Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map ${\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })\to E^{2}(\mathbb {C} \mathbf {P} ^{1})}$is surjective. An element of ${\displaystyle E^{2}(\mathbb {C} \mathbf {P} ^{\infty })}$that restricts to the canonical generator of the reduced theory ${\displaystyle {\widetilde {E}}^{2}(\mathbb {C} \mathbf {P} ^{1})}$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 ${\displaystyle \pi _{3}E=\pi _{5}E=\cdots }$, 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 ${\displaystyle \operatorname {H} ^{2}(\mathbb {C} \mathbf {P} ^{\infty };R)\simeq \operatorname {H} ^{2}(\mathbb {C} \mathbf {P} ^{1};R)}$.
• 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

${\displaystyle \mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty }\to \mathbb {C} \mathbf {P} ^{\infty },([x],[y])\mapsto [xy]}$

where ${\displaystyle [x]}$denotes a line passing through x in the underlying vector space ${\displaystyle \mathbb {C} [t]}$of ${\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}$. This is the map classifying the tensor product of the universal line bundle over ${\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}$. Viewing

${\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n})=\varprojlim R[t]/(t^{n+1})=R[\![t]\!],\quad R=\pi _{*}E}$,

let ${\displaystyle f=m^{*}(t)}$be the pullback of t along m. It lives in

${\displaystyle E^{*}(\mathbb {C} \mathbf {P} ^{\infty }\times \mathbb {C} \mathbf {P} ^{\infty })=\varprojlim E^{*}(\mathbb {C} \mathbf {P} ^{n}\times \mathbb {C} \mathbf {P} ^{m})=\varprojlim R[x,y]/(x^{n+1},y^{m+1})=R[\![x,y]\!]}$

and one can show, using properties of the tensor product of line bundles, it is a formal group law (e.g., satisfies associativity).

• Chromatic homotopy theory

• M. Hopkins, Complex oriented cohomology theory and the language of stacks
• J. Lurie, Chromatic Homotopy Theory (252x)