cohomology

# Contents

## Idea

The generalized (Eilenberg-Steenrod) cohomology theory called $\mathop{tmf}$ is the one represented by the spectrum that is obtained as the homotopy limit of the spectra of all elliptic cohomology theories.

The abbreviation $\mathop{tmf}$” stands for the ring of topological modular forms as this is the cohomology ring that $\mathop{tmf}$ assigns, essentially, to the point.

One of the greatest recent achievements in algebraic topology is the construction of the $\mathop{tmf}$ spectrum as the global section of a certain (infinity,1)-sheaf of commutative ring spectra over the moduli stack of elliptic curves.

From this sheaf, one can recover the Adams-type spectral sequence associated to $\mathop{tmf}$. According to SEC, this sheaf is actually the structure sheaf of the moduli stack classifying “oriented elliptic curves” over commutative ring spectra, or, to be in the correct variance, over derived affine schemes.

## Properties

### Inclusion of circle 2-bundles

Write $B^2 U(1) \simeq K(\mathbb{Z},3)$ for the abelian ∞-group whose underlying homotopy type is the classifying space for circle 2-bundle. Write $\mathbb{S}[B^2 U(1)]$ for its ∞-group ∞-ring.

###### Proposition

There is a canonical homomorphism of E-∞ rings

$\mathbb{S}[B^2 U(1)] \to tmf \,.$
###### Remark

This means that every circle 2-bundle ($U(1)$-bundle gerbe) given by a modulating map $\chi \colon X \to B^2 U(1)$ determines a class represented by

$X \stackrel{\chi}{\to} B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf$

in the $tmf$-generalized cohomology of its base space $X$.

### Witten genus and string orientation

The $tmf$-spectrum is the codomain of the Witten genus, or rather of its refinements to the string orientation of tmf

$\sigma : M String \to tmf \,.$

### Chromatic filtration

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomology$Ell_E$
tmftmf spectrum
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$

### Anderson self-duality

The spectrum $tmf$ is self-dual under Anderson duality, more precisley $tmf[1/2]$ is Anderson-dual to $\Sigma^{21} tmf[1/2]$ (Stojanoska 11, theorem 13.1)

## Construction as the global sections of a derived structure sheaf

There is a way to “construct” the tmf-spectrum as the E-∞ ring of global sections of a structured (∞,1)-topos whose underlying space is essentially the moduli stack of elliptic curves. We sketch some main ideas of this construction.

### The context – derived geometry over formal duals of $E_\infty$-rings

The discussion happens in the context of derived geometry in the (∞,1)-topos $\mathbf{H}$ over a small version of the (∞,1)-site of formal duals of E-∞ rings (ring spectra). This is equipped with some subcanonical coverage. For $R \in E_\infty Ring$ we write $Spec R$ for its image under the (∞,1)-Yoneda embedding $(E_\infty Ring)^{op} \hookrightarrow \mathbf{H}$.

###### Observation

The terminal object in $\mathbf{H}$ is the formal dual of the sphere spectrum

$* \simeq Spec \mathbb{S} \,.$

Because the sphere spectrum is the initial object in $E_\infty Ring$.

### Coverings by the Thom spectrum

The crucial input for the entire construction is the following statement.

###### Fact

The formal dual of the Thom spectrum $M U$ is a well-supported object in $\mathbf{H}$, in that the morphism

$Spec M U \to *$

to the terminal object in $\mathbf{H}$ is an effective epimorphism.

This means that $Spec M U$ plays the role of a cover of the point. This allows to do some computations with ring spectra locally on the cover $Spec M U$ . Since $M U^*$ is the Lazard ring, this explains why formal group laws show up all over the place.

To see this, first notice that the problem of realizing $R = tmf$ or any other ring spectrum as the ring of global sections on something has a tautological solution : almost by definition (see generalized scheme) there is an $E_\infty$-ring valued structure sheaf $\mathcal{O}Spec(R)$ on $Spec R$ and its global sections is $R$. So we have in particular

$tmf \simeq \mathcal{O}Spec tmf \,.$

In order to get a less tautological and more insightful characterization, the strategy is now to pass on the right to the $Spec M U$-cover by forming the (∞,1)-pullback

$\array{ Spec tmf \times Spec M U &\to& Spec tmf \\ \downarrow && \downarrow \\ Spec M U &\to& * \simeq Spec \mathbb{S} } \,.$

The resulting Cech nerve is a groupoid object in an (∞,1)-category given by

$\cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec tmf \times Spec M U \times Spec M U \stackrel{\to}{\to} Spec tmf \times Spec M U$

which by formal duality is

$\cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (tmf \wedge M U \wedge M U) \stackrel{\to}{\to} Spec ( tmf \wedge M U)$

where the smash product $\wedge$ of ring spectral over the sphere spectrum $\mathbb{S}$ is the tensor product operation on function algebras formally dual to forming products of spaces.

As a groupoid object this is still equivalent to just $Spec tmf$.

### Decategorification: the ordinary moduli stack of elliptic curves

To simplify this we take a drastic step and apply a lot of decategorification: by applying the homotopy group (∞,1)-functor to all the $E_\infty$-rings involved these are sent to graded ordinary rings $\pi_*(tmf)$, $\pi_*(M U)$ etc. The result is an ordinary simplicial scheme

$\cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(tmf \wedge M U \wedge M U)) \stackrel{\to}{\to} Spec ( \pi_*(tmf \wedge M U)) \,,$

which remembers the fact that its structure rings are graded by being equipped with an action of the multiplicative group $\mathbb{G} = \mathbb{A}^\times$ (see line object).

This general Ansatz is discussed in (Hopkins).

This simplicial scheme, which is degreewise the formal dual of a graded ring of generalized homology-groups one can show is in fact a groupoid, hence a stack: effectively the moduli stack of elliptic curves. $\mathcal{M}_{ell}$. See (Henriques).

In fact if in this construction one replaced $Spec tmf$ by the point, one obtains the simplicial scheme

$\cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(M U \wedge M U)) \stackrel{\to}{\to} Spec ( \pi_*(M U))$

which one finds is the moduli stack $\mathcal{M}_{fg}$ of formal group laws.

### Explicit computation of homotopy groups by a spectral sequence

Now, a priori these underived stacks remember little about the original derived schemes $Spec tmf$ etc. They may not even carry any $E_\infty$-ring valued structure sheaf anymore (though some of them do).

If they do carry an $E_\infty$-ring valued structure sheaf $\mathcal{O}$, one can compute the homotopy groups of its global sections by a spectral sequence

$H^p(\mathcal{M}_{ell}, \pi_q(\mathcal{O})) \Rightarrow \pi_{p+q} \mathcal{O}(\mathcal{M}_{ell}) \,.$

But it turns out that even if the derived structure sheaf does not exist, this spectral sequence may still converge and may still compute the homotopy groups of the ring spectrum that one started with. This gives one way to compute the homotopy groups of $tmf$.

For the case of $tmf$ one finds that the homotopy sheaves $\pi_q(\mathcal{O}(\mathcal{M}_{ell}))$ are simple: they vanish in odd degree and are tensor powers $\omega^{\otimes k}$ of the canonical line bundle $\omega$ in even degree $2 k$, where the fiber of $\omega$ over an elliptic curve is the tangent space of that curve at its identity element. A section of $\omega^{\otimes k}$ is a modular form of weight $k$. So the whole problem of computing the homotopy groups of $tmf$ boils down to computing the abelian sheaf cohomology of the moduli stack of elliptic curves with coefficients in these abelian groups of modular forms — and then examining the resulting spectral sequence.

This can be done quite explicitly in terms of a long but fairly elementary computation in ordinary algebra. A detailed discussion of this computation is in (Henriques)

## References

A course that goes through the construction of tmf and the computation of its homotopy groups is

More details are at

and in the series of lecture notes linked to there.

The refinement of the Witten genus to a morphism of E-∞ rings to $tmf$, hence the string orientation of tmf is due to

Discussion of twisted cohomology with coefficients in $tmf$ is in section 8 of
The self-Anderson duality of $tmf$ is discussed in