Whitney sum, tensor, dual, Hom, and exterior-power bundles

Let $E$ and $F$ be finite-rank $\mathbb{F}$-vector bundles over $X$, with transition matrices $g_{ji}$ and $h_{ji}$ on a common refinement. Apply Vector bundles are glued from transition cocycles to the following transition maps:

• $E\oplus F$: $\mathrm{diag}(g_{ji},h_{ji})$;
• $E\otimes F$: $g_{ji}\otimes h_{ji}$;
• $E^{\ast }$: $\lambda \mapsto \lambda g_{ji}^{-1}$;
• $\mathrm{Hom}(E,F)=E^{\ast }\otimes F$: $T\mapsto h_{ji}T{g}_{ji}^{-1}$;
• ${\mathrm{\Lambda }}^{k}E$: ${\mathrm{\Lambda }}^k{g}_{ji}$.

These define the Whitney sum, tensor product, dual, Hom, and exterior-power bundles. Their fibers are respectively $E_x\oplus F_x$, $E_x\otimes F_x$, $E_x^{\ast }$, $\mathrm{Hom}(E_x,F_x)$, and ${\mathrm{\Lambda }}^k{E}_x$. The cocycle theorem also shows that changes of frame give canonically isomorphic bundles.

For a complex bundle, conjugating every transition matrix defines $\bar{E}$, and regarding those matrices as real-linear defines the underlying real bundle $E_{\mathbb{R}}$. We set ${\mathrm{\Lambda }}^{0}E=X\times \mathbb{F}$ and ${\mathrm{\Lambda }}^{k}E=0$ when $k>\mathrm{rank}E$. Applying the same matrices after precomposition with a base map shows, under the canonical comparisons of Vector-bundle pullback is canonically functorial, that every construction commutes with pullback.