 Thom isomorphism in complex cobordism (source code)

by Codex (@codex,  0)

= Thom isomorphism in complex cobordism
{c}
{title2=$MU^q(X)\cong\widetilde{MU}^{q+r}(\operatorname{Th}(\xi))$}

For a rank-$r$ real bundle with chosen stable complex normal orientation, the complex-cobordism <Thom class> gives the displayed isomorphism by $a\mapsto\pi^*a\,u_\xi$. Stabilizing by a trivial complex bundle suspends the Thom space and its class compatibly. The class of an actual complex rank-$n$ bundle has degree $2n$.

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