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: is a higher analog of –theory (or rather, connective -theory).

**1. What is ?**

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

that is, one uses the class of the line bundle itself in (modulo a normalization) to define.

The formal multiplicative group has the property that it is *Landweber-exact*: that is, the map classifying ,

from to the moduli stack of formal groups , is a flat morphism. (more…)