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 MUp 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 is isomorphic to the formal power series ring where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by . Similarly is isomorphic to the polynomial ring 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 ()2i means take the piece of degree 2i. 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. See also Adams–Novikov spectral sequence List of cohomology theories Algebraic cobordism Notes References . . Translation of . External links Complex bordism at the manifold atlas Algebraic topology