Complex cobordism

Formal group laws

John Milnor (1960) and Sergei Novikov (1960, 1962) showed that the coefficient ring ${\displaystyle {\pi }_{\ast }(\mathrm{MU})}$ (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring ${\displaystyle \mathbb{Z}[x_1,x_2,\dots ]}$ on infinitely many generators ${\displaystyle x_i\in {\pi }_{2i}(\mathrm{MU})}$ of positive even degrees.

Write ${\displaystyle {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}}$ 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 ${\displaystyle \mu :{\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}\times {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}\to {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}.}$ A complex orientation on an associative commutative ring spectrum E is an element x in ${\displaystyle E^2({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})}$ whose restriction to ${\displaystyle E^2({\mathbb{C}\mathbb{P}}^1)}$ 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

${\displaystyle E^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})=E^{\ast }(\text{point})[[x]]}$

${\displaystyle E^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})\times E^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }})=E^{\ast }(\text{point})[[x\otimes 1,1\otimes x]]}$

and ${\displaystyle {\mu }^{\ast }(x)\in E^{\ast }(\text{point})[[x\otimes 1,1\otimes x]]}$ is a formal group law over the ring ${\displaystyle E^{\ast }(\text{point})={\pi }^{\ast }(E)}$.

Complex cobordism has a natural complex orientation. Daniel 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.

Brown–Peterson cohomology

Complex cobordism over the rationals can be reduced to ordinary cohomology over the rationals, so the main interest is in the torsion of complex cobordism. It is often easier to study the torsion one prime at a time by localizing MU at a prime p; roughly speaking this means one kills off torsion prime to p. The localization MU_p of MU at a prime p splits as a sum of suspensions of a simpler cohomology theory called Brown–Peterson cohomology, first described by Brown & Peterson (1966). In practice one often does calculations with Brown–Peterson cohomology rather than with complex cobordism. Knowledge of the Brown–Peterson cohomologies of a space for all primes p is roughly equivalent to knowledge of its complex cobordism.

Conner–Floyd classes

The ring ${\displaystyle {\mathrm{MU}}^{\ast }(BU)}$ is isomorphic to the formal power series ring ${\displaystyle {\mathrm{MU}}^{\ast }(\text{point})[[c{f}_1,c{f}_2,\dots ]]}$ where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by Conner & Floyd (1966).

Similarly ${\displaystyle {\mathrm{MU}}_{\ast }(BU)}$ is isomorphic to the polynomial ring ${\displaystyle {\mathrm{MU}}_{\ast }(\text{point})[[{\beta }_1,{\beta }_2,\dots ]]}$

Cohomology operations

The Hopf algebra MU_*(MU) is isomorphic to the polynomial algebra R[b₁, b₂, ...], where R is the reduced bordism ring of a 0-sphere.

The coproduct is given by

${\displaystyle \psi (b_k)=\sum _{i+j=k}(b{)}_{2i}^{j+1}\otimes b_j}$

where the notation ()_2i means take the piece of degree 2i. This can be interpreted as follows. The map

${\displaystyle x\to x+b_1{x}^2+b_2{x}^3+\cdots }$

is a continuous automorphism of the ring of formal power series in x, and the coproduct of MU_*(MU) gives the composition of two such automorphisms.