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…)