the product ${X \times [0, 1]}$, then ${E|_{X \times \left\{1\right\}}}$ and ${E|_{X \times \left\{0\right\}}}$ are isomorphic. In fact, we will show that the isomorphism class of ${E|_{X \times \left\{t\right\}}}$ (considered as an equivariant bundle on ${X}$) is locally constant in ${t}$, which is enough.

To do this, we just need to show that ${E|_{X \times \left\{t\right\}}}$ is isomorphic to ${E|_{X \times \left\{0\right\}}}$ for ${t \simeq 0}$. To get this, we can consider the vector bundle ${F}$ on ${X \times [0, 1]}$ given by ${F= p^* i_0^* E}$ where ${p}$ is the projection ${X \times [0, 1] \rightarrow X}$ and ${i_0: X \hookrightarrow X \times [0, 1]}$ is the inclusion. In other words, ${F}$ is rigged such that ${F|_{X \times \left\{t\right\}} \simeq E|_{X \times \left\{0\right\}}}$ for all ${t}$.

Now ${F}$ and ${E}$ are isomorphic when restricted to ${X \times \left\{0\right\}}$. So there is an equivariant section of ${\hom(F, E)|_{X \times \left\{0\right\}}}$ which is an isomorphism on each fiber. The claim is that we can extend it to an equivariant section in some neighborhood of ${X \times \left\{0\right\}}$ which will have to be an isomorphism in some neighborhood of the form ${X \times [0, \epsilon)}$ for ${\epsilon > 0}$; this is what we want. But, well, using standard techniques, we can extend it to a section in some ${G}$-invariant neighborhood (e.g. something of the form ${X \times [0, \epsilon')}$ for ${\epsilon > 0}$), and then by averaging we can make the section equivariant. $\Box$

3. K-theory

Anyway, with these preliminaries established, we can now say that we have defined a functor ${K_G(\cdot)}$ from compact ${G}$-spaces to abelian groups, which descends to the homotopy category of equivariant spaces. It is easy to see from the definition as a Grothendieck group that, as in the nonequivariant case, ${K_G(X)}$ is actually a commutative ring.

We can interpret some of the earlier examples of ${G}$-vector bundles in terms of K-theory. For instance, we have seen that there is an equivalence of categories between equivariant vector bundles on ${G/H}$ and ${H}$-representations; thus, ${K_G(G/H)}$ is the representation ring ${R(H)}$. When ${H = G}$, then this is saying that ${K_G(\ast)}$ is the representation ring ${R(G)}$; of course, an equivariant vector bundle on ${\ast}$ is just a ${G}$-representation.

In general, there are many interesting relations between equivariant K-theory and ordinary K-theory. To start with, as there is a forgetful functor from equivariant vector bundles to vector bundles, there is a natural ring-homomorphism

$\displaystyle K_G(X) \rightarrow K(X)$

and more generally, whenever ${H \rightarrow G}$ is a morphism of compact Lie groups, there is a map

$\displaystyle K_G(X) \rightarrow K_H(X).$

More interesting is the relation between the equivariant K-theory of a space and the K-theory of the quotient. Namely, let ${X}$ be a ${G}$-space, and consider ${X/G}$ as an ordinary space. There is a map

$\displaystyle X \rightarrow X/G$

which enables one to pull back a vector bundle on ${X/G}$ to an ordinary vector bundle to ${X}$. In fact, the functor factors through the category of equivariant bundles on ${X}$; that is, there is a natural map

$\displaystyle K(X/G) \rightarrow K_G(X)$

because the pull-back of a bundle on ${X/G}$ to ${X}$ automatically acquires an equivariant structure by general nonsense. This map is in general very far from being an isomorphism, for instance if ${X = \ast}$! However, we have:

Proposition 6 The map ${K(X/G) \rightarrow K_G(X)}$ is an isomorphism if ${G}$ acts freely on ${X}$.

Proof: In fact, we need to define an inverse map sending equivariant vector bundles ${P \rightarrow X}$ to vector bundles over ${X/G}$. Here we can send ${P \rightarrow X}$ to ${P/G \rightarrow X/G}$. Since all the identifications made in ${P/G}$ are made between distinct fibers (by freeness of the action), this defines a vector bundle on ${X/G}$, and this is the inverse to the above construction. I’m skipping some details. $\Box$

In general, the above proposition suggests that the failure of ${K(X) \rightarrow K_G(X)}$ to be an isomorphism might be rectified if instead of taking quotients ${X/G}$, one took “homotopy quotients” ${X \times EG/G}$ where ${EG }$ is the total space for the universal principal ${G}$-bundle. This is in fact (essentially) the case, and is the content of the Atiyah-Segal completion theorem. One has to be careful, because taking homotopy quotients does not preserve compactness.

Anyway, so far I’ve only covered the material in the first couple of pages; there’s much more to say on this topic in the future.