Thom isomorphism in complex cobordism ($M{U}^q(X)\cong {\widetilde{MU}}^{q+r}(\mathrm{Th}(\xi ))$)

For a rank-$r$ real bundle with chosen stable complex normal orientation, the complex-cobordism Thom class gives the displayed isomorphism by $a\mapsto {\pi }^{\ast }a{u}_{\xi }$. 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 $2n$.