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I have some questionss about the construction of the complex bordism ring MU and would appreciate every answer:

  1. I have read that the multiplication in MU is given by the tensor product of vector bundles $BU(n) \times BU(m) \rightarrow BU(n+m)$. How does this tensor product yield us a map $MU(n) \wedge MU(m) \rightarrow MU(n+m) $, I do not get how we can go from $BU$ to $MU$?

  2. What is the unit for the multiplication on MU?

I would be happy, if you could give me an answer.

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    To complement/summarize Tyler's answer: $MU$ is the Thom spectrum of the universal (0-dimensional virtual) complex vector bundle over $BU$, and its product map is induced from the map $BU \times BU \rightarrow BU$ guaranteed by the external tensor product with itself of the universal bundle. (But this only works up to homotopy. Presumably you could use the $E_\infty$-structure on $BU$ to tighten things up...)2012-07-18

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