Complex Cobordism - Formal Group Laws

Formal Group Laws

Milnor (1960) and Novikov (1960, 1962) showed that the coefficient ring π_*(MU) (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring Z on infinitely many generators x_i ∈ π_2i(MU) of positive even degrees.

Write CP∞ for infinite dimensional complex projective space, which is the classifying space for complex line bundles, so that tensor product of line bundles induces a map μ:CP∞× CP∞ → CP∞. A complex orientation on an associative commutative ring spectrum E is an element x in E2(CP∞) whose restriction to E2(CP1) is 1, if the latter ring is identified with the coefficient ring of E. A spectrum E with such an element x is called a complex oriented ring spectrum.

If E is a complex oriented ring spectrum, then

and μ*(x) ∈ E*(point)] is a formal group law over the ring E*(point) = π*(E).

Complex cobordism has a natural complex orientation. Quillen (1969) showed that there is a natural isomorphism from its coefficient ring to Lazard's universal ring, making the formal group law of complex cobordism into the universal formal group law. In other words, for any formal group law F over any commutative ring R, there is a unique ring homomorphism from MU*(point) to R such that F is the pullback of the formal group law of complex cobordism.