Chromotopy

Eilenberg--Mac Lane spaces in the chromatic picture

In the 60s, Ravenel and Wilson constructed and executed a program to compute the K(n)-homology of Eilenberg-Mac Lane spaces, so they could study a geometric conjecture of Conner and Floyd about certain homology classes in complex K-theory. Their argument is fairly complex, but, in the end, it follows a typical inductive pattern.

First, they compute $K(n)_* B\mathbb{Z}/p^j$ by realizing $B\mathbb{Z}/p^j$ as the fiber of the map $\mathbb{C}P^\infty \xrightarrow{p^j} \mathbb{C}P^\infty$. To say the action of this map on homology is well-understood is a gross understatement; this object is the most basic part of the link between topology and the theory of formal groups, and Morava K-theory is specifically constructed so that the algebra structure on $K(n)_* \mathbb{C}P^\infty$ is prescribed by very simple formulas. The exact method is to realize that fiber as a fibration itself, giving a fiber sequence

$$K(n)_* S^1 \to K(n)_* B\mathbb{Z}/p^j \to K(n)_* \mathbb{C}P^\infty,$$

whose Serre spectral sequence has a very simple $E_2$-page. (This is the same mechanism that results in the Gysin sequence, since every $S^1$-bundle over $\mathbb{C}P^\infty$ is oriented.)

With this base object in hand, they induct up. The object $K(\mathbb{Z}/p^j, q)$ is itself realized as a sort of bar construction from $K(\mathbb{Z}/p^j, q-1)$. Of course, homology itself is commonly computed using a bar construction, and the interaction of these two bar constructions yields a spectral sequence which takes as input some mangled version of $K(n)_* K(\mathbb{Z}/p^j, q-1)$ and converges to $K(n)_* K(\mathbb{Z}/p^j, q)$. There are certainly nontrivial differentials in this spectral sequence, and I haven’t figured out where they claim they come from yet, but they’re all special cases of some general construction, and miraculously everything works out. The key to structuring their answer is to recognize that two appropriately product-friendly ring spectra E and F produce an object called a Hopf ring by taking homology: $E_* F_* = \bigoplus_j E_* \Omega^{\infty-*} F$. If you have an isomorphism

$$E_*(\Omega^{\infty-n} F \times \Omega^{\infty-m} F) E_* \Omega^{\infty-n} F \otimes E_* \Omega^{\infty-m} F,$$

then the ring object maps on F induce a ring object structure on $E_* F_*$ in the category of $E_*$-comodules. Morava K-theory and Eilenberg-Mac Lane spectra fit into this picture, and Ravenel-Wilson basically says that $K(n)_* K(\mathbb{Z}/p^j, q)$ is given by all q-fnew strings of Hopf ring products of elements of $K(n)_* B\mathbb{Z}/p^j$, modulo the symmetry relation that

$$a_1 \circ \cdots \circ a_q = sgn(\sigma) a_{\sigma 1} \circ \cdots \circ a_{\sigma q}.$$

Finally, we take a limit in j to compute the homology of $K(\mathbb{Z}/p^\infty, q)$, which p-locally agrees with $K(\mathbb{Z}, q)$, which is our real goal.

While naming elements is necessary to complete computations in topology, to interpret the results it’s almost universally necessary to come up with global statements that don’t require naming elements; there’s too much information otherwise. To do this, we use the language of formal groups; write $X_E = \mathrm{spec}\,E_* X$ and $X^E = \mathrm{spf}\,E^* X$ for the schemes represented by the E-(co)homology of X. Taking the fiber of the map $\mathbb{C}P^\infty \xrightarrow{p^j} \mathbb{C}P^\infty$ corresponds to selecting the subscheme of $p^j$-torsion points of the formal group associated to Morava K-theory: $\mathbb{C}P^\infty_{K(n)}[p^j]$. The statement above about q-fnew strings and so forth corresponds to the statement that, as formal schemes,

$$K(\mathbb{Z}/p^j, q) = \Lambda^q \mathbb{C}P^\infty_{K(n)}[p^j],$$

and this identification is compatible with the Hopf ring product in the sense that, as a Hopf ring, the Morava K-theory of Eilenberg-Mac Lane spaces is the alternating algebra on $S^1_{K(n)}$. Not such an awful description!

Of course, we can easily make things awful by asking for an infinitessimal amount more. The formal group law attached to Morava K-theory, called the Honda formal group law, is nice because it is in some sense the simplest formal group law of height n over an $\mathbb{F}_p$-algebra – in a sense, it corresponds to selecting a point on the moduli stack of formal groups. The next move is to expand to an infinitessimal neighborhood of this point; this corresponds to studying formal groups over complete, local rings A, together with specified maps $\mathbb{F}_p \to A$ which pulls the formal group law over A back to the Honda formal group law. Such an object is called a deformation of the Honda formal group. Lubin and Tate studied this problem, also in the 60s, and found that the category of such deformations has a terminal object, called the Lubin-Tate formal group law, and Morava demonstrated that this picture can also be realized in homotopy theory: there is a spectrum $E_n$, called Morava E-theory, whose associated formal group is the Lubin-Tate formal group law. The structure of the moduli stack of p-typical formal groups itself says roughly that the open substack of formal groups of height at most n is represented by basically the same object – it’s a formal completion of Morava E-theory, and one corresponding spectrum is a finite free module over the other. So, really, we asking for quite a lot more information than Ravenel and Wilson were.

Still, Lubin and Tate give us a foothnew by describing the Lubin-Tate formal group law modulo the square of the maximal ideal of the complete, local ring over which it lives. Working modulo the square of the maximal ideal is enough to tell where the ‘new’ pieces of the ring $(E_n)_* K(\mathbb{Z}/p^j, q)$ will go, and after that all that’s left to answer is whether powers of these elements continue to exist unobstructed, or if new relations (especially nilpotence!) start to crop up. Both of these steps are, of course, harder than those two sentences make them sound – even with the guidance of Ravenel and Wilson’s work, the algebra involved is expansive enough to be nearly unmanageable. To illustrate: for a complex oriented spectrum E, the comodule $E_* \mathbb{C}P^\infty = E_* \{ \beta_i \}_{i=0}^\infty$ comes with the structure of a multiplicative module. Writing $\beta(s) = \sum_{i=0}^\infty \beta_i s^i$ in the ring $E_* \mathbb{C}P^\infty [[s]]$, we have the relation $\beta(s) \ast \beta(t) = \beta(s +_E t)$, where $+_E$ is the formal group addition coming from $E^* \mathbb{C}P^\infty$. Modulo p, we can compute “multiplication by p” in this multiplicative structure using $\beta(s)^{\ast p} = \beta([p]_E s)$ and the fact that $\beta(s)^{\ast p} = \sum_{i=0}^\infty \beta_i^{\ast p} s^i$. In the case that p is not equal to zero, the distribution in $\beta(s)^{\ast p}$ is not nearly so polite, and $\beta_i^{\ast p}$ is vastly harder to compute.

But, when p is taken to be zero, the computation becomes quite feasible, nevermind the rest of the maximal ideal in the local ring. The whole thing to turns into a game of trying to compute as much as possible, as delicately as possible, and seeing if it’s enough to say something meaningful. I’ve got the whole summer to chip away at it, so we’ll see how it goes. Should all this go through (though my suspicion is that it won’t, at least not in the next half-year), the next step is to think about Lurie’s machine, which allows the construction of cohomology theories corresponding to the next ‘delocalization’ step in the moduli stack of formal groups. I trust Matt and Neil on this, and they both suspect that, at least for Morava E-theory, the same fact will turn out to be true: they conjecture

$$K(\mathbb{Z}, *)_{E_n} = \Lambda^* \mathbb{C}P^\infty_{E_n}[p^\infty].$$

We’ll see!

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