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.
1. Equivariant cohomology theories
In order to make this proof work, the authors use the language of equivariant cohomology theories (which is also how Atiyah and Segal did it). There is an equivariant cohomology theory , equivariant -theory, defined on finite complexes: of a finite -CW complex (i.e., a complex obtained by attaching “equivariant cells” of the form ) is the Grothendieck group of -equivariant vector bundles on . Then
because an equivariant vector bundle on a point is just a -representation. Moreover, : this follows because
as the -theory on a space with trivial -action is determined in terms of ordinary -theory.
Moreover, one has the following property of equivariant -theory: given a -space on which acts freely, so that the map
is a principal -bundle over , then there is a canonical isomorphism
This is something that one expects for an equivariant cohomology theory. With this in mind, we can interpret the map constructed above as the map
at least modulo the fact that is an infinite complex. So, the completion theorem is saying that is the completion of , -adically, or that the map (induced by ) is an isomorphism after -adic completion. Note that is a non-equivariant homotopy equivalence (though not an equivariant one).
The Atiyah-Segal theorem thus takes the following more general form.
Theorem 2 Let be a -space which is non-equivariantly contractible. Then .
It’s not immediately obvious what this means, or how it implies the previous version of the theorem; this will be spelled out below in the language of pro-groups. Note that it is not true on a non-equivariantly contractible -space, we should expect an equivariant cohomology theory to vanish.
Adams, Haeberly, Jacokwski, and May use category theoretic arguments to reduce this result to a computation with the Thom isomorphism. One problem in proving a statement like this is that is not a cohomology theory, because completion is not exact in general for non-finitely generated modules. Moreover, cohomology theories tend to be somewhat inconvenient in the first place for dealing with infinite things, whereas homology theories are much better behaved: most reasonable homology theories will commute with direct limits. The authors of this paper found a way to treat like a homology theory. This allows one to bring in a battery of inductive techniques to prove a statement such as Theorem 2 above.
2. Pro-object valued cohomology theories
The first step is to make sense of equivariant -theory on infinite complexes. We could do this as in the non-equivariant case, by considering “representable -theory,” i.e. by finding the equivariant space that represents on finite complexes and then defining the associated representable functor to be -theory in general. To deal with problems of infinite complexes, though, it is convenient to use pro-groups.
Definition 9 Let be a (reduced) cohomology functor defined on pointed finite complexes, taking values in , the category of finitely generated abelian groups. We define a functor
which sends a pointed space to the pro-group as ranges over all the finite subcomplexes of .
We can also say this in the language of -categories. We started with a functor between -categories
from the opposite to the -category of pointed finite spaces (i.e., finite cell complexes), taking values in the -category of finitely generated abelian groups. (The latter just happens to be an ordinary category.) Then, we extend to a functor of -categories
Now can be taken either in the -categorical sense or in the 1-categorical sense; both give the same result since we started with an ordinary category. However, really is taken in the -categorical sense, and it’s the -category . This is a reflection of the fact that
Anyway, we get a functor of -categories
and since the latter is an ordinary category, this is the same thing as a functor from the homotopy categoryinto pro-abelian groups.
Proposition 10 is a cohomology theory (with values in ) on all spaces.
Proof: To say that is a cohomology theory means that for any cofiber sequence in pointed spaces, we have an exact sequence
We can see this by noting that it is true (by assumption) if are finite, and that any map is a filtered colimit of maps of finite complexes.
For our purposes, it will be necessary to do this in a slightly more general context. Let be a commutative ring (always assumed noetherian). Consider a cohomology theory
from finite -equivariant spectra into finitely generated -modules. We can use the same technology to extend this to a cohomology theory
For instance, we can do this for equivariant -theory, where is the representation ring of . This is the first step: we have constructed a version of equivariant -theory on all equivariant spectra which is pro-module valued and which extends the old functor on finite spectra. This assumes a good -category of -equivariant spectra, a point which I’ll ignore: my understanding is that many homotopy theorists have built such an -category.
We are also going to interpret completion in this manner. This way, it will make sense to consider as a cohomology theory taking values in pro-modules.
Let be a noetherian ring, as above, and let be an ideal. We will define an exact functor (“completion”) from to itself. By the universal property of , to give an exact functor
is equivalent to giving an exact functor . That is, we need to interpret the ordinary completion of an -module as a pro-module, and then we can interpret the completion of a pro -module as one too. This functor is
where we regard the family as a pro-module.
Proposition 11 The functor is exact.
This is essentially the argument in commutative algebra that completion is an exact functor; note that it applies since we are only working with finitely generated modules. Now, though, we find that we get an exact functor of -adic completion .
This is the precisely the language we need for the Atiyah-Segal theorem. Namely, we have a cohomology theory
which we extend to a functor, as above, from all -spectra into the pro-category of modules over . By general nonsense, this functor sends filtered colimits of -spaces into filtered limits of pro-modules. This is a lot better than ordinary cohomology theories, which don’t do this: one has a Milnor exact sequence for a countable union, for instance. Next, we can compose this functor with -adic completion, constructed above, to get a cohomology theory:
This means that we get to think of the -adic completion of as a type of cohomology theory, and it’s even one which sends all filtered colimits into filtered limits.
Our goal is to show that this cohomology theory annihilates any -spectrum which is non-equivariantly contractible.
4. Isotropy separation
The construction of a “cohomology theory” (which really behaves like a homology theory to the opposite category) corresponding to a completion of -theory means that we can apply various inductive and categorical techniques to Theorem 2. Namely, given a homology theory on spectra, the collection of objects it annihilates is closed under colimits (in spectra) and translations. From this, we might hope to show that if our homology theory annihilates certain carefully chosen objects, it annihilates a much larger class. The following result will be useful in this regard.
I’m going to assume the existence of a stable, presentable -category of -equivariant spectra, which is generated under colimits of the translates of the suspension spectra of , for a subgroup. This -category is symmetric monoidal under the smash product, denoted . I am pretty sure that this has been done, but I’m not really familiar with the details.
Proposition 12 (Isotropy separation argument) The -category of -spectra which are non-equivariantly contractible is generated under colimits by translations of the following objects:
- , for a non-equivariantly contractible -spectrum and a proper subgroup.
- , for a chosen infinite-dimensional -representation such that .
Proof: Let be any -spectrum which is non-equivariantly contractible. Let be as above, and consider a cofiber sequence
where is the cofiber of . It thus suffices to show that both and (which are both nonequivariantly contractible), live in the sub -category of -spectra generated by the above objects.
- is built up as a colimit of translations of for a subgroup. It thus suffices to show that belongs to for any . When is the whole thing, this spectrum is . When is proper, then we already said that smashed with anything nonequivariantly contractible (like ) is in .
- has contractible -fixed points (since is an isomorphism on -fixed points), so it lives in the subcategory generated by translates of for proper. It follows that lives in the subcategory generated by for proper. Since is non-equivariantly contractible, it follows that these objects are all in .
The strategy will be to apply this result, together with an induction on the group and a calculation of the equivariant -theory of infinite spheres, to deduce the Atiyah-Segal theorem.
Corollary 13 Let be an abelian category, and let be a homology functor which commutes with filtered colimits. Suppose vanishes on and on whenever is a non-equivariantly contractible -spectrum, for . Then vanishes on all non-equivariantly contractible -spectra.
Proof: In fact, the subcategory of equivariant spectra annihilated by is a thick subcategory of -spectra (i.e., closed under finite colimits and translations), and is closed under direct sums. It is thus closed under all colimits. By the previous result, this category contains all non-equivariantly contractible spectra.
5. The proof of Theorem 2
We can now prove the Atiyah-Segal completion theorem in the form of Theorem 2 above. Let be the extension to -spectra of the pro-group valued cohomology theory . So actually behaves like a homology theory to the opposite category, rather than a cohomology theory. Moreover, so does the completed version : that is, the functor sends filtered colimits of -spectra to filtered limits of pro-modules over .
Our strategy will be to apply the above ideas, but also add in an induction on the group . We can assume, since the category of compact Lie groups is artinian, that the completion theorem is true for any . By the above isotropy separation argument, we are now reduced to checking two things:
- , for any non-equivariantly contractible spectrum and .
- , where is as above.
Let’s first verify the first claim. This follows from the isomorphism in pro -modules,
which is true for spaces, and which gets extended by general nonsense to arbitrary -spectra. This isomorphism in turn implies the isomorphsim
which is not totally formal. One has to use a theorem of Segal that the -adic topology (for the augmentation ideal) on is the same as the -adic topology. See this paper.
Consequently, the first claim is a consequence of the inductive hypothesis (that Theorem 2 holds for any ) and the above isomorphisms.
The second claim is the real computation (remarkably, the only real computation) needed in the whole argument. Namely, the strategy is to compute the non-completed as a pro-group; this is the formal limit as runs through a filtration of by finite-dimensional subrepresentations.
Now, we have that
by the equivariant version of the Thom isomorphism. So the pro-group corresponding to is really a pro-group
However, the maps in this inverse system are not identities: the Thom isomorphisms give different results. In fact, we have a commutative diagram
where the successive maps are given by multiplication by the relative Euler class (the alternating sum of the exterior powers of ). The key point is that these Euler classes are in the augmentation ideal.
The following lemma then shows that the functor annihilates , and completes the proof of the theorem 2.
Lemma 14 Let be a pro- module such that the image of is contained in for each . Then the formal -adic completion (in pro-modules) is zero.
Proof: In fact, by a cofinality argument, the formal -adic completion is the formal inverse system
and all the successive maps in this sequence are zero.
This proves Theorem 2.
6. Proof of the completion theorem
It remains to prove the Atiyah-Segal completion theorem. We know that the map
is a non-equivariant equivalence of spaces, so Theorem 2 applied to the cofiber gives us an isomorphism of pro-modules
Now, observe that the pro-module is unchanged under -completion; that’s because is a filtered limit of the -theory of -free finite subcomplexes. The -theory of a -free complex , being the -theory of , is annihilated by the augmentation ideal. Consequently
Consequently, Theorem 2 gives an isomorphism of pro-modules
Using the Milnor exact sequence, we can thus get the (ordinary, abelian-group valued) -theory of as desired.