ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 9

To summarize, we have maps

{ MU

∗

→

E∗

}

Quillen

{ MU → E }

❡❡❡❡❡❡

❨❨❨❨❨❨❨❨❨

FGL(E∗)

where

E∗

can be any graded ring. Given a formal group law, can we construct a

complex oriented cohomology theory with that formal group law? We will return

to this question in Section 4 and see that in certain cases we can construct such a

cohomology theory.

First, we discuss formal group laws from the slightly different viewpoint of

complex genera. A genus is some multiplicative bordism invariant associated to

manifolds. There are two main types of genera, and this is due to the description

of the cobordism groups from (1).

Definition 2.5. A complex genus is a ring homomorphism

ϕ : MU

∗

→ R.

An oriented genus (or usually just genus) is a ring homomorphism

ϕ :

MSO∗

⊗ Q → R,

where R is a Q-algebra. More explicitly, ϕ(M) only depends on the cobordism class

of M and satisfies

ϕ(M1 M2) = ϕ(M1) + ϕ(M2), ϕ(M1 × M2) = ϕ(M1)ϕ(M2).

Quillen’s theorem implies there is a 1-1 correspondence between formal group

laws over R and complex genera over R. We introduce some common terminology

which will make this correspondence more concrete.

First, a homomorphism between formal group laws F

f

→ G (over R) is a power

series f(x) ∈ R x such that

f(F (x1,x2)) = G(f(x1),f(x2)).

If f is invertible then it is considered an isomorphism, and f is a strict isomorphism

if f(x) = x + higher order terms.

Example 2.6. We could have chosen complex orientation of K-theory so that

c1(L) = [L] − 1 as opposed to 1 − [L]. The resulting formal group law would

have been F (x1,x2) = x1 + x2 + x1x2, which is also sometimes defined as the

multiplicative formal group law. These two formal group laws are (non-strictly)

isomorphic, with isomorphism given by f(x) = −x. Our original choice, though,

coincides with the Todd genus and with conventions in index theory.

Remark 2.7. In general, our formal group law depends on the particular com-

plex orientation. Two different orientations will lead to an isomorphism between

the formal group laws. More abstractly, to any complex orientable theory is canon-

ically associated a formal group. The choice of orientation gives a coordinate for

the formal group, and the formal group expanded in this coordinate is the formal

group law.