group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
This is a sub-entry of
see there for background and context.
This entry reviews basics of periodic multiplicative cohomology theories and their relation to formal group laws.
Next:
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
whereas in the second case we get
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
such that under the map
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
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: $E^{2k+1}(X) = 0$.
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
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
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
this $f$ is called a formal group law if the following conditions are satisfied
commutativity $f(x,y) = f(y,x)$
identity $f(x,0) = x$
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
The Lazard ring is the “universal formal group law”. it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$
and relatins as follows
$a_{i j} = a_{j i}$
$a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$
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
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
periodic complex cobordism cohomology theory is given by
we get a canonical orientation? from
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
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).