(This is the first in a series of posts on the Hopkins-Miller theorem; this post is primarily motivational.)

Let ${K}$ be the functor of complex ${K}$-theory. Then ${K}$ is the first serious “extraordinary” cohomology theory one tends to encounter, and historically it has provided a useful language to express problems such as obtaining the right language for index theory.

One thing that you might want with a new exotic thing like ${K}$, though, is to be able to see better that maps ${f: A \rightarrow B}$ that are not nullhomotopic are in fact not nullhomotopic. For instance, any map of spheres

$\displaystyle f: S^r \rightarrow S^t$

for ${r \neq t}$ induces the zero map in ordinary homology, but such an ${f}$ can be far from being nullhomotopic. So homology can’t say much (at least at this level) about the homotopy groups of spheres.

Unfortunately, ${K}$-theory doesn’t help much more either. If ${f: S^r \rightarrow S^t}$ is any map between spheres for ${r \neq t}$, then ${K^*(f): K^*(S^t) \rightarrow K^*(S^r)}$ is zero: this is a consequence of the fact that the stable homotopy groups of spheres are torsion, while the ${K}$-groups of spheres are torsion-free. Another way of saying this is that if you think of ${K}$-theory as a ring spectrum, then the Hurewicz map

$\displaystyle \pi_* S \rightarrow \pi_* K$

is zero (except on ${\pi_0}$).

However, it turns out that we can, with a little additional effort, manufacture a cohomology theory from ${K}$ with a much better Hurewicz homomorphism. The observation is that ${K}$-theory, as a spectrum, admits a ${\mathbb{Z}/2}$-action. (more…)