(This is the first in a series of posts on the Hopkins-Miller theorem; this post is primarily motivational.)
Let be the functor of complex -theory. Then 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 , though, is to be able to see better that maps that are not nullhomotopic are in fact not nullhomotopic. For instance, any map of spheres
for induces the zero map in ordinary homology, but such an can be far from being nullhomotopic. So homology can’t say much (at least at this level) about the homotopy groups of spheres.
Unfortunately, -theory doesn’t help much more either. If is any map between spheres for , then is zero: this is a consequence of the fact that the stable homotopy groups of spheres are torsion, while the -groups of spheres are torsion-free. Another way of saying this is that if you think of -theory as a ring spectrum, then the Hurewicz map
is zero (except on ).
However, it turns out that we can, with a little additional effort, manufacture a cohomology theory from with a much better Hurewicz homomorphism. The observation is that -theory, as a spectrum, admits a -action. (more…)