Complex cobordism

Summary

Complex cobordism $MU^*$ is the generalized cohomology theory represented by the Thom spectrum $MU$ of the universal complex vector bundles. Its coefficient ring is the cobordism ring of stably complex manifolds, a polynomial ring over $\Z$ with one generator in each positive even degree. By a theorem of Quillen this ring carries the universal formal group law. This link makes complex cobordism the central theory of modern stable homotopy theory.