(This is the first in a series of posts on the Hopkins-Miller theorem; this post is primarily motivational.)

Let {K} be the functor of complex {K}-theory. Then {K} is the first serious “extraordinary” cohomology theory one tends to encounter, and historically it has provided a useful language to express problems such as obtaining the right language for index theory.

One thing that you might want with a new exotic thing like {K}, though, is to be able to see better that maps {f: A \rightarrow B} that are not nullhomotopic are in fact not nullhomotopic. For instance, any map of spheres

\displaystyle f: S^r \rightarrow S^t

for {r \neq t} induces the zero map in ordinary homology, but such an {f} can be far from being nullhomotopic. So homology can’t say much (at least at this level) about the homotopy groups of spheres.

Unfortunately, {K}-theory doesn’t help much more either. If {f: S^r \rightarrow S^t} is any map between spheres for {r \neq t}, then {K^*(f): K^*(S^t) \rightarrow K^*(S^r)} is zero: this is a consequence of the fact that the stable homotopy groups of spheres are torsion, while the {K}-groups of spheres are torsion-free. Another way of saying this is that if you think of {K}-theory as a ring spectrum, then the Hurewicz map

\displaystyle \pi_* S \rightarrow \pi_* K

is zero (except on {\pi_0}).

However, it turns out that we can, with a little additional effort, manufacture a cohomology theory from {K} with a much better Hurewicz homomorphism. The observation is that {K}-theory, as a spectrum, admits a {\mathbb{Z}/2}-action.

1. KR-theory

On the level of cohomology theories, we have an involution on the group {K^0(X)} for each finite space {X}, coming from complex conjugation {\xi \mapsto \overline{\xi}} of vector bundles.

Proposition 1 The above action of {\mathbb{Z}/2} on {K^0(X)} for each space {X} comes from a {\mathbb{Z}/2}-action on the spectrum representing {K}-theory.

By the Brown representability theorem, we should think of a {\mathbb{Z}/2}-equivariant spectrum as a cohomology theory on {\mathbb{Z}/2}-equivariant spaces. This cohomology theory is “{KR}-theory,” or Real {K}-theory. This is an idea that goes back to Atiyah’s paper{K}-theory and reality,” and the capitalization of “Real” is intentional.

Definition 2 Real vector bundle on a {\mathbb{Z}/2}-equivariant space {X} is a complex vector bundle {E \rightarrow X}. Moreover, we require an antilinear action of {\mathbb{Z}/2} on {E}, such that the map {E \rightarrow X} is a morphism of equivariant spaces.

In other words, if we denote the involution on {X} by {\tau}, then we are required to have antilinear maps

\displaystyle E_{x} \simeq E_{\tau x}

which fit together into a map of topological spaces {E \rightarrow E}. In Atiyah’s paper, he shows that Real vector bundles can be used to define a cohomology theory on {\mathbb{Z}/2}-equivariant spaces.

So, for instance, if the involution on {X} is trivial, then (by a special case of Galois descent) a Real vector bundle on {X} is the same as a real vector bundle on {X}. If {X = T \times \mathbb{Z}/2} for {T} a non-equivariant space, then a Real vector bundle on {X} is the same as a complex vector bundle on {T}.

Now the whole point of this digression, and one of the major applications of the Hopkins-Miller theorem, is that given an equivariant spectrum, taking homotopy fixed points is a way to manufacture new and potentially interesting ones. For instance, we’ve seen that {K}-theory cannot do some of the things we might ask of an “extraordinary” cohomology theory; can its homotopy fixed point spectrum {K^{h \mathbb{Z}/2}} do better?

Example: The homotopy fixed point spectrum {K^{h \mathbb{Z}/2}} is {KO}-theory: that is, real {K}-theory. For a finite space {X}, {KO^0(X)} is the Grothendieck group of stable real vector bundles on {X}.

The idea here is that if making {K} into a {\mathbb{Z}/2}-equivariant spectrum meant inventing a new equivariant cohomology theory {KR}, then taking homotopy fixed points means restricting {KR} to spaces with trivial {\mathbb{Z}/2}-action. But on a space with trivial {\mathbb{Z}/2}-action, {KR} is literally the Grothendieck group of real vector bundles, as above.

Edit: I forgot to mention, somewhat carelessly, that the Hurewicz image of KO-theory is much better than the Hurewicz image of K-theory, because there is torsion in \pi_* KO. For instance, the Hopf map \eta, and its square, are nonzero in KO-theory. In general, in dimensions 1, 2 \mod 8, Adams showed that there is a nontrivial Hurewicz image in KO-theory.

Now {K}-theory and {KO}-theory are easier examples than the one relevant to the Hopkins-Miller theorem, because concrete geometric models (via vector bundles) are known for both of them. The spectra relevant for Hopkins-Miller are a fair bit more algebraic, though see the answer to this MathOverflow question for some interesting ideas.

2. Landweber’s exact functor theorem

Let {E} be a complex-oriented cohomology theory. As we saw in the (unfinished) series of posts on complex cobordism, the complex orientation of {E} allows one to define a theory of Chern classes in {E}-cohomology for complex vector bundles. However, the theory of Chern classes is not necessarily well-behaved with respect to tensor products: that is, for line bundles {\mathcal{L}, \mathcal{L}'}, we do not have

\displaystyle c_1( \mathcal{L} \otimes \mathcal{L}') = c_1( \mathcal{L}) + c_1(\mathcal{L}')

(as we do in ordinary cohomology), but rather we have

\displaystyle c_1( \mathcal{L} \otimes \mathcal{L}') = F( c_1( \mathcal{L}) , c_1(\mathcal{L}')),

where {F} is a formal group law over the ring {\pi_* E = E^{-*}(\ast)}.

The amazing result of Quillen was that, when one works with the cohomology theory {MU} of complex bordism (which, for relatively formal reasons, turns out to be the “universal” complex-oriented theory), then the formal group law we get is actually the universal one: that is, there is a canonical isomorphism

\displaystyle \pi_* MU \simeq L,

for {L} the Lazard ring.

My understanding is that a “good” explanation of Quillen’s theorem is not known to working homotopy theorists. It is proven using a fairly explicit computation in the Adams spectral sequence for {MU} and a look at the (far simpler) formal group law over {H\mathbb{Z} \wedge MU}. (Quillen had a later more elementary and less homotopy-theoretic proof.) Still, without such an explanation, we can still infer from the result that the connection between formal group laws and cohomology theories is pretty deep. And that maybe we might go in the other direction — maybe we might start with a formal group law, and get a cohomology theory.

So let’s say we have a ring {R} (appropriately graded) together with a formal group law {F(x,y) \in R[[x, y]]} over {R}.

Goal: Construct a complex-oriented cohomology theory {E^*} with {E^*(\ast) = R} and such that the formal group law of {E} is precisely {F(x, y)}.

How might we do this? The formal group law {F(x,y)} over {R} is classified by a map {\pi_* MU \rightarrow R}, by Quillen’s theorem. So we might try to manufacture our cohomology theory directly from {MU}: that is, we might set

\displaystyle E^*(X) = MU^*(X) \otimes_{\pi_* MU} R.

This is not generally going to work, because the tensor product is only right-exact, so {E^*(X)} as defined above cannot a priori necessarily be expected to satisfy the required exactness assumptions.

Theorem 3 (Landweber) Suppose {R} and {F(x,y)} satisfy the following property. For each prime {p}, let {v_i} denote the coefficient of {x^{p^i}} in the {p}-series {[p]_F(x) \in R[[x]]}. Suppose

\displaystyle p, v_1, v_2, \dots \in R

is a regular sequence in {R}, for each prime {p}. Then {E_*(X) = MU_*(X) \otimes_{\pi_* MU} E_*} defines a homology theory on spectra. Alternatively, {E^*(X) = MU^*(X) \otimes_{\pi_* MU} E_*} defines a cohomology theory on finite complexes, and its formal group law is precisely {F}.

By the homology version of the Brown representability theorem, we can use the above “exact functor theorem” to produce new spectra.

The regular sequence condition in Landweber’s theorem seems a little strange at first. Landweber originally proved it by studying the category of finitely presented comodules over the “Hopf algebroid” {(MU_*, MU_* MU)}: in other words, the {MU} analog of categories of modules over the Steenrod algebra. He showed that, while {MU_*} has a big category of modules, the category of modules admitting “{MU}-operations” is a lot smaller. Landweber proved:

Theorem 4 (Invariant prime ideal theorem) Let {\mathfrak{p} \subset MU_*} be a prime ideal which is invariant under the action of {MU}-cooperations. Then {\mathfrak{p}} is of the form

\displaystyle (p, v_1, v_2, \dots , v_n)

for some {p} and {n} (possibly infinite).

Using a version of primary decomposition for comodules, Landweber proved the exact functor theorem by reducing to showing that any finitely presented comodule admits a filtration whose subquotients were of the form {MU/(p, v_1, \dots, v_n)} for some {n}; this is how the regularity condition enters.

The modern take on Landweber’s results is rather different; it was later observed that the conditions of the exact functor theorem are in fact a flatness condition. In other words, the Landweber condition is precisely requiring that the map

\displaystyle \mathrm{Spec} R \rightarrow M_{FG}

to the stack {M_{FG}} of formal groups (notice: not formal group laws!) is a flat morphism, and the precise nature of the condition comes from the rather unusual picture of {M_{FG}} drawn for instance in these notes. This is a beautiful story, which is explained in Jacob Lurie’s course notes or Paul Goerss’s paper.

So, using the Landweber exact functor theorem, we can get a whole host of new complex-oriented cohomology theories. For instance, we can get {K}. (We can’t get ordinary cohomology, though.) Unfortunately, complex-oriented cohomology theories won’t let us see the stable homotopy groups {\pi_* S} any more: the problem is that the Hurewicz map

\displaystyle \pi_* S \rightarrow \pi_* E

for any complex-oriented cohomology theory {E}, factors through

\displaystyle \pi_* S \rightarrow \pi_* MU \rightarrow \pi_* E

and the first map is zero since {\pi_* MU} is torsion-free.

However, motivated by the discussion at the beginning of the post, perhaps we might get a more powerful theory if we took homotopy fixed points of a Landweber-exact theory under a suitable group action. The fact that we can do this, and that we can in fact do this in the setting of structured ring spectra, is the content of the Hopkins-Miller theorem.

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