Complex orientation

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 Atiyah (1961) using the Thom spectrum.

The complex bordism ${\displaystyle MU^{*}(X)}$ of a space ${\displaystyle X}$ is roughly the group of bordism classes of manifolds over ${\displaystyle X}$ 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.

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