cohomology

# Contents

## Idea

An elliptic cohomology theory is a type of generalized (Eilenberg-Steenrod) cohomology theory associated with the datum of an elliptic curve.

Even (weakly) periodic multiplicative generalized (Eilenberg-Steenrod) cohomology theories $A$ are characterized by the formal group whose ring of functions $A(\mathbb{C}P^\infty)$ is the cohomology ring of $A$ evaluated on the complex projective space $\mathbb{C}P^\infty$ and whose group product is induced from the canonical morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that describes the tensor product of complex line bundles under the identification $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$.

There are precisely three types of such formal group laws:

An elliptic cohomology theory is an eve periodic multiplicative generalized (Eilenberg-Steenrod) cohomology theory whose corresponding formal group is an elliptic curve, hence which is represented by an elliptic spectrum.

Then Goerss-Hopkins-Miller-Lurie theorem shows that the assignment of generalized (Eilenberg-Steenrod) cohomology theories to elliptic curves lifts to an assignment of representing spectra in a structure preserving way.

The homotopy limit of this assignment functor, i.e. the “gluing” of all spectra representing all elliptic cohomology theories is the spectrum that represents the cohomology theory called tmf.

## Properties

### Genera, the elliptic genus and relation to string theory

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

The following is rough material originating from notes taken live (and long ago), to be polished. See also at elliptic genus and Witten genus

Some topological invariants of manifolds that are of interest:

we restricted attention to closed connected smooth manifolds $X$

• the Euler characteristic $e(X) \in \mathbb{Z}$

• takes all values in $\mathbb{Z}$

• is the obstruction to the existence of a nowhere vanishing vector field on $X$:

$(e(X)= 0) \Leftrightarrow (\exists v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)$
• signature $sgn(X)$

this is the obstruction to $X$ being cobordant to a fiber bundle over the circle:

$X$ is bordant to a fiber bundle over $S^1$ precisely if $sgn(X) = 0$

• when $X$ has a spin structure

the index of the Dirac operator $D$:

$ind D_X \in \mathbb{Z}$
$\alpha(D) \in \left\{ \array{ \mathbb{Z} & dim X = 0 mod 4 \\ \mathbb{Z}_2 & dim X = 1, 2 mod 8 \\ 0 & otherwise } \right.$

theorem (Gromov-Lawson / Stolz) let $dim X \geq 5$ and

then $X$ admits a Riemannian metric of positive scalar curvature precisely when $\alpha(X) = 0$

These invariants share the following properties:

• they are additive under disjoint union of manifolds

• they are multiplicative under cartesian product of manifolds

• $e(X) mod 2, sgn(X), ind(D_X)$ all vanish when $X$ is a boundary, $\exists W : X = \partial W$, which means that $X$ is cobordant to the empty manifold $\emptyset$.

in other words, these invariants are genera, namely ring homomorphisms

$\Omega \to R$

form the cobordism ring $\Omega$ to some commutative ring $R$

• good genera are those which reflect geometric properties of $X$.

• now for $X$ a topological space consider the cobordism ring over $X$:

$\Omega(X) := \{(M,f)| f : M \stackrel{cont}{\to X}\}/_{bordism}$

where addition and multiplication are again given by disjoint union and cartesian product.

this assignment of rings to topological spaces is a generalized homology theory: cobordism homology theory

question given a genus $\Omega \to R$, can we find a homology theory $R(-)$ with $R = R(pt)$ its homology ring over the point and such that it all fits into a natural diagram

$\array{ \Omega &\to& R \\ \uparrow && \uparrow \\ \Omega(X) &\stackrel{\rho}{\to}& R(X) }$

This would be a parameterized extension $\rho = R(-)$ of $R$ .

Now let $X$ be a closed manifold.

consider $u_X : X \to K(\pi_1(X),1)$ (on the right an Eilenberg-MacLane space) which is the classifying map for the universal cover

$u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to} \pi_1(K(\pi_1(X), 1))$

then consider

$\rho_X[X, u_X] \in R(K(\pi_1(X),1))$

theorem (Julia Weber)

take the Euler characteristic mod 2, $Eu(X)$

$\array{ \Omega^0 &\stackrel{Eu(M)\cdot t^{dim M}}{\to}& \mathbb{Z}_2[t] \\ \uparrow && \uparrow \\ \Omega^0(X) &\to& Eu(X) & \simeq H_\bullet(X; \mathbb{Z}_2[t]) }$

for $X$ smooth we have then:

$Eu_X[X, id] = Poincare dual of total Stiefel-Whitney class$

theorem (Minalta)

something analogous for signature genus

$\array{ \Omega_\bullet^{SO} &\to& Sig_\bullet(X) }$

$sign_X[X,u] \in sig_\bullet(K) \otimes \mathbb{Q}$

this is the Novikov higher signature

now the same for the $\alpha$-genus

$\array{ \Omega_{X}^{Spin} &\stackrel{\alpha}{\to}& KO_\bullet(pt) \\ \uparrow && \uparrow \\ \Omega_\bullet^{Spin} &\to& KO_\bullet(X) }$

now towards elliptic genera: recall the notion of string structure of a manifold $X$: a lift of the structure map $X \to \mathcal{B}O(n)$ through the 4th connected universal cover $\mathcal{B}String(n) := \mathcal{B}O(n)\langle 4\rangle \to \mathcal{B} O$:

so consider String manifolds and the bordism ring $\Omega_\bullet^{String}$ of String manifold, let $M_\bullet$ be the ring of integral modular forms, then there is a genus – the Witten genus $W$

$\array{ \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet \\ \uparrow && \uparrow \\ \Omega_\bullet^{String}(X) &\to& M_\bullet(X) \\ &\searrow& \\ && tmf_\bullet(X) }$

where $\Omega_\bullet^{String}(pt) \to tmf_\bullet(pt)$ is surjective

conjecture (Stolz conjecture)

If a String manifold $Y$ has a positive Ricci curvature metric, then the Witten genus vanishes.

The attempted “Proof” of this is the motivation for the Stolz-Teichner-program for geometric models for elliptic cohomology:

“Proof” If $Y$ is String, then the loop space $L Y$ is has spin structure, so if $Y$ has positive Ricci curvature the $L Y$ has positive scalar curvature which implies by the above that $ind^{S^1} D_{L Y} = 0$ which by the index formula is the Witten genus.

### Equivariant elliptic cohomology and loop group representations

The analog of the orbit method with equivariant K-theory replaced by equivariant elliptic cohomology yields (aspects of) the representation theory of loop groups. (Ganter 12)

### 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

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$Calabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

## References

### General

The concept of elliptic cohomology originates around

motivated by Serge Ochanine’s concept of elliptic genus and by Edward Witten’s QFT/string theoretic explanation of the Ochanine genus and of the Witten genus (as the partition functions of the type II superstring and the heterotic superstring, respectively).

Accounts include

The concept of an elliptic spectrum representing an elliptic cohomology theory is due to

Modern accounts of (equivariant) elliptic cohomology in terms of stable homotopy theory include

Further discussion of equivariant elliptic cohomology and the relation to loop group representation theory is in

Surveys include

• Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn 2015 (web)

Discussion of generalization to higher chromatic homotopy theory is discussed in

• Douglas Ravenel, Toward higher chromatic analogs of elliptic cohomology pdf

• Douglas Ravenel, Toward higher chromatic analogs of elliptic cohomology II, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1{36 (pdf, pdf slides)

### Relation to quantum field theory

The elliptic genus and Witten genus were understood as the large volume limit of the partition function of the superstring in

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

The following reference discuss aspects of the construction of elliptic cohomology / tmf in terms of quantum field theory. See also at Witten genus.

• P Hu, Igor Kriz, Conformal field theory and elliptic cohomology, Advances in Mathematics, Volume 189, Issue 2, 20 December 2004, Pages 325–412 (pdf)

• Igor Kriz, Hisham Sati, M-theory, type IIA superstrings, and elliptic cohomology, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345–394 (Euclid, arXiv:hep-th/0404013)

A proposal for a construction via FQFT is discussed at

The case of elliptic cohomology associated with the Tate curve is discussed in

Revised on March 20, 2015 12:55:27 by Urs Schreiber (80.92.246.195)