ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 7
In addition to the above proposition, the splitting principle carries over, and
the class c1 uniquely determines isomorphisms
× · · · ×
x1, · · · , xn
c1, · · · , cn ,
where ck ∈
is the k-th elementary symmetric polynomial in the vari-
ables xi. This gives us a theory of Chern classes analogous to the one in ordinary
2. Formal group laws and genera
A complex orientation of E determines Chern classes for complex vector bun-
dles. As in ordinary cohomology, the Chern classes satisfy the properties of natu-
rality and additivity. In ordinary cohomology, the first Chern class of a product of
line bundles is given by
c1(L1 ⊗ L2) = c1(L1) + c1(L2).
For a general complex-oriented cohomology theory, this relation no longer holds
and leads to an interesting structure.
The universal tensor product is classified by
ξ ⊗ ξ ξ
The induced map in cohomology,
F (x1,x2) ←− c1
where F (x1,x2) is a formal power series in two variables over the ring
c1(L1 ⊗ L2) = F (c1(L1),c1(L2)).
This formal power series F is an example of a formal group law over the graded
Definition 2.1. A formal group law over a ring R is a formal power series
F ∈ R x1,x2 satisfying the following conditions:
• F (x, 0) = F (0,x) = x (Identity)
• F (x1,x2) = F (x2,x1) (Commutativity)
• F (F (x1,x2),x3) = F (x1,F (x2,x3)) (Associativity)
If R is a graded ring, we require F to be homogeneous of degree 2 where |x1| =
|x2| = 2.
One easily verifies that the power series giving c1(L1 ⊗ L2) is a formal group
law. The three properties in the definition follow immediately from the natural
transformations which give the identity, commutativity, and associativity properties
of the tensor product.