Before moving on, I’d like to work out the analog for real-oriented cohomology theories (which is tangential to the rest of the story, though). This is considerably less interesting, but perhaps it’s a toy example of the ideas explained in the last few posts without the full-blown machinery of the Adams spectral sequence and so forth.

So, let’s state what the analogous ideas are in the real-oriented context:

  1. real-oriented ring spectrum {E} is a ring spectrum together with a functorial, multiplicative choice of Thom classes for real vector bundles; equivalently, there is a morphism of ring spectra

    \displaystyle MO \rightarrow E.

    That is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

  2. If {E} is real-oriented, then {\pi_* E} is a {\mathbb{Z}/2}-vector space, and all the usual computations of {H_*( BO; \mathbb{Z}/2)} and so forth work just fine for {E}, and there is a theory of Stiefel-Whitney classes in {E}-cohomology.
  3. Given a real-oriented spectrum {E}, we thus have {E^*(\mathbb{RP}^\infty) = \pi_* E [[t]]} where {t} is the Stiefel-Whitney class of the tautological bundle. Similiarly, {E^*(\mathbb{RP}^\infty \times \mathbb{RP}^\infty) = \pi_* E [[t_1, t_2]]}.

Since {\mathbb{RP}^\infty} is the classifying space for real line bundles, there is a monoidal product

\displaystyle \mathbb{RP}^\infty \times \mathbb{RP}^\infty \rightarrow \mathbb{RP}^\infty,

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over {\pi_* E}. This formal group law {f(\cdot, \cdot) \in \pi_* E [[x, y]]} has the property that if {\mathcal{L}_1, \mathcal{L}_2} are two line bundles over a finite-dimensional space {X}, then in {E^*(X)},

\displaystyle w_1( \mathcal{L}_1 \otimes \mathcal{L}_2) = f( w_1(\mathcal{L}_1), w_1(\mathcal{L}_2)).

So far everything has been analogous to the complex-oriented case, but there is an extra feature here which changes the picture drastically. Namely, when one works with realline bundles {\mathcal{L}}, we have that {\mathcal{L}^{\otimes 2}} is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law {f} must satisfy

\displaystyle f(x, x) = 0.

This is not satisfied by, say, the multiplicative formal group law. (And in fact, {KO}-theory—the natural candidate for this—is not real-oriented, only spin-oriented.)

With this in mind, the analog of Quillen’s theorem for {MU} becomes:

Theorem 1 (Quillen) The formal group law for {MO} is the universal formal group law over a satisfying {f(x, x) = 0}. (more…)

The next goal in this series of posts is to understand fully the computation of the complex cobordism ring {\pi_* MU}, which is due to Milnor. To start with, though, let’s consider a simpler example: the spectrum {MO}. The spectrum {MO} is obtained in the same way that {MU} is, but with the Thom spaces of the universal {n}-dimensional real bundles over {BO(n)}. In other words, if {\xi_n \rightarrow BO(n)} is the universal {n}-dimensional bundle, we define

\displaystyle MO(n) = \Sigma^{-n} T(\xi_n),

(where T means Thom space) and define maps

\displaystyle MO(n) \rightarrow MO(n+1)

coming from the map {BO(n) \rightarrow BO(n+1)} classifying {\xi_n \oplus \mathbb{R}}. The homotopy colimit of the sequence

\displaystyle MO(0) \rightarrow MO(1) \rightarrow \dots

is a spectrum and, for the same reasons as {MU}, is in fact a commutative ring spectrum (even an {E_\infty} ring spectrum).

One reason to care about {MO} is the following result:

Theorem 1 (Thom) The homotopy groups {\pi_* MO} (a graded ring) is the cobordism ring of unoriented manifolds.

In other words, to describe {\pi_* MO}, we can also use geometry: an element of {\pi_n MO} is a compact {n}-manifold {M} modulo the relation of cobordism: two {n}-manifolds {M, M'} are cobordant if there is an {(n+1)}-manifold-with-boundary {W} such that {\partial W = M \sqcup M'}. From a geometric point of view, it is thus interesting to determine {\pi_* MO}. It turns out that we can do so using homotopy-theoretic, algebraic methods; this was what Thom showed.

How might we compute the homotopy groups? In general, the homotopy groups of any space are extremely difficult to compute, but in this case we can get a complete answer for two reasons: first, it’s stable homotopy groups we’re after, not just homotopy groups; second, much more importantly, {MO} is a fairly simple spectrum.

In fact, we have:

Theorem 2 (Thom) {MO} is a wedge of Eilenberg-MacLane spectra {H\mathbb{Z}/2} (and its shifts) and {\pi_* MO = \mathbb{Z}/2[x_2, x_4, \dots]} is a polynomial ring on variables {x_i} for all {i} such that {i+1} is not a power of {2}.

This is the theorem I’d like to discuss today. It is definitely a much easier computation than that of {\pi_* MU}, but it will be a toy example. (more…)