ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 7

In addition to the above proposition, the splitting principle carries over, and

the class c1 uniquely determines isomorphisms

E∗(BU(n))

∼

=

(E∗(CP∞

× · · · ×

CP∞))Σn

∼

=

(

E∗(pt)

x1, · · · , xn

)Σn

∼

=

E∗(pt)

c1, · · · , cn ,

where ck ∈

E2k(BU(n))

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

cohomology.

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

CP∞

×

CP∞ CP∞.

The induced map in cohomology,

E∗(CP∞)

⊗

E∗(CP∞) ←−E∗(CP∞)

F (x1,x2) ←− c1

where F (x1,x2) is a formal power series in two variables over the ring

E∗,

gives the

universal formula

c1(L1 ⊗ L2) = F (c1(L1),c1(L2)).

This formal power series F is an example of a formal group law over the graded

ring

E∗.

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.