nLab A Survey of Elliptic Cohomology - cohomology theories

This is a sub-entry of

• A Survey of Elliptic Cohomology

see there for background and context.

This entry reviews basics of periodic multiplicative cohomology theories and their relation to formal group laws.

Next:

• A Survey of Elliptic Cohomology - formal groups and cohomology

rough notes from a talk

the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention

A complex oriented cohomology theory (meant is here and in all of the following a generalized (Eilenberg-Steenrod) cohomology) is one with a good notion of Thom classes, equivalently first Chern class for complex vector bundle

(this “good notion” will boil down to certain extra assumptions such as multiplicativity and periodicity etc. What one needs is that the cohomology ring assigned by the cohomology theory to $\mathbb{C}P^\infty \simeq \mathcal{B}U(1)$ is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).

ordinary Chern class lives in integral cohomology $H^*(-,\mathbb{Z})$

or in K-theory $K^*(-)$ where for a vector bundle $V$ we would set $c_1(V) := ([V]-1)\beta$ where $\beta$ is the Bott generator.

In the first case we have that under tensor product of vector bundles the class behaves as

$$c_1(V\otimes W) = c_1(V) + c_1(W)$$

whereas in the second case we get

$$c_1(V \otimes W) = c_1(V)c_1(W)\beta^{-1} + c_1(V) + c_1(W) \,.$$

In general we will get that the Chern class of a tensor product is given by a certain power series $E^*(pt)$

not all formal group laws arises this way. the Landweber criterion gives a condition under which there is a cohomology theory

definition of complex-orientation

there is an

$$x \in \tilde E^2(\mathbb{C}P^\infty)$$

such that under the map

$$\tilde E^2(\mathbb{C}P^\infty) \to \tilde E^2(\mathbb{C}P^1) \simeq \tilde E^2(S^1) \simeq E^0({*})$$

induced by

$\mathbb{C}P^1 \to \mathbb{C}P^\infty$

we have $x \mapsto 1$

remark this also gives Thom classes since $\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma$ is a homotopy equivalence

$$\tilde E^2((\mathbb{C}P^\infty)^\gamma) \simeq \tilde E^2((\mathbb{C}P^\infty)) \ni X$$

Thom iso $\tilde H^{*+2}(X^\gamma) \simeq H^*(X)$

…

(here and everywhere the tilde sign is for reduced cohomology)

definition (Bott element and even periodic cohomology theory)

• An even cohomology theory is one whose odd cohomology rings vanish: $E^{2k+1}({*}) = 0$.

• A periodic cohomology theory is one with a Bott element $\beta \in E^2({*})$ which is invertible (under multiplication in the cohomology ring of the point)

  so that gives an isomorphism $(-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*})$

Periodic cohomology theories are complex-orientable. $E^*(\mathbb{C}P^\infty)$ can be calculated using the Atiyah-Hirzebruch spectral sequence

$$H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)$$

notice that since $\mathbb{C}P^\infty$ is homotopy equivalent to the classifying space $\mathcal{B}U(1)$ (which is a topological group) it has a product on it

$$\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$$

which is the one that induces the tensor product of line bundles classified by maps into $\mathbb{C}P^\infty$.

on (at least on even periodic cohomology theories) this induces a map of the form

$$\array{ \mathbb{C}P^\infty \times \mathbb{C}P^\infty &\to& \mathbb{C}P^\infty \\ E({*})[[x,y]] &\leftarrow& E(*)[[t]] \\ f(x,y) &\leftarrow |& t }$$

this $f$ is called a formal group law if the following conditions are satisfied

1. commutativity $f(x,y) = f(y,x)$

2. identity $f(x,0) = x$

3. associtivity f(x,f(y,z)) = f(f(x,y),z)

remark the second condition implies that the constant term in the power series $f$ is 0, so therefore all these power series are automatically invertible and hence there is no further need to state the existence of inverses in the formal group. So these $f$ always start as

$$f(x,y) = x + y + \cdots$$

The Lazard ring is the “universal formal group law”. it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$

$$L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)$$

and relatins as follows

1. $a_{i j} = a_{j i}$

2. $a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$

3. the obvious associativity relation

the universal formal group law we get from this is the power series in $x,y$ with coefficients in the Lazard ring

$$\ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]] \,.$$

remark the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements $x$. But the underlying formal group is independent of this choice and well defined.

For any ring $S$ with formal group law $g(x,y) \in power series in x,y with coefficients in S$ there is a unique morphism $L \to S$ that sends $\ell$ to $g$.

remark Quillen’s theorem says that the Lazard ring is the ring of complex cobordisms

some universal cohomology theories $M U$ is the spectrum for complex cobordism cohomology theory. The corresponding spectrum is in degree $2 n$ given by

$$M U(2n) = Thom \left( standard associated bundle to universal bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right)$$

periodic complex cobordism cohomology theory is given by

$$M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U$$

we get a canonical orientation? from

$$\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty)$$

this is the universal even periodic cohomology theory with orientation

Theorem (Quillen) the cohomology ring $M P(*)$ of periodic complex cobordism cohomology theory over the point together with its formal group law is naturally isomorphic to the universal Lazard ring with its formal group law $(L,\ell)$

how one might make a formal group law $(R,f(x,y))$ into a cohomology theory

use the classifying map $M P({*}) \to R$ to build the tensor product

$$E^n(X) := M P^n(X) \otimes_{M P({*})} R$$

this construction could however break the left exactness condition. However, $E$ built this way will be left exact of the ring morphism $M P({*}) \to R$ is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger).