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.

In general, computing the ${\mathrm{tmf}}$-homology is difficult, especially since the homotopy groups of ${\mathrm{tmf}}$ are so difficult, but if one works with the non-connective version ${\mathrm{Tmf}}$ (a spectrum whose connective cover is ${\mathrm{Tmf}}$), then the ${\mathrm{Tmf}}$-homology of a given spectrum ${X}$ is essentially an amalgamation of the elliptic homology of ${X}$, modulo differentials in a spectral sequence. In the case of ${DA(1)}$, a key step in the paper is a modular description of the elliptic homology of ${DA(1)}$ (roughly, it is the ring classifying ${\Gamma_1(3)}$-structures on the elliptic curve), which makes possible the computation of the groups ${\mathrm{Tmf}_*(DA(1))}$. In fact, the spectrum ${\mathrm{Tmf} \wedge DA(1)}$ is almost certainly the spectrum of “topological modular forms of level 3,” although I do not believe that a description of this spectrum is in print. (Away from the prime 2, this appears in Vesna Stojanoska‘s thesis.)

In this blog post, I’d like to sketch the strategy of the computation used in the paper.

1. The different flavors of ${\mathrm{tmf}}$

In a previous post, I described the spectrum ${\mathrm{TMF}}$ of (periodic) topological modular forms, which was obtained as a homotopy limit of various elliptic spectra. Namely, one had a sheaf ${\mathcal{O}^{\mathrm{top}}}$ of ${E_\infty}$-ring spectra on the étale site of the moduli stack ${M_{ell}}$ of elliptic curves. For every affine étale morphism

$\displaystyle \mathrm{Spec} R \rightarrow M_{ell} ,$

classifying an elliptic curve over ${R}$, one had an elliptic spectrum ${\mathcal{O}^{\mathrm{top}}(\mathrm{Spec} R)}$, which was an ${E_\infty}$-algebra whose formal group was identified with the formal group of that elliptic curve. (The stack ${M_{ell}}$ is a Deligne-Mumford stack: that is, there are enough étale morphisms into ${M_{ell}}$ from actual schemes.) The spectrum ${\mathrm{TMF}}$ was defined as the global sections of this sheaf. In other words, it was the elliptic cohomology associated to the “universal” elliptic curve — but the only way to define that was taking an inverse limit over a stack.

The descent spectral sequence provides a map

$\displaystyle \mathrm{TMF}_* \rightarrow MF_*[\Delta^{-1}],$

into the ring of ${MF_*}$ integral modular forms (with grading doubled), with the modular discriminant ${\Delta}$ inverted since we are working with smooth elliptic curves. The map is not surjective, although it is an isomorphism with ${6}$ inverted. Integrally, only ${\Delta^{24}}$ survives, making ${\mathrm{TMF}}$ a 576-periodic ring spectrum.

The above data can be thought of as a structure sheaf for the étale site of ${M_{ell}}$, except it takes values in ${E_\infty}$-rings instead of ordinary commutative rings. The POV of derived algebraic geometry suggests that one should think of this as a sort of derived algebraic (Deligne-Mumford) stack whose structure sheaf is in fact the sheaf ${\mathcal{O}^{\mathrm{top}}}$ of elliptic spectra. Given this, and given the description as a ringed ${\infty}$-topos, one may ask about the moduli interpretation is of this “derived stack”: this was found by Jacob Lurie and is sketched in his survey on elliptic cohomology.

But there are other variants of topological modular forms. It turns out that it is possible to extend the sheaf of ${E_\infty}$-rings from the étale site of ${M_{ell}}$ to the étale site of the compactification ${M_{\bar{ell}}}$ of “generalized” elliptic curves that are allowed to have a nodal singularity. Such generalized elliptic curves also have formal groups; for a nodal cubic, it is given by the formal multiplicative group. One defines

$\displaystyle \mathrm{Tmf} = \Gamma( M_{\bar{ell}}, \mathcal{O}^{top});$

this is a non-periodic ring spectrum, since ${\Delta}$ is no longer invertible over the compactified stack ${M_{\bar{ell}}}$. The spectrum ${\mathrm{tmf}}$ is defined by

$\displaystyle \mathrm{tmf} = \tau_{\geq 0} \mathrm{Tmf},$

that is, it is the connective cover. This is the smallest of the various things called ${\mathrm{tmf}}$, and it is much smaller than taking ${\tau_{\geq 0}( \mathrm{TMF})}$.

2. The complex ${DA(1)}$ and level 3 structures

In general, the homotopy groups of ${\mathrm{Tmf}}$ are quite complicated; there is considerable torsion at the primes 2 and 3. (This paper of Tilman Bauer describes the calculation of the connective cover.) The homotopy groups of ${\mathrm{Tmf}}$ are calculated via a spectral sequence

$\displaystyle H^i(M_{\bar{ell}}, \omega^j) \implies \pi_{2j-i} \mathrm{Tmf},$

where ${\omega}$ is the line bundle on ${M_{\bar{ell}}}$ that assigns to an elliptic curve the dual of its Lie algebra. The reason the homotopy groups of ${\mathrm{Tmf}}$ are so complicated is that the cohomology of the moduli stack ${M_{\bar{ell}}}$ is very messy. However, it’s possible that smashing with a finite spectrum could simplify the homotopy groups. For instance, in ${K}$-theory, it’s a classical theorem of Wood that ${KO \wedge \mathbb{CP}^2 \simeq KU}$: that is, one can get from the (comparatively messy) homotopy groups of ${KO}$-theory to the very simple ones of unitary ${K}$-theory ${KU}$.

In general, for an even finite spectrum ${X}$ (that is, a connective spectrum with finitely generated homology), the elliptic homology of ${X}$ naturally lives as a quasi-coherent sheaf on ${M_{\bar{ell}}}$. In other words, for every elliptic curve ${C \rightarrow \mathrm{Spec} R}$ classified by an étale map ${\mathrm{Spec} R \rightarrow M_{\bar{ell}}}$, one can form the associated elliptic homology theory ${E}$ (that is, the sections of the sheaf of elliptic spectra over ${\mathrm{Spec} R}$), and the resulting ${E_0( X)}$ as ${E}$ varies defines a vector bundle ${\mathcal{V}}$ on the moduli stack of elliptic curves. One then has a descent spectral sequence

$\displaystyle H^i(M_{\bar{ell}}, \mathcal{V} \otimes \omega^j) \implies \pi_{2j-i} ( \mathrm{Tmf} \wedge X),$

which is the descent spectral sequence for ${\mathrm{Tmf} \wedge X}$. In other words, one has this sheaf ${\mathcal{O}^{\mathrm{top}}}$ of ${E_\infty}$-rings over the étale site of ${M_{\bar{ell}}}$, and then one smashes it with ${X}$ to get a sheaf of modules whose global sections give ${\mathrm{Tmf} \wedge X}$. The descent spectral sequence one gets is as above.

The above spectral sequence is related to the Adams-Novikov spectral sequence, which for an even spectrum ${X}$ produces a vector bundle ${\mathcal{V}}$ on the moduli stack of formal groups ${M_{FG}}$, and runs ${H^i(M_{FG}, \mathcal{V} \otimes \omega_j) \implies \pi_{2j-i} X}$.

The basic observation here is that while the cohomology of the structure sheaf of ${M_{\bar{ell}}}$ is very complicated, the cohomology of vector bundles on it can be much simpler. For example, at the prime ${2}$, there is an eight-fold cover of the moduli stack ${M_{ell}}$,

$\displaystyle (M_{ell})_1(3) \rightarrow M_{ell},$

where the stack ${(M_{ell})_1(3)}$ classifies elliptic curves together with a nonzero point of order 3. The pushforward of the structure sheaf along this cover gives a rank eight vector bundle on ${M_{ell}}$. The stack ${(M_{ell})_1(3)}$ is much simpler than ${M_{ell}}$: it is, up to ${\mathbb{G}_m}$-action, affine. To see this, given an elliptic curve with a point of order 3, we can move the point of order 3 to ${(0, 0)}$, which means ${(0, 0)}$ is an inflection point. Hence the cubic equation must have the form

$\displaystyle y^2 + a_1 y + a_3 xy = x^3,$

and the only isomorphisms between such cubic curves come from ${(x,y) \mapsto (u^2 x, u^3 y)}$. It follows that the moduli stack of elliptic curves with a point of order 3 can be described as ${\mathrm{Spec} \mathbb{Z}_{(2)}[a_1, a_3][\Delta^{-1}]/\mathbb{G}_m}$ where ${u \in \mathbb{G}_m}$ acts by ${a_1 \mapsto u a_1, a_3 \mapsto u^3 a_3}$. In particular, since ${\mathbb{G}_m}$-actions just keep track of gradings, the stack ${(M_{ell})_1(3)}$ is basically affine, for our purposes, and we have a nice vector bundle on ${M_{ell}}$ with trivial cohomology and lots of sections.

The vector bundle actually extends to ${M_{\bar{ell}}}$ as well; in fact, an explicit calculation shows that ${\mathrm{Spec} \mathbb{Z}_{(2)}[a_1, a_3]/\mathbb{G}_m}$ provides an eight-fold flat cover of the even larger moduli stack of all cubic curves (which are allowed to have a cuspidal singularity).

The main step in the paper is to show that this eight-dimensional bundle on ${M_{\bar{ell}}}$ is realizable as the elliptic homology of an eight-cell complex. This complex is denoted ${DA(1)}$; it is a 2-local finite spectrum whose cohomology, as a module over the Steenrod algebra, can be drawn as:

Although it’s generally not possible to realize even a small module over the Steenrod algebra by a spectrum (cf. the Hopf invariant one problem), the complex ${DA(1)}$ can be built fairly explicitly by attaching cells.

How does one compute the elliptic homology of ${DA(1)}$? The main point is to understand the “cooperations”: that is, one needs to know not what the elliptic homology of ${DA(1)}$ is for one elliptic curve (which is easy to determine; it’s projective of rank eight), but what it is in a functorial manner. In the paper, the key step is to observe that, for formal reasons, the vector bundle can be extended over the stack ${M_{cub}}$ of cubic curves; over this stack, the fiber over the cuspidal cubic ${y^2 = x^3}$ encodes the mod 2 homology. This fiber turns out to play a special role in the theory of vector bundles over ${M_{cub} }$, by a form of Nakayama’s lemma: since the entire stack “contracts” onto this cuspidal point, it’s the key place from which to extract data.

3. Truncated Brown-Peterson spectra

The above work shows that the ${\mathrm{Tmf}}$-homology of a certain eight-cell complex ${DA(1)}$ is tractable; in fact, it is the cohomology of the stack ${\mathrm{Spec} \mathbb{Z}_{(2)}[a_1, a_3]/\mathbb{G}_m}$. At least additively, one can get an evaluation of the homotopy groups

$\displaystyle (\mathrm{Tmf}_*(DA(1)))_{\geq 0} \simeq \mathbb{Z}_{(2)}[a_1, a_3] ;$

the negative homotopy groups are similar (and dual to these). The very simple answer is, of course, in sharp contrast to the complicated homotopy groups of ${\mathrm{Tmf}}$, and arise because the stack of elliptic curves with ${\Gamma_1(3)}$-structure is much simpler (e.g., it has cohomological dimension one) than the moduli stack of elliptic curves.

The next step from here is to appeal to a somewhat mysterious fact: the homotopy groups of ${\mathrm{Tmf}}$ have a gap in dimensions ${[-21, 0]}$, while the 8 cell complex ${DA(1)}$ is sufficiently small that one has

$\displaystyle \mathrm{tmf} \wedge DA(1) = \tau_{\geq 0}( \mathrm{Tmf} \wedge DA(1)).$

In particular, this computes the ${\mathrm{tmf}}$-homology of ${DA(1)}$ as well; that’s surprising because ${\mathrm{tmf}}$ itself doesn’t have a similar moduli interpretation and, at least a priori, it’s not clear how to compute the ${\mathrm{tmf}}$-homology of anything. (It turns out that there is a tractable Adams-Novikov spectral sequence for ${\mathrm{tmf}}$, but that requires some work to set up; one approach is presented in this paper.)

The homotopy groups of ${\mathrm{tmf} \wedge DA(1)}$ are precisely those of the spectrum ${BP\left \langle 2\right\rangle}$, obtained from the Brown-Peterson spectrum ${BP}$ with

$\displaystyle BP_* \simeq \mathbb{Z}_{(2)}[v_1, v_2, \dots, ],$

by taking the quotient by the regular sequence ${v_3, v_4, \dots}$. Since the choice of the generators ${v_i}$ is not canonical, it’s preferable to say “a form of ${BP\left \langle 2\right\rangle}$.” That doesn’t prove that ${\mathrm{tmf} \wedge DA(1)}$ is in fact a form of ${BP\left \langle 2\right\rangle}$, but some additional work can be used to produce a map from ${\mathrm{tmf} \wedge DA(1) }$ to the connective cover of a quotient of ${( \mathrm{Tmf} \wedge MU)}$, which one can check is in fact a form of ${BP\left \langle 2\right\rangle}$. In other words, one concludes the folk theorem

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

which is a ${\mathrm{tmf}}$-analog of Wood’s theorem ${ko \wedge \Sigma^{-2}\mathbb{CP}^2 \simeq ku}$ in (connective) real and complex ${K}$-theory.

From here, the evaluation of the homology can be done as follows. The homology of ${DA(1)}$ is known, by definition. The homology of ${BP\left \langle 2\right\rangle}$ can be calculated explicitly from the homology of ${BP}$. Putting this together, one finds that ${H_*(\mathrm{tmf}; \mathbb{Z}/2) \subset H_*(BP; \mathbb{Z}/2)}$, and one has its graded dimension. In general, the graded dimension is enough to pin down ${H_*(\mathrm{tmf}; \mathbb{Z}/2)}$; however, the homology of a ring spectrum is a comodule algebra over the dual Steenrod algebra, and it’s very hard to write down comodule subalgebras of ${H_*(BP ; \mathbb{Z}/2)}$. In fact, a little bit of Hopf algebra technology is enough to pin down ${H_*(\mathrm{tmf}; \mathbb{Z}/2)}$.