Complex cobordism

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.
The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum.
Spectrum of complex cobordism.
The complex bordism formula_1 of a space formula_2 is roughly the group of bordism classes of manifolds over formula_2 with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows.
The space formula_4 is the Thom space of the universal formula_5-plane bundle over the classifying space formula_6 of the unitary group formula_7. The natural inclusion from formula_7 into formula_9 induces a map from the double suspension formula_10 to formula_11. Together these maps give the spectrum formula_12; namely, it is the homotopy colimit of formula_4.
Examples: formula_14 is the sphere spectrum. formula_15 is the desuspension formula_16 of formula_17.
The nilpotence theorem states that, for any ring spectrum formula_18, the kernel of formula_19 consists of nilpotent elements. The theorem implies in particular that, if formula_20 is the sphere spectrum, then for any formula_21, every element of formula_22 is nilpotent (a theorem of Goro Nishida). (Proof: if formula_23 is in formula_24, then formula_23 is a torsion but its image in formula_26, the Lazard ring, cannot be torsion since formula_27 is a polynomial ring. Thus, formula_23 must be in the kernel.)
Formal group laws.
 and showed that the coefficient ring formula_29 (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring formula_30 on infinitely many generators formula_31 of positive even degrees.
Write formula_32 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 formula_33 A complex orientation on an associative commutative ring spectrum "E" is an element "x" in formula_34 whose restriction to formula_35 
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 formula_38 is a formal group law over the ring formula_39.
Complex cobordism has a natural complex orientation. 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 . 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 formula_40 is isomorphic to the formal power series ring formula_41 where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by .
Similarly formula_42 is isomorphic to the polynomial ring formula_43
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
where the notation ()2"i" means take the piece of degree 2"i". This can be interpreted as follows. The map
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.