Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map E2(ℂ𝐏∞)->E2(ℂ𝐏1) is surjective. An element of E2(ℂ𝐏∞) that restricts to the canonical generator of the reduced theory E~2(ℂ𝐏1) is called a complex orientation. The notion is central to Quillen's work relating cohomology to formal group laws.[citation needed] If E is an even-graded theory meaning π3E=π5E=⋯, 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 H2(ℂ𝐏∞;R)≃H2(ℂ𝐏1;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

ℂ𝐏∞×ℂ𝐏∞->ℂ𝐏∞,([x],[y])↦[xy]

where [x] denotes a line passing through x in the underlying vector space ℂ[t] of ℂ𝐏∞. This is the map classifying the tensor product of the universal line bundle over ℂ𝐏∞. Viewing

E*(ℂ𝐏∞)=lim<-E*(ℂ𝐏n)=lim<-R[t]/(tn+1)=R[[t]],R=π*E,

let f=m*(t) be the pullback of t along m. It lives in

E*(ℂ𝐏∞×ℂ𝐏∞)=lim<-E*(ℂ𝐏n×ℂ𝐏m)=lim<-R[x,y]/(xn+1,ym+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).

## See also

* Chromatic homotopy theory

## References

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

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