Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map $$E^2(\mathbb{C}\mathbf{P}^\infty) \to E^2(\mathbb{C}\mathbf{P}^1)$$ is surjective. An element of $$E^2(\mathbb{C}\mathbf{P}^\infty)$$ that restricts to the canonical generator of the reduced theory $$\widetilde{E}^2(\mathbb{C}\mathbf{P}^1)$$ 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 $$\pi_3 E = \pi_5 E = \cdots$$, 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 $$\operatorname{H}^2(\mathbb{C}\mathbf{P}^\infty; R) \simeq \operatorname{H}^2(\mathbb{C}\mathbf{P}^1;R)$$.
 * 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
 * $$\mathbb{C}\mathbf{P}^\infty \times \mathbb{C}\mathbf{P}^\infty \to \mathbb{C}\mathbf{P}^\infty, ([x], [y]) \mapsto [xy]$$

where $$[x]$$ denotes a line passing through x in the underlying vector space $$\mathbb{C}[t]$$ of $$\mathbb{C}\mathbf{P}^\infty$$. This is the map classifying the tensor product of the universal line bundle over $$ \mathbb{C}\mathbf{P}^\infty $$. Viewing
 * $$E^*(\mathbb{C}\mathbf{P}^\infty) = \varprojlim E^*(\mathbb{C}\mathbf{P}^n) = \varprojlim R[t]/(t^{n+1}) = R[\![t]\!], \quad R =\pi_* E $$,

let $$f = m^*(t)$$ be the pullback of t along m. It lives in
 * $$E^*(\mathbb{C}\mathbf{P}^\infty \times \mathbb{C}\mathbf{P}^\infty) = \varprojlim E^*(\mathbb{C}\mathbf{P}^n \times \mathbb{C}\mathbf{P}^m) = \varprojlim R[x,y]/(x^{n+1},y^{m+1}) = R[\![x, y]\!]$$

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