I’ve just uploaded to arXiv my paper “The homology of {\mathrm{tmf}},” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod {2} cohomology of the spectrum {\mathrm{tmf}} of (connective) topological modular forms, as a module over the Steenrod algebra: one has

\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,

where {\mathcal{A}} is the Steenrod algebra and {\mathcal{A}(2) \subset \mathcal{A}} is the 64-dimensional subalgebra generated by {\mathrm{Sq}^1, \mathrm{Sq}^2,} and { \mathrm{Sq}^4}. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of {\mathrm{tmf}}; one has a spectral sequence

\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.

Since {\mathcal{A}(2) \subset \mathcal{A}} is finite-dimensional, the entire {E_2} page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the {E_2} page for {\mathrm{tmf}} is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of {H^*(\mathrm{tmf}; \mathbb{Z}/2)} in this paper is based on a certain eight-cell (2-local) complex {DA(1)}, with the property that

\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,

where {BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )} is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum {BP\left \langle 2\right\rangle} is a complex-orientable ring spectrum, so that computations with it (instead of {\mathrm{tmf}}) become much simpler. In particular, one can compute the cohomology of {BP\left \langle 2\right\rangle} (e.g., from the cohomology of {BP}), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of {\mathrm{tmf}}. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

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…)