Complex cobordism

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 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 ${\displaystyle \Sigma ^{\infty -2}\mathbb {C} \mathbf {P} ^{\infty }}$ of ${\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}$.

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