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 the category of vector bundles on . This is an additive category, where all exact sequences split; one can thus define the Grothendieck group of this category. This is obtained by taking the free abelian group on all symbols for a vector bundle on , and quotienting by the relations whenever there is an isomorphism . In other words, K-theory is the “group completion” of the monoid of isomorphism classes of vector bundles on . Moreover, becomes a ring because one has a tensor product operation on the category of vector bundles, which clearly commutes with direct sums.
The point of Segal’s paper is to generalize this to the equivariant setting. Namely, let be a compact Lie group. A -space will be a space with a continuous -action . Usually, we will want to be compact. So let’s assume this for the present post.
Definition 1 A -vector bundle on is a vector bundle where has the structure of a -space such that is equivariant. Moreover, we require that the maps on fibers for be morphisms of vector spaces (i.e. linear).
Clearly, there is a category of -vector bundles on and -equivariant maps; an equivariant map between -vector bundles is simply a map of vector bundles which is -equivariant.
The basic example is when is acted upon trivially by . In that case, we can think of a -vector bundle as a continuously varying family of -representations, as all the fibers will be equipped with an action of . An example of how we can get such equivariant vector bundles is to start with an ordinary vector bundle , and take the tensor power . This is canonically endowed with the structure of an -vector bundle if is given the trivial -action.
Another useful example is when for a closed subgroup . In this case, we claim:
Proposition 2 There is an equivalence of categories between -vector bundles on and finite-dimensional -representations.
Proof: To see the idea, note that if is a -vector bundle, then the fiber over the distinguished element (the image of the identity in ) is acted on by , and becomes an -representation. So there is a functor from equivariant (-) vector bundles on to -representations given by sending to . One has to check that this is an equivalence of categories, but this is because one can easily check that any vector bundle over is given of the form (where the product over denotes a natural identification). I won’t spell out all the details.
The category of equivariant vector bundles on a -space naturally forms an additive category. As before, we can define:
Definition 3 The equivariant K-group of is the Grothendieck group of equivariant vector bundles on .
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 -vector bundles on given by (i.e. such that the fibers are all exact) necessarily splits.
Proof: To see this, we have to show that is a split injection. One way to see this is to choose a complement to ; this will then map isomorphically to and we will get an isomorphism .
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 . Then we could take to be the orthogonal complement to in . 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 ), and averaging under the -action to get an equivariant section, which will be an equivariant hermitian metric.
The group is a contravariant functor in under equivariant maps of spaces , because for such a map there is an additive functor from equivariant bundles on to equivariant bundles on .
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 be an equivariant vector bundle over the -space ; then we let be the vector space of all sections , and we let be the subspace of equivariant sections. Note that has a -action, while is just a vector space (or it has the trivial action).
Now, is actually a topological vector space, even a Banach space. Given a section , we can define, using the normalized Haar measure on ,
this map is just the projection onto the subspace of fixed points of the -action. Here we genuinely use the compactness of to use the existence of an invariant measure on 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 be equivariantly homotopic equivariant maps of -spaces. Then as maps .
Proof: As usual, we can reduce to the case of the two imbeddings . In this case, we essentially have to show the following. If is a vector bundle on the product , then and are isomorphic. In fact, we will show that the isomorphism class of (considered as an equivariant bundle on ) is locally constant in , which is enough.
To do this, we just need to show that is isomorphic to for . To get this, we can consider the vector bundle on given by where is the projection and is the inclusion. In other words, is rigged such that for all .
Now and are isomorphic when restricted to . So there is an equivariant section of which is an isomorphism on each fiber. The claim is that we can extend it to an equivariant section in some neighborhood of which will have to be an isomorphism in some neighborhood of the form for ; this is what we want. But, well, using standard techniques, we can extend it to a section in some -invariant neighborhood (e.g. something of the form for ), and then by averaging we can make the section equivariant.
Anyway, with these preliminaries established, we can now say that we have defined a functor from compact -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, is actually a commutative ring.
We can interpret some of the earlier examples of -vector bundles in terms of K-theory. For instance, we have seen that there is an equivalence of categories between equivariant vector bundles on and -representations; thus, is the representation ring . When , then this is saying that is the representation ring ; of course, an equivariant vector bundle on is just a -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
and more generally, whenever is a morphism of compact Lie groups, there is a map
More interesting is the relation between the equivariant K-theory of a space and the K-theory of the quotient. Namely, let be a -space, and consider as an ordinary space. There is a map
which enables one to pull back a vector bundle on to an ordinary vector bundle to . In fact, the functor factors through the category of equivariant bundles on ; that is, there is a natural map
because the pull-back of a bundle on to automatically acquires an equivariant structure by general nonsense. This map is in general very far from being an isomorphism, for instance if ! However, we have:
Proposition 6 The map is an isomorphism if acts freely on .
Proof: In fact, we need to define an inverse map sending equivariant vector bundles to vector bundles over . Here we can send to . Since all the identifications made in are made between distinct fibers (by freeness of the action), this defines a vector bundle on , and this is the inverse to the above construction. I’m skipping some details.
In general, the above proposition suggests that the failure of to be an isomorphism might be rectified if instead of taking quotients , one took “homotopy quotients” where is the total space for the universal principal -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.