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 of a space is roughly the group of bordism classes of manifolds over 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 is the Thom space of the universal -plane bundle over the classifying space of the unitary group . The natural inclusion from into induces a map from the double suspension to . Together these maps give the spectrum ; namely, it is the homotopy colimit of . Examples: is the sphere spectrum. is the desuspension of . The nilpotence theorem states that, for any ring spectrum , the kernel of consists of nilpotent elements. The theorem implies in particular that, if is the sphere spectrum, then for any , every element of is nilpotent (a theorem of Goro Nishida). (Proof: if is in , then is a torsion but its image in , the Lazard ring, cannot be torsion since is a polynomial ring. Thus, must be in the kernel.) Formal group laws and showed that the coefficient ring (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring on infinitely many generators of positive even degrees. Write 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 A complex orientation on an associative commutative ring spectrum E is an element x in whose restriction to 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 is a formal group law over the ring . Complex cobordism has a natural