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})} 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 formal group laws.

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 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 ) {\displaystyle \operatorname {H} ^{2}(\mathbb {C} \mathbf {P} ^{\infty };R)\simeq \operatorname {H} ^{2}(\mathbb {C} \mathbf {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 ] {\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).