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…)