10 CORBETT REDDEN

Over a Q-algebra, any formal group law is uniquely (strictly) isomorphic to the

additive formal group law F+. We denote this isomorphism by logF and its inverse

by expF :

F

logF

F+

expF

The isomorphism logF can be solved by the following:

f(F (x1,x2)) = f(x1) + f(x2)

∂

∂x2

x,0

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

∂

∂x2

x,0

(f(x1) + f(x2))

f (x)

∂F

∂x2

(x, 0) = 1

logF (x) = f(x) =

x

0

dt

∂F

∂x2

(t, 0)

(2)

Going from the third to fourth line involves inverting a power series, so one must

work over a Q-algebra. If R is torsion-free, then R → R ⊗ Q is an injection, and

we lose no information in considering logF instead of F itself.

Over the ring MU

∗

⊗ Q, the universal formal group law FMU coming from

complex cobordism has the particularly nice logarithm

logFMU (x) =

n≥0

[CPn]

n + 1

xn.

Therefore, a formal group law F (or a complex genus) induced by ϕ : MU

∗

→ R

has a logarithm

logF (x) =

n≥0

ϕ([CPn])

n + 1

xn.

While (modulo torsion in R) the logarithm encodes the value of a genus on any

complex manifold, in practice it is diﬃcult to decompose the bordism class of a

manifold into projective spaces. However, there is an easier approach to calculating

genera due to work of Hirzebruch.

Proposition 2.8. (Hirzebruch) For R a Q-algebra, there are bijections

{Q(x) = 1 + a1x +

a2x2

+ · · · ∈ R x } ←→ {ϕ : MU

∗

⊗ Q → R}

{Q(x) = 1 +

a2x2

+

a4x4

+ · · · ∈ R x | aodd = 0} ←→ {ϕ :

MSO∗

⊗ Q → R}

The first bijection is given by the following construction. Given Q(x), to a

complex line bundle L → X assign the cohomology class

ϕQ(L) := Q(c1(L)) ∈

H∗(X;

R).

Using the splitting principle, ϕQ extends to a stable exponential characteristic class

on all complex vector bundles. The complex genus ϕ generated by Q(x) is then

defined by

ϕ(M) := ϕQ(TM), [M] ∈ R,

where M is a stably almost complex manifold, , is the natural pairing between

cohomology and homology, and [M] is the fundamental class (an almost complex