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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
)
{\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })\to E^{2}(\mathbb {C} \mathbf {P} ^{1})}
E
2
(
C
P
∞
)
{\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })}
E
~
2
(
C
P
1
)
{\displaystyle {\widetilde {E}}^{2}(\mathbb {C} \mathbf {P} ^{1})}
complex orientation . The notion is central to Quillen's work relating cohomology to formal group laws .[citation needed
If
π
3
E
=
π
5
E
=
⋯
{\displaystyle \pi _{3}E=\pi _{5}E=\cdots }
E is complex-orientable.
Examples:
An ordinary cohomology with any coefficient ring R is complex orientable, as
H
2
(
C
P
∞
;
R
)
≃
H
2
(
C
P
1
;
R
)
{\displaystyle \operatorname {H} ^{2}(\mathbb {C} \mathbf {P} ^{\infty };R)\simeq \operatorname {H} ^{2}(\mathbb {C} \mathbf {P} ^{1};R)}
A complex K -theory, denoted by K , is complex-orientable, as
π
3
K
=
π
5
K
=
⋯
=
0
{\displaystyle \pi _{3}K=\pi _{5}K=\cdots =0}
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
]
{\displaystyle \mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty }\to \mathbb {C} \mathbf {P} ^{\infty },([x],[y])\mapsto [xy]}
where
[
x
]
{\displaystyle [x]}
x in the underlying vector space
C
[
t
]
{\displaystyle \mathbb {C} [t]}
C
P
∞
{\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}
E
∗
(
C
P
∞
)
=
lim
←
E
∗
(
C
P
n
)
=
lim
←
R
[
t
]
/
(
t
n
+
1
)
=
R
[
[
t
]
]
,
R
=
π
∗
E
=
⊕
π
2
n
E
{\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n})=\varprojlim R[t]/(t^{n+1})=R[\![t]\!],\quad R=\pi _{*}E=\oplus \pi _{2n}E}
let
f
=
m
∗
(
t
)
{\displaystyle f=m^{*}(t)}
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
]
]
{\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n}\times \mathbb {C} \mathbf {P} ^{m})=\varprojlim R[x,y]/(x^{n+1},y^{m+1})=R[\![x,y]\!]}
and one can show it is a formal group law (e.g., satisfies associativity).
See also
References
![{\displaystyle \mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty }\to \mathbb {C} \mathbf {P} ^{\infty },([x],[y])\mapsto [xy]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c614db1b01f15c09195dd0144473ce030ca2a65d)
![{\displaystyle [x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07548563c21e128890501e14eb7c80ee2d6fda4d)
![{\displaystyle \mathbb {C} [t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d83161b277d877a8dd4ad77c9884af86de11c2e4)
![{\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n})=\varprojlim R[t]/(t^{n+1})=R[\![t]\!],\quad R=\pi _{*}E=\oplus \pi _{2n}E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df1f24609d661254e08538424e667bd1188397e1)
![{\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n}\times \mathbb {C} \mathbf {P} ^{m})=\varprojlim R[x,y]/(x^{n+1},y^{m+1})=R[\![x,y]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c60771f248d7e9c5487636040d0b4238f30c302)
