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…)