Chromotopy

Complex orientations and the Steenrod algebra

One of the most incredible facts I learned last semester was a result of Ravenel and Wilson from the 70s (later revisited and formalized by Goerss in the very late 80s) about the space of maps between two complex orientable spectra.

A complex oriented cohomology theory is a ring-valued cohomology theory E, together with a distinguished element $x \in E^2 \mathbb{C}P^\infty$ so that $E^* \mathbb{C}P^\infty \cong (\pi_* E)[\![x]\!]$. What makes complex oriented cohomology theories interesting is that they provide a notion of Chern classes very similar to the Chern classes associated to ordinary cohomology, and characteristic classes are in general an extremely useful approach to the study of vector bundles. Given the class x above, a complex oriented cohomology theory E associates to a vector bundle V over a space X a sequence of E-Chern classes $c^E_n V \in E^{2n} X$ satisfying the following properties:

• Normalization: Let $\mathbb{L}$ denote the complex line bundle associated to $\mathbb{C}^\infty \to \mathbb{C}P^\infty$, and write $c^E(V) = \sum_i c_i^E(V)$. Then $c^E(\mathbb{L}) = 1 + x$.
• Naturality: Given a vector bundle V over a space X and a map $f: X \to Y$, we have the equality $c^E(f^* V) = f^* c^E(V)$, where the left $f^*$ denotes pullback of bundles and the right $f^*$ denotes the induced map on cohomology.
• Splitting: Given two vector bundles V and W over a space X, we have the equality $c^E(V \oplus W) = c^E(V) \cdot c^E(W)$, where the multiplication on the right hand side is that of power series.

A famous result on the geometry of vector bundles is that every vector bundle can be pulled back to a sum of line bundles along a map that is injective on cohomology, and so these axioms actually determine the Chern classes for all vector bundles.

Part of the reason K-theory gets so much interest is that the K-theory of a space is not just a group under direct sum of virtual bundles but actually a ring with multiplication given by tensor product of bundles. However, our axioms don’t say anything about the tensor product of bundles, but they do give us the tools we need to study it, since the tensor product of line bundles comes with a universal example: the map $f: \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ classifying the tensor product $\mathbb{L} \otimes \mathbb{L}$. Applying the axioms, we simply compute:

$$c_1^E(\mathbb{L} \otimes \mathbb{L}) = c_1^E(f^* \mathbb{L}) = f^* c_1^E(\mathbb{L}) = f^*(x) = F(u, v)\in (\pi_* E)[\![u, v]\!],$$

where F(u, v) is just defined to be the image of x under the map induced by f in E-cohomology. F itself is an interesting gadget, since it inherits several properties of the tensor product, namely:

• Symmetry: F(u, v) = F(v, u),
• Normality: F(u, 0) = 0, and
• Associativity: F(F(u, v), w) = F(u, F(v, w)).

F is called a formal group law over $\pi_* E$, and is a very interesting invariant associated to a complex oriented cohomology theory.

An even more interesting invariant, constructed in exactly the same way, is the more complicated formal group from which the formal group law descends: $\mathrm{spf}\,E^* \mathbb{C}P^\infty$. For the uninitiated, one approach to algebraic geometry is to think of rings R as the functors $\mathrm{Hom}(R, -)$ that they represent. This assignment from rings to functors is written $\mathrm{spec}$, and it adds in the limits that are missing in the original category of rings (by dualizing and adding colimits). This is very much akin to easing the study of manifnews by adding their missing colimits — for instance, the corner $* \leftarrow S^1 \hookrightarrow S^2$ has no pushout in the category of manifnews, but in some sense because those maps exist, the pushout should morally exist as an object of interest, and the same is true for rings. Things are made even more complicated because our cohomology rings $E^* X$ inherit a topology that comes from filtering the CW-complex X by its finite subcomplexes $X_\alpha$. We define

$$\mathrm{spf}\,E^* X = \mathrm{colim}_\alpha\,\mathrm{spec}\,E^* X_\alpha,$$

a sort of scheme-with-topology, called a formal scheme.

This object carries all the data of the formal group law associated to the complex orientation of E – and more, via this topology. What’s neat about this object is the incredible amount of data about E that it encodes. The result of Ravenel and Wilson that I wanted to get to is, given two complex oriented spectra E and F, the E-cohomology of F (under certain niceness conditions) can be described algebro-geometrically as

$$E_* F = \mathrm{Iso}(\mathrm{spf}\,E^*\mathbb{C}P^\infty, \mathrm{spf}\,F^*\mathbb{C}P^\infty).$$

This is a truly incredible result, but I personally started to feel its impact when I found it could be used to describe the structure of the Steenrod algebra. Recall that the (mod p) Steenrod algebra is defined as the cohomology operations of $H\mathbb{F}_p$, or equivalently the cohomology ring $H\mathbb{F}_p^* H\mathbb{F}_p$. In the 50s, Milnor famously noticed that the linear dual of the Steenrod algebra is a lot easier to handle than the Steenrod algebra itself, and since we’re working with coefficients in a field, there’s little work to be done to see that this coincides with the homology ring $(H\mathbb{F}_p)_* H\mathbb{F}_p$. Integral cohomology, and hence cohomology with coefficients in $\mathbb{F}_p$, is complex orientable, and in fact it satisfies the niceness conditions needed to employ the algebraic geometry gestured at above — we just need to compute the formal group $\mathrm{spf}\,H\mathbb{F}_p^* \mathbb{C}P^\infty$. It turns out that this formal group is well known, it goes by the name “the additive formal group,” denoted $\widehat{\mathbb{G}_a}$, and it comes with the formal group law $F_{\mathbb{G}_a}(x, y) = x + y$. This formal group comes with the action

$$\widehat{\mathbb{G}_a}(R) = \{r \in R \mid \exists n , r^n = 0\},$$

with addition inherited from R.

The ring representing $\widehat{\mathbb{G}_a}$ is $\mathbb{Z}[\![x]\!]$, and an automorphism of $\widehat{\mathbb{G}_a}$ corresponds to a power series F such that $F(x+y) = F(x) + F(y)$. Over $\mathbb{F}_p$, it’s easy to check that F must be of the form

$$F(x) = \sum_{i=0}^\infty c_{p^i} x^{p^i},$$

and the space of such power series is represented by the ring $\mathcal{A}_* = \mathbb{F}_p[\xi_1, \xi_2, \ldots]$. The Hopf algebra structure map $\Delta: \mathcal{A}_* \to \mathcal{A}_* \otimes \mathcal{A}_*$ arises from the composition of these automorphisms, which we calculate as follows:

$$\begin{array}{rl} f(x) = \sum_i a_i x^{2^i}, & g(x) = \sum_j b_j x^{2^j}, \\ f(g(x)) & = f \left( \sum_i b_i x^{2^i} \right) \\ & = \sum_j a_j \left( \sum_i b_i x^{2^i} \right)^{2^j} \\ & = \sum_j a_j \left( \sum_i b_i^{2^j} x^{2^{i+j}} \right) \\ & = \sum_n \left( \sum_{j=0}^n a_j b_{n-j}^{2^j} \right) x^{2^n}, \end{array}$$

and hence

$$\Delta(\xi_i) = \sum_{j=0}^n \xi_j \otimes \xi_{n-j}^{2^j},$$

which is the usual cryptic relation among the duals of the Steenrod squares, laid bare as a trivial consequence of composing certain kinds of power series. Neat!

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