cohomology

# Contents

## Idea

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

There are precisely three types of such formal group laws:

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

A theorem proven by Goerss-Hopkins-Miller and later in a different way by Jacob Lurie 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 and the elliptic genus

rough material , to be polished

Some topological invariants of manifolds that are of interest:

we restricted attention to closed connected smooth manifolds $X$

• the Euler characteristic $e\left(X\right)\in ℤ$

• takes all values in $ℤ$

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

$\left(e\left(X\right)=0\right)⇔\left(\exists v\in \Gamma \left(TX\right):\forall x\in X:v\left(x\right)\ne 0\right)$(e(X)= 0) \Leftrightarrow (\exists v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)
• signature $\mathrm{sgn}\left(X\right)$

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 $\mathrm{sgn}\left(X\right)=0$

• when $X$ has a spin structure

the index of the Dirac operator $D$:

$\mathrm{ind}{D}_{X}\in ℤ$ind D_X \in \mathbb{Z}
$\alpha \left(D\right)\in \left\{\begin{array}{cc}ℤ& \mathrm{dim}X=0\mathrm{mod}4\\ {ℤ}_{2}& \mathrm{dim}X=1,2\mathrm{mod}8\\ 0& \mathrm{otherwise}\end{array}$\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 $\mathrm{dim}X\ge 5$ and

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

These invariants share the following properties:

• they are additive under disjoint union of manifolds

• they are multiplicative under cartesian product of manifolds

• $e\left(X\right)\mathrm{mod}2,\mathrm{sgn}\left(X\right),\mathrm{ind}\left({D}_{X}\right)$ all vanish when $X$ is a boundary, $\exists W:X=\partial W$, which means that $X$ is cobordant? to the empty manifold $\varnothing$.

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

$\Omega \to R$\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 $X$:

$\Omega \left(X\right):=\left\{\left(M,f\right)\mid f:M\stackrel{\mathrm{cont}}{\to X}\right\}{/}_{\mathrm{bordism}}$\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 $R\left(-\right)$ with $R=R\left(\mathrm{pt}\right)$ its homology ring? over the point and such that it all fits into a natural diagram

$\begin{array}{ccc}\Omega & \to & R\\ ↑& & ↑\\ \Omega \left(X\right)& \stackrel{\rho }{\to }& R\left(X\right)\end{array}$\array{ \Omega &\to& R \\ \uparrow && \uparrow \\ \Omega(X) &\stackrel{\rho}{\to}& R(X) }

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

Now let $X$ be a closed manifold.

consider ${u}_{X}:X\to K\left({\pi }_{1}\left(X\right),1\right)$ (on the right an Eilenberg-MacLane space) which is the classifying map for the universal cover

${u}_{*}{\pi }_{1}\left(X\right)\stackrel{{\simeq }_{\mathrm{canon}}}{\to }{\pi }_{1}\left(K\left({\pi }_{1}\left(X\right),1\right)\right)$u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to} \pi_1(K(\pi_1(X), 1))

then consider

${\rho }_{X}\left[X,{u}_{X}\right]\in R\left(K\left({\pi }_{1}\left(X\right),1\right)\right)$\rho_X[X, u_X] \in R(K(\pi_1(X),1))

theorem (Julia Weber)

take the Euler characteristic mod 2, $\mathrm{Eu}\left(X\right)$

$\begin{array}{ccc}{\Omega }^{0}& \stackrel{\mathrm{Eu}\left(M\right)\cdot {t}^{\mathrm{dim}M}}{\to }& {ℤ}_{2}\left[t\right]\\ ↑& & ↑\\ {\Omega }^{0}\left(X\right)& \to & \mathrm{Eu}\left(X\right)& \simeq {H}_{•}\left(X;{ℤ}_{2}\left[t\right]\right)\end{array}$\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:

${\mathrm{Eu}}_{X}\left[X,\mathrm{id}\right]=\mathrm{Poincare}\mathrm{dual}\mathrm{of}\mathrm{total}\mathrm{Stiefel}-\mathrm{Whitney}\mathrm{class}$Eu_X[X, id] = Poincare dual of total Stiefel-Whitney class

theorem (Minalta)

someting analogous for signature genus

$\begin{array}{ccc}{\Omega }_{•}^{\mathrm{SO}}& \to & {\mathrm{Sig}}_{•}\left(X\right)\end{array}$\array{ \Omega_\bullet^{SO} &\to& Sig_\bullet(X) }

${\mathrm{sign}}_{X}\left[X,u\right]\in {\mathrm{sig}}_{•}\left(K\right)\otimes ℚ$

this is the Novikov higher signature

now the same for the $\alpha$-genus

$\begin{array}{ccc}{\Omega }_{X}^{\mathrm{Spin}}& \stackrel{\alpha }{\to }& {\mathrm{KO}}_{•}\left(\mathrm{pt}\right)\\ ↑& & ↑\\ {\Omega }_{•}^{\mathrm{Spin}}& \to & {\mathrm{KO}}_{•}\left(X\right)\end{array}$\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 $X\to ℬO\left(n\right)$ through the 4th connected universal cover $ℬ\mathrm{String}\left(n\right):=ℬO\left(n\right)⟨4⟩\to ℬO$:

so consider String manifolds and the bordism ring ${\Omega }_{•}^{\mathrm{String}}$ of String manifold, let ${M}_{•}$ be the ring of integral modular forms, then there is a genus – the Witten genus $W$

$\begin{array}{ccc}{\Omega }_{•}^{\mathrm{String}}& \stackrel{W}{\to }& {M}_{•}\\ ↑& & ↑\\ {\Omega }_{•}^{\mathrm{String}}\left(X\right)& \to & {M}_{•}\left(X\right)\\ & ↘& \\ & & {\mathrm{tmf}}_{•}\left(X\right)\end{array}$\array{ \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet \\ \uparrow && \uparrow \\ \Omega_\bullet^{String}(X) &\to& M_\bullet(X) \\ &\searrow& \\ && tmf_\bullet(X) }

where ${\Omega }_{•}^{\mathrm{String}}\left(\mathrm{pt}\right)\to {\mathrm{tmf}}_{•}\left(\mathrm{pt}\right)$ is surjective

conjecture (Höhn, Stolz)

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 $LY$ is has spin structure, so if $Y$ has positive Ricci curvature the $LY$ has positive scalar curvature which implies by the above that ${\mathrm{ind}}^{{S}^{1}}{D}_{LY}=0$ 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 levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $Hℤ$
0th Morava K-theory$K\left(0\right)$
1complex K-theorycomplex K-theory spectrum $\mathrm{KU}$
first Morava K-theory$K\left(1\right)$
first Morava E-theory$E\left(1\right)$
2elliptic cohomology${\mathrm{Ell}}_{E}$
tmftmf spectrum
second Morava K-theory$K\left(2\right)$
second Morava E-theory$E\left(2\right)$
algebraic K-theory of KU$K\left(\mathrm{KU}\right)$
$n$$n$th Morava K-theory$K\left(n\right)$
$n$th Morava E-theory$E\left(n\right)$
$n+1$algebraic K-theory applied to chrom. level $n$$K\left({E}_{n}\right)$ (red-shift conjecture)
$\infty$

## References

Construction of elliptic cohomology / tmf by FQFT is discussed at (2,1)-dimensional Euclidean field theories and tmf. The case of elliptic cohomology associated with the Tate curve is discussed in

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

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)

Revised on November 12, 2013 12:33:31 by Urs Schreiber (188.200.54.65)