Contents

cohomology

# Contents

## Idea

The generalized (Eilenberg-Steenrod) cohomology theory/spectrum called $tmf$ – for topological modular forms – is in a precise sense the union of all elliptic cohomology theories/elliptic spectra (Hopkins 94).

More precisely, $tmf$ is the homotopy limit in E-∞ rings of the elliptic spectra of all elliptic cohomology theories, parameterized over the moduli stack of elliptic curves $\mathcal{M}_{ell}$. That such a parameterization exists, coherently, in the first place is due to the Goerss-Hopkins-Miller theorem. In the language of derived algebraic geometry this refines the commutative ring-valued structure sheaf $\mathcal{O}$ of the moduli stack of elliptic curves to an E-∞ ring-valued sheaf $\mathcal{O}^{top}$, making $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ a spectral Deligne-Mumford stack, and $tmf$ is the E-∞ ring of global sections of that structure sheaf (Lurie).

The construction of $tmf$ has motivation from physics (string theory) and from chromatic homotopy theory:

1. from string theory. Associating to a space, roughly, the partition function of the spinning string/superstring sigma-model with that space as target spacetime defines a genus known as the Witten genus, with coefficients in ordinary modular forms. Now, the interesting genera typically appear as the values on homotopy groups (the decategorification) of orientations of multiplicative cohomology theories; for instance the A-hat genus, which is the partition function of the spinning particle/superparticle is a shadow of the Atiyah-Bott-Shapiro Spin structure-orientation of the KO spectrum. Therefore an obvious question is which spectrum lifts this classical statement from point particles to strings. The spectrum $tmf$ solves this: there is a String structure orientation of tmf such that on homotopy groups it reduces to the Witten genus of the superstring (Ando-Hopkins-Rezk 10).

Mathematically this means for instance that $tmf$-cohomology classes help to detect elements in the string cobordism ring. Physically it means that the small aspect of string theory which is captured by the Witten genus is realized more deeply as part of fundamental mathematics (chromatic stable homotopy theory, see the next point) and specifically of elliptic cohomology. Since the full mathematical structure of string theory is still under investigation, this might point the way:

A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. (Witten 87, very last sentence)

2. from chromatic homotopy theory. The symmetric monoidal stable (∞,1)-category of spectra (finite spectra) has its prime spectrum parameterized by prime numbers $p$ and Morava K-theory spectra $K(n)$ at these primes, for natural numbers $n$. The level $n$ here is called the chromatic level. In some sense the part of this prime spectrum at chromatic level 0 is ordinary cohomology and that at level 1 is topological K-theory. Therefore an obvious question is what the part at level 2 would be, and in some sense the answer is $tmf$. (This point of view has been particularly amplified in the review (Mazel-Gee 13) of the writeup of the construction in (Behrens 13), which in turn is based on unpublished results based on (Hopkins 02)). For purposes of stable homotopy theory this means for instance that $tmf$ provides new tools for computing more homotopy groups of spheres via an Adams-Novikov spectral sequence.

## Definition

Write

(Here $\mathcal{M}_{cub}$ is obatined by furthermor adding also the cuspidal cubic curve, hence we have canonical maps $\mathcal{M}_{ell}\to \mathcal{M}_{\overline{ell}}\to \mathcal{M}_{cusp} \to \mathcal{M}_{FG}$).

The Goerss-Hopkins-Miller theorem equips these three moduli stacks with E-∞ ring-valued structure sheaves $\mathcal{O}^{top}$ (and by Lurie (Survey) that makes them into spectral Deligne-Mumford stacks which are moduli spaces for derived elliptic curves etc.)

The $tmf$-spectrum is defined to be the $E_\infty$-ring of global sections of $\mathcal{O}^{op}$ (in the sense of derived algebraic geometry, hence the homotopy limit of $\mathcal{O}^{top}$ over the etale site of $\mathcal{M}$). More precisely one sets

• $TMF \coloneqq \Gamma(\mathcal{M}_{ell}, \mathcal{O}^{top})$;

• $Tmf \coloneqq \Gamma(\mathcal{M}_{\overline{ell}}, \mathcal{O}^{top})$;

• $tmf \coloneqq$ the connective cover of $Tmf$ (also $\simeq \Gamma(\mathcal{M}_{\overline{cub}}, \mathcal{O}^{top})$ (Hill-Lawson 13, p. 2 (?)).

## Constructions

### Decomposition via Arithmetic fracture squares

We survey here some aspects of the explicit construction in (Behrens 13), a review is also in (Mazel-Gee 13),

The basic strategy here is to use arithmetic squares in order to decompose the problem into smaller more manageable pieces.

Write $\overline{\mathcal{M}_{ell}}$ for the compactified moduli stack of elliptic curves. In there one finds the pieces

$\array{ \overline{\mathcal{M}_{ell}} &\stackrel{\iota_{p}}{\leftarrow}& (\overline{\mathcal{M}_{ell}})_p \\ {}^{\mathllap{\iota_{\mathbb{Q}}}}\uparrow \\ (\overline{\mathcal{M}_{ell}})_{\mathbb{Q}} }$

given by rationalization

$(\overline{\mathcal{M}_{ell}})_{\mathbb{Q}} = \overline{\mathcal{M}_{ell}} \underset{Spec(\mathbb{Z})}{\times} Spec(\mathbb{Q})$

(hence this is the moduli of elliptic curves over the rational numbers) and by p-completion

$(\overline{\mathcal{M}_{ell}})_p = (\overline{\mathcal{M}_{ell}}) \underset{Spec(\mathbb{Z})}{\times} Spf(\mathbb{Z}_p)$

for any prime number $p$, where $\mathbb{Z}_p$ denotes the p-adic integers and $Spf(-)$ the formal spectrum. (Hence this is the moduli of elliptic curves over p-adic integers).

This induces the arithmetic square decomposition which realizes $\mathcal{O}^{top}$ as the homotopy fiber product in

$\array{ \mathcal{O}^{top} &\to& \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p \\ \downarrow && \downarrow^{\mathrlap{L_{\mathbb{Q}}}} \\ (\iota_{\mathbb{Q}})_\ast \mathcal{O}^{top}_{\mathbb{Q}} &\stackrel{\alpha_{arith}}{\to}& \left( \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p \right)_{\mathbb{Q}} }$

Here $\mathcal{O}^{top}_{\mathbb{Q}}$ can be obtained directly, and to obtain $\mathcal{O}^{top}_p$ one uses in turn another fracture square, now decomposing via K(n)-localization into $K(1)$-local and $K(2)$-local pieces.

(…)

### Stacks from spectra

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.

The idea is that the formal dual of the complex cobordism Thom spectrum $M U$ is in a suitable sense a covering

$Spec M U \to Spec \mathbb{S}$

of the terminal object in $\mathbf{H}$. (See at Adams spectral sequence – As derived descent)

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

$\cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec(tmf) \times Spec(MU) \times Spec(MU) \stackrel{\to}{\to} Spec(tmf) \times Spec(MU)$

which by formal duality is

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

where the smash product $\wedge$ of ring spectra 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 of formal group laws $\mathcal{M}_{fg}$.

#### 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)

### With Level structure

The moduli stack of elliptic curves has covers by that of elliptic curves with level structure $\Gamma$. Under some conditions these covers inherit derived structure sheaves $\mathcal{O}^{top}$ and hence induce spectra of “topological forms with level structure”, $tmf(\Gamma)$ (Mahowald-Rezk 09). For more on this see at modular equivariant elliptic cohomology.

For instance $tmf_0(2)$ (for the congruence subgroup which preserves an NS-R spin structure on elliptic curves over the complex numbers) is the elliptic spectrum $Ell$ of (Landweber-Ravenel-Stong 93), see at tmf0(2).

Discussion of level structure also governs the relation of tmf to K-theory, see at Maps to K-theory and Tate K-theory.

## Properties

### Homotopy groups

The first few homotopy groups of $tmf$ are (Hopkins 02, section 4.3)

$k$01234567891011121314
$\pi_k(tmf)$$\mathbb{Z}$$\mathbb{Z}/2\mathbb{Z}$$\mathbb{Z}/2\mathbb{Z}$$\mathbb{Z}/24\mathbb{Z}$00$\mathbb{Z}/2\mathbb{Z}$0$\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$$(\mathbb{Z}/2\mathbb{Z})^2$$\mathbb{Z}/6\mathbb{Z}$0$\mathbb{Z}$$\mathbb{Z}/3\mathbb{Z}$$\mathbb{Z}/2\mathbb{Z}$

The canonical map $\mathbb{S} \to tmf$ (the unit of the E-∞ ring structure) induces in degree $\leq 6$ an isomorphism with the stable homotopy groups of spheres.

### 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 $X$.

### Maps to K-theory and to Tate K-theory

The inclusion of the compactification point (representing the nodal curve but being itself the cusp of $\mathcal{M}_{\overline{ell}}$) into the compactified moduli stack of elliptic curves $\mathcal{M}_{\overline{ell}}$ is equivalently the inclusion of the moduli stack of 1-dimensional tori $\mathcal{M}_{1dtori} = \mathcal{M}_{\mathbb{G}_m}$ (Lawson-Naumann 12, Appendix A)

$\mathcal{M}_{\mathbb{G}_m} \simeq \mathbf{B}\mathbb{Z}_2 \longrightarrow \mathcal{M}_{\overline{ell}} \to \mathcal{M}_{FG}$

and pullback of global sections of Goerss-Hopkins-Miller-Lurie theorem-wise $E_\infty$-ring valued structure sheaves yields maps

$KO \longleftarrow \longleftarrow \mathbb{S}$

exhibiting KO $= \Gamma(\mathcal{M}_{\mathbb{G}_m}, \mathcal{O}^{top})$.

At least after 2-localization the canonical double cover of the compactification of $\mathcal{M}_{\mathbb{G}_m} \simeq \mathbf{B}\mathbb{Z}_2$ similarly yields under $\Gamma(-,\mathcal{O}^{top})$ the inclusion of $ko$ as the $\mathbb{Z}_2$-homotopy fixed points of $ku$ (see at KR-theory for more on this)

$\array{ ku_{(2)} \\ \uparrow \\ ko_{(2)} }$

and combined with the above this comes with maps from $tmf$ by restriction along the inclusion of the nodal curve cusp as

$\array{ ku_{(2)} & \longleftarrow & tmf_1(3)_{(2)} \\ \uparrow && \uparrow \\ ko_{(2)} & \longleftarrow & tmf_{(2)} } \,,$

(Lawson-Naumann 12, theorem 1.2), where $tmf_1(3)$ denotes topological modular forms with level-3 structure (Mahowald-Rezk 09).

Moreover, including not just the nodal curve cusp but its formal neighbourhood, which is the Tate curve, there is analogously a canonical map of $E_\infty$-rings

$tmf \longrightarrow KO[ [ q ] ]$

to Tate K-theory (this is originally asserted in Ando-Hopkins-Strickland 01, details are in Hill-Lawson 13, appendix A).

### 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 with value in topological modular forms

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

The original Witten genus is the value of the composite of this with the map to Tate K-theory on homotopy groups. (Ando-Hopkins-Rezk 10)

### Chromatic filtration

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

### 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)

### Modular equivariant versions

See at modular equivariant elliptic cohomology and at Tmf(n).

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

coveringby of level-n structures (modular curve)
$\ast = Spec(\mathbb{Z})$$\to$$Spec(\mathbb{Z}[ [q] ])$$\to$$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack$\mathcal{M}_{1dTori}$$\hookrightarrow$$\mathcal{M}_{Tate}$$\hookrightarrow$$\mathcal{M}_{\overline{ell}}$ (M_ell)$\hookrightarrow$$\mathcal{M}_{cub}$$\to$$\mathcal{M}_{fg}$ (M_fg)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$KU$KU[ [q] ]$elliptic spectrumcomplex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf(KO $\hookrightarrow$ KU) = KR-theoryTate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)tmf$\mathbb{S}$

## References

The idea of a generalized cohomology theory with coefficients the ring of topological modular forms providing a home for the refined Witten genus of

• Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)

and produced as a homotopy limit of elliptic cohomology theories over the moduli stack of elliptic curves was originally announced, as joint work with Mark Mahowald and Haynes Miller, in

• Michael Hopkins, section 9 of Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)

(There the spectrum was still called “$eo_2$” instead of “$tmf$”.) The details of the definition then appeared in

A central tool that goes into the construction is the Goerss-Hopkins-Miller theorem, see there for references on that.

Expositions include

An actual detailed account of the construction of $tmf$ (via decomposition by arithmetic squares) is spelled out in

• Mark Behrens, Notes on the construction of $tmf$, 2013 (pdf)

A complete account of the computation of the homotopy groups of $tmf$ (following previous unpublished computations) is in

• Tilman Bauer, Computation of the homotopy groups of the spectrum $tmf$ (pdf)

A survey of how this works is in

• Akhil Mathew, The homotopy groups of $TMF$ (pdf)

(This presents as an instructive much simpler but analogous case the construction of KO in analogy to the construction of $tmf$, more details on this are in Mathew 13, section 3.)

and course notes that go through the construction of tmf and the computation of its homotopy groups are here:

The non-connective version of this is discussed in

Supplementary material graphically displaying parts of these intricate computations is in

• the graded ring $tmf^\ast(pt)$ (pdf);

the spectral sequence used to compute it (pdf);

whose $E_2$-page is $Ext_{A(2)}(\mathbb{F}_2, \mathbb{F}_2)$ where $A(2)$ is displayed here: pdf;

the spectral sequence that computes $Tmf^\ast(pt)$ (pdf)

The $\mathbb{Z}_2$-homology of $tmf$ is discussed in

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
Topological modular forms with level N-structure$tmf(N)$ – is discussed in
The self-Anderson duality of $tmf$ is discussed in (Stojanoska 11).