= Rank-two complex vector bundles over the four-sphere
{title2=$\operatorname{Vect}_{\mathbb C}^2(S^4)\cong\mathbb Z$}
The contractible universal frame bundle gives $\pi_4(BU_2)\cong\pi_3(U_2)$ and simple connectedness of $BU_2$. The determinant fibration $SU_2\to U_2\to S^1$ identifies $\pi_3(U_2)$ with $\pi_3(SU_2)$. Since $SU_2\cong S^3$, this is $\mathbb Z$. Hence the <classification of complex vector bundles by a Grassmannian> gives the displayed bijection of isomorphism classes; fixed-rank classes do not acquire a group operation from Whitney sum.
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