nLab A Survey of Elliptic Cohomology

This entry is about the text

• Jacob Lurie, A Survey of Elliptic Cohomology, in: Algebraic Topology, Abel Symposia Volume 4, 2009, pp 219-277 (pdf, doi:10.1007/978-3-642-01200-6_9)

It

• reviews basics of elliptic cohomology

• indicates the construction of the tmf-spectrum

• and discusses this in the context of equivariant cohomology.

The central theorem is

• the realization of the moduli space of elliptic curves as a structured (∞,1)-topos

• with a structure (∞,1)-sheaf of E-∞ ring-valued functions

• such that the tmf-spectrum is the E-∞ ring of global sections of this structure sheaf.

See also Elliptic Cohomology I and Chromatic Homotopy Theory.

Table of Contents

1. Summary

   The following entry has some paragraphs that summarize central ideas.

   1. gluing all elliptic cohomology theories to the tmf spectrum

   2. interpretation in terms of higher topos theory

2. Partial surveys

   These links point to pages that contain notes on aspects of the theory that are in the style of and originate from a seminar on A Survey of Elliptic Cohomology:

   1. A Survey of Elliptic Cohomology - cohomology theories

   2. A Survey of Elliptic Cohomology - formal groups and cohomology

   3. A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

   4. A Survey of Elliptic Cohomology - elliptic curves

   5. A Survey of Elliptic Cohomology - equivariant cohomology

   6. A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations

   7. A Survey of Elliptic Cohomology - A-equivariant cohomology

   8. A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

   9. A Survey of Elliptic Cohomology - towards a proof

   10. A Survey of Elliptic Cohomology - compactifying the derived moduli stack

   11. A Survey of Elliptic Cohomology - descent ss and coefficients

3. towards geometric models

   These links point to pages that have an exposition of the Stolz-Teichner program for constructing geometric models for elliptic cohomology.

• Outline of the constructions and statements

  • Axiomatic field theories and their motivation from topology

  • (1,1)-dimensional Euclidean field theories and K-theory

  • (2,1)-dimensional Euclidean field theories and tmf

• Definitions

  • bordism categories following Stolz-Teichner

  • (2,1)-dimensional Euclidean field theories

• Geometric models for elliptic cohomology – Axiomatic field theories and their motivation from topology

Contents

Here is the table of contents of the Survey reproduced. Behind the links are linked keyword lists for relevant terms.

Summary

The text starts with showing or recalling that

• the collection of all elliptic cohomology theories and

• the gluing of their representing spectra into the single tmf spectrum

is best understood in terms of global sections of the structure sheaf of functions on the refinement of the moduli space of all elliptic curves to a structured (∞,1)-topos.

Then it uses this higher topos theoretic derived algebraic geometry perspective to analyze further properties of elliptic cohomology theories, in particular their refinements to equivariant cohomology.

Gluing all elliptic cohomology theories to the tmf spectrum

The triple of generalized (Eilenberg-Steenrod) cohomology theories

1. periodic ordinary integral cohomology

2. complex K-theory

3. elliptic cohomology

constitutes the collection of all possible generalized (Eilenberg-Steenrod) cohomology theories with the extra property that they are

• multiplicative

and

• periodic.

It so happens that all multiplicative periodic generalized Eilenberg-Steenrod cohomology theories $A$ are characterized by the formal group (an infinitesimal group) whose ring of functions is the cohomology ring $A(\mathbb{C}P^\infty)$ obtained by evaluating $A$ on the complex projective space $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$ – the classifying space for complex line bundles – and whose group product is induced from the morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that representes the tensor product of complex line bundles.

There are precisely three different types of such formal groups:

• the additive formal group (a single one)

• the multiplicative formal group (a single one)

• a formal group defined by an elliptic curve (many).

The first case corresponds to periodic integral cohomology. The second corresponds to complex K-theory. Each element in the third family corresponds to one flavor of elliptic cohomology.

It is therefore natural to subsume all elliptic cohomology theories into one single cohomology theory. This is the theory called tmf.

It turns out that the right way to formalize what “subsume” means in the above sentence involves formulating the way in which an elliptic cohomology theory is associated to a given elliptic curve in the correct higher categorical language:

The collection of all 1-dimensional elliptic curves forms a generalized space $M_{1,1}$ – a stack – defined by the property that it is the classifying space for elliptic curves in that elliptic curves over a ring $R$ correspond to classifying maps $\phi : Spec R \to M_{1,1}$.

Then the classical assignment of an elliptic cohomology theory to an elliptic curve is an assignment

$$\{\phi : Spec R \to M_{1,1}\} \to CohomologyTheories \,.$$

We may think of maps $Spec R \to M_{1,1}$ as picking certain subsets of the generalized space $M_{1,1}$ and of morphisms

$$\array{ Spec(R) &&\to&& Spec(R') \\ & \searrow && \swarrow \\ && M_{1,1} }$$

as maps between such subsets. Hence the assignment of cohomology theories to elliptic curves is much like a sheaf of cohomology theories on the moduli space (stack) of elliptic curves.

In order to glue all elliptic cohomology theories in some way one would like to take something like the category of elements of this sheaf, i.e. its homotopy limit. In order to say what that should mean, one has to specify the suitable nature of the codomain, the collection of “all cohomology theories”.

As emphasized at generalized (Eilenberg-Steenrod) cohomology, the best way to do this is to identify a generalized (Eilenberg-Steenrod) cohomology theory with the spectrum that represents it. It is and was well known how to do this for each elliptic curve separately. What is not so clear is how this can be done coherently for all elliptic curves at once: we need a lift of the above cohomology-theory-valued sheaf to a sheaf of representing spectra

$$\array{ && Spectra \\ & {}^{?}\nearrow & \;\;\;\downarrow^{represent} \\ \{\phi : Spec R \to M_{1,1}\} &\to& CohomologyTheories } \,.$$

In this generality this turns out to be a hard problem. But by definition here we are really interested just in the special case where all cohomology theories in question are multiplicative cohomology theories and where hence all spectra in question are commutative ring spectra

$$\array{ && CommRingSpectra \\ & {}^{O_{M^{der}}}\nearrow & \downarrow \\ \{\phi : Spec R \to M_{1,1}\} &\to& MultiplicativeCohomologyTheories } \,.$$

As indicated, this problem does turn out to have a solution: Goerss, Hopkins and Miller showed that the desired lift denoted $O_{M^{der}}$ above exists – the Goerss-Hopkins-Miller theorem

Accordingly, one can then obtain the tmf spectrum as the homotopy limit of this sheaf of E-∞ rings $O_{M^{der}}$. Recall from the discussion at limit in a quasi-category that such a homotopy limit computes global sections. It is an $\infty$-version of computing sections in a Grothendieck construction, really, as described there.

Interpretation in terms of higher topos theory

What is noteworthy about the above construction is that, as the notation above suggests, sheaves of E-infinity rings generalize sheaves of rings as thery are familiar from the theory of ringed spaces, where they are called structure sheaves.

Accordingly, the morphism $O_{M^{der}}$ makes the moduli space of elliptic curves into a structured (∞,1)-topos.

This perspective embeds the theory of elliptic cohomology and of the tmf spectrum as an application into the general context of higher topos theory and derived algebraic geometry.

equivariant elliptic cohomology

• equivariant elliptic cohomology

Contents

1. Elliptic Cohomology

1.1 Cohomology Theories

• cohomology

  • generalized (Eilenberg-Steenrod) cohomology

  • integral cohomology

  • K-theory

  • elliptic cohomology

• multiplicative cohomology theory

• periodic cohomology theory

• topological modular form

1.2 Formal Groups from Cohomology Theories

• formal group

1.3 Elliptic Cohomology

• elliptic curve

• elliptic cohomology

• E-∞ ring

2 Derived Algebraic geometry

• derived algebraic geometry

  • higher topos theory

  • higher algebra

2.1 $E_\infty$ rings

• commutative algebra in an (∞,1)-category

• E-∞ ring

• commutative ring spectrum

2.2 Derived Schemes

• structured (∞,1)-topos

• derived scheme

3 Derived Group Schemes and Orientations

• group scheme

3.1 Orientations of the Multiplicative Group

3.2 Orientations of the Additive Group

3.3 The Geometry of Preorientations

3.4 Equivariant $A$-Cohomology for Abelian Groups

3.5 The Nonabelian Case

4 Oriented Elliptic Curves

4.1 Construction of the Moduli Stack

4.2 The Proof of Theorem 4.4: The Local Case

4.3 Elliptic Cohomology near $\infty$

5 Applications

5.1 2-Equivariant Elliptic Cohomology

• equivariant elliptic cohomology

5.2 Loop Group Representations

• loop group

5.3 String Orientation

• string structure

• sigma-orientation

5.4 Higher Equivariance

5.5 Elliptic Cohomology and Geometry

Further references

Lots of literature on modular forms is collected at

• Nora Ganter, Topological modular forms literature list

An introduction to and survey of the Goerss-Hopkins-Miller-Lurie theorem is in

• Paul Goerss, Topological modular forms (after Hopkins, Miller, and Lurie) Séminaire BOURBAKI Mars 2009 61ème année, 2008-2009, no 1005(2009)(arXiv)

which has grown out of

• Topological Algebraic Geometry - A Workshop.

A good bit of details is in

• David Gepner, Homotopy topoi and equivariant elliptic cohomology, 1999 (pdf)