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 } ←→ : MU

Q R}
{Q(x) = 1 +
a2x2
+
a4x4
+ · · · R x | aodd = 0} ←→ :
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
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