Complex-oriented cohomology theory

In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, a complex-orientable cohomology theory is a <a href="/facts/Multiplicative_cohomology_theory/J7FJzTNh">multiplicative cohomology theory</a> 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})}
 
 is surjective. An element of 
 
 
 
 
 E
 
 2
 
 
 (
 
 C
 
 
 
 P
 
 
 ∞
 
 
 )
 
 
 {\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })}
 
 that restricts to the canonical generator of the reduced theory 
 
 
 
 
 
 
 
 E
 ~
 
 
 
 
 2
 
 
 (
 
 C
 
 
 
 P
 
 
 1
 
 
 )
 
 
 {\displaystyle {\widetilde {E}}^{2}(\mathbb {C} \mathbf {P} ^{1})}
 
 is called a complex orientation. The notion is central to Quillen's work relating cohomology to <a href="/facts/Formal_group_law/xUnVSCC5">formal group laws</a>.
If E is an even-graded theory meaning 
 
 
 
 
 π
 
 3
 
 
 E
 =
 
 π
 
 5
 
 
 E
 =
 ⋯
 
 
 {\displaystyle \pi _{3}E=\pi _{5}E=\cdots }
 
, then E is complex-orientable. This follows from the <a href="/facts/Atiyah%E2%80%93Hirzebruch_spectral_sequence/2cwxuanD">Atiyah–Hirzebruch spectral sequence</a>. 
Examples:

<ul><li>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)}
 
.</li>
<li>Complex K-theory, denoted KU, is complex-orientable, as it is even-graded. (<a href="/facts/Bott_periodicity_theorem/GqgJRllU">Bott periodicity theorem</a>)</li>
<li><a href="/facts/Complex_cobordism/tAkFKIL1">Complex cobordism</a>, whose spectrum is denoted by MU, is complex-orientable.</li></ul>
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]}
 
 denotes a line passing through x in the underlying vector space 
 
 
 
 
 C
 
 [
 t
 ]
 
 
 {\displaystyle \mathbb {C} [t]}
 
 of 
 
 
 
 
 C
 
 
 
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}
 
. This is the map classifying the tensor product of the universal line bundle over 
 
 
 
 
 C
 
 
 
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}
 
. Viewing

E
 
 ∗
 
 
 (
 
 C
 
 
 
 P
 
 
 ∞
 
 
 )
 =
 
 
 lim
 ←
 
 
 ⁡
 
 E
 
 ∗
 
 
 (
 
 C
 
 
 
 P
 
 
 n
 
 
 )
 =
 
 
 lim
 ←
 
 
 ⁡
 R
 [
 t
 ]
 
 /
 
 (
 
 t
 
 n
 +
 1
 
 
 )
 =
 R
 [
 
 [
 t
 ]
 
 ]
 ,
 
 R
 =
 
 π
 
 ∗
 
 
 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}
 
,
let 
 
 
 
 f
 =
 
 m
 
 ∗
 
 
 (
 t
 )
 
 
 {\displaystyle 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
        ]
        
        ]
      
    
    {\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, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).

Complex-oriented cohomology theory open-in-new

Complex-oriented cohomology theory