Complex Cobordism - Spectrum of Complex Cobordism

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.