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

(where means Thom space) and define maps

coming from the map classifying . The homotopy colimit of the sequence

is a spectrum and, for the same reasons as , is in fact a commutative ring spectrum (even an ring spectrum).

One reason to care about is the following result:

Theorem 1 (Thom)The homotopy groups (a graded ring) is the cobordism ring of unoriented manifolds.

In other words, to describe , we can also use geometry: an element of is a compact -manifold modulo the relation of *cobordism*: two -manifolds are cobordant if there is an -manifold-with-boundary such that . From a geometric point of view, it is thus interesting to determine . 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, is a fairly simple spectrum.

In fact, we have:

Theorem 2 (Thom)is a wedge of Eilenberg-MacLane spectra (and its shifts) and is a polynomial ring on variables for all such that is not a power of .

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