(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…)