The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: {\mathrm{tmf}} is a higher analog of {KO}theory (or rather, connective {KO}-theory).

1. What is {\mathrm{tmf}}?

The spectrum of (real) {KO}-theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex {K}-theory. As a ring spectrum, {K} is complex orientable, and it corresponds to the formal group {\hat{\mathbb{G}_m}}: the formal multiplicative group. Along with {\hat{\mathbb{G}_a}}, the formal multiplicative group {\hat{\mathbb{G}_m}} is one of the few “tautological” formal groups, and it is not surprising that {K}-theory has a “tautological” formal group because the Chern classes of a line bundle {\mathcal{L}} (over a topological space {X}) in {K}-theory are defined by

\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];

that is, one uses the class of the line bundle {\mathcal{L}} itself in {K^0(X)} (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying {\hat{\mathbb{G}_m}},

\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},

from {\mathrm{Spec} \mathbb{Z}} to the moduli stack of formal groups {M_{FG}}, is a flat morphism. (more…)

(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…)