nLab
elliptic cohomology

Idea

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

There are precisely three types of such formal group laws:

• the simple additive group structure – this corresponds to standard integral cohomology given by the Eilenberg-MacLane spectrum;

• the multiplicative group that corresponds to complex K-theory

• the formal group law on elliptic curve.

An elliptic cohomology theory is a periodic multiplicative generalized (Eilenberg-Steenrod) cohomology theory whose corresponding formal group is an elliptic curve.

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.

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 $e(X)\in \mathbb{Z}$

  • takes all values in $\mathbb{Z}$

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

    $$(e(X)=0)\iff (\exists v\in \Gamma (TX):\forall x\in X:v(x)\ne 0)$$(e(X)= 0) \Leftrightarrow (\exists v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)

• signature? $\mathrm{sgn}(X)$

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

• when $X$ has a spin structure

  the index of the Dirac operator? $D$:

  $$\mathrm{ind}{D}_X\in \mathbb{Z}$$ind D_X \in \mathbb{Z}
  $$\alpha (D)\in \{\begin{array}{cc}\mathbb{Z}& \dim X=0\mathrm{mod}4\\ {\mathbb{Z}}_2& \dim X=1,2\mathrm{mod}8\\ 0& \mathrm{otherwise}\end{array}$$\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 $\dim X\ge 5$ 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

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

  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 $X$:

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

  $$\begin{array}{ccc}\Omega & \to & R\\ \uparrow & & \uparrow \\ \Omega (X)& \stackrel{\rho }{\to }& R(X)\end{array}$$\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 $u_X:X\to K({\pi }_1(X)\mathrm{,1})$ (on the right an Eilenberg-MacLane space) which is the classifying map for the universal cover

  $$u_{*}{\pi }_1(X)\stackrel{{\simeq }_{\mathrm{canon}}}{\to }{\pi }_1(K({\pi }_1(X),1))$$u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to} \pi_1(K(\pi_1(X), 1))

  then consider

  $${\rho }_X[X,u_X]\in R(K({\pi }_1(X)\mathrm{,1}))$$\rho_X[X, u_X] \in R(K(\pi_1(X),1))

  theorem (Julia Weber)

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

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

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

  theorem (Minalta)

  someting analogous for signature genus

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

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

  this is the Novikov higher signature

  now the same for the $\alpha$-genus

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

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? $W$–

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

where ${\Omega }_{•}^{\mathrm{String}}(\mathrm{pt})\to {\mathrm{tmf}}_{•}(\mathrm{pt})$ 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?.

References

• Jacob Lurie, A Survey of Elliptic Cohomology