In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map
E
2
(
C
P
∞
)
→
E
2
(
C
P
1
)
is surjective. An element of
E
2
(
C
P
∞
)
that restricts to the canonical generator of the reduced theory
E
~
2
(
C
P
1
)
is called a complex orientation. The notion is central to Quillen's work relating cohomology to formal group laws.
If E is an even-graded theory meaning
π
3
E
=
π
5
E
=
⋯
, then E is complex-orientable. This follows from the Atiyah–Hirzebruch spectral sequence.
Examples:
An ordinary cohomology with any coefficient ring R is complex orientable, as
H
2
(
C
P
∞
;
R
)
≃
H
2
(
C
P
1
;
R
)
.
Complex K-theory, denoted KU, is complex-orientable, as it is even-graded. (Bott periodicity theorem)
Complex cobordism, whose spectrum is denoted by MU, is complex-orientable.
A complex orientation, call it t, gives rise to a formal group law as follows: let m be the multiplication
C
P
∞
×
C
P
∞
→
C
P
∞
,
(
[
x
]
,
[
y
]
)
↦
[
x
y
]
where
[
x
]
denotes a line passing through x in the underlying vector space
C
[
t
]
of
C
P
∞
. This is the map classifying the tensor product of the universal line bundle over
C
P
∞
. Viewing
E
∗
(
C
P
∞
)
=
lim
←
E
∗
(
C
P
n
)
=
lim
←
R
[
t
]
/
(
t
n
+
1
)
=
R
[
[
t
]
]
,
R
=
π
∗
E
,
let
f
=
m
∗
(
t
)
be the pullback of t along m. It lives in
E
∗
(
C
P
∞
×
C
P
∞
)
=
lim
←
E
∗
(
C
P
n
×
C
P
m
)
=
lim
←
R
[
x
,
y
]
/
(
x
n
+
1
,
y
m
+
1
)
=
R
[
[
x
,
y
]
]
and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).
