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 brownâpeterson 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 mu*(x) of a space x is roughly the group of bordism classes of manifolds over x 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 mu(n) is the thom space of the universal n-plane bundle over the classifying space bu(n) of the unitary group u(n). the natural inclusion from u(n) into u(n+1) induces a map from the double suspension s2mu(n) to mu(n+1). together these maps give the spectrum mu; namely, it is the homotopy colimit of mu(n). examples: mu(0) is the sphere spectrum. mu(1) is the desuspension \sigma^{\infty -2} \mathbb{c}\mathbf{p}^\infty of \mathbb{c}\mathbf{p}^\infty. the nilpotence theorem states that, for any ring spectrum r, the kernel of \pi_* r \to \operatorname{mu}_*(r) consists of nilpotent elements.http://www.math.harvard.edu/~lurie/252xnotes/lecture25.pdf the theorem implies in particular that, if s is the sphere spectrum, then for any n > 0, every element of \pi_n s is nilpotent (a theorem of nishida). (proof: if x is in \pi_n s, then x is a torsion but its image in \operatorname{mu}_*(s) \simeq l, the lazard ring, cannot be torsion since l is a polynomial ring. thus, x must be in the kernel.) ==formal group laws==  and  showed that the coefficient ring Ï*(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[x1, x2,...] on infinitely many generators x'i â Ï2i(mu) of positive even degrees. write cpâ 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 Î¼:cpâ&times; cpâ â cpâ. a complex orientation on an associative commutative ring spectrum e is an element x in e2(cpâ) whose restriction to e2(cp1) is 1, if the latter ring is identified with the coefficient ring of e. a spectrum e with such
Algebraic_topology