Vector Bundles

Let $\xi :E(\xi )\to B$ and $\eta :E(\eta )\to C$ be a vector bundles. A bundle morphism $\xi \to \eta$ over $\phi :B\to C$ is a commutative diagram$$E(\xi )E(\eta )BC\Phi \xi \eta \phi$$with a map $\Phi$ which is fibrewise linear. If $\Phi$ is bijective on fibres, then we call the bundle morphism a bundle map. Thus we have categories of vector bundles with bundle morphism or bundle maps as morphisms.

命题 1. A bundle map over the identity is a bundle isomorphism.

证明.

证明. 略.

$\square$

命题 2. If the previous diagram is a pullback in $\mathrm{TOP}$ and $\eta$ a vector bundle, then there exists a unique structure of a vector bundle on $\xi$ such that the diagram is a bundle.

证明.

证明. 略.

$\square$

Let $q:E\to B$ be a right $G$-principal bundle and $V$ an $n$-dimensional representation of $G$. Then the associated bundle $p:E{\times }_{G}V\to B$ is an $n$-dimensional vector bundle. A bundle chart $\phi :q^{-1}(U)\to U\times G$ for $q$ induces a bundle chart $q^{-1}{\times }_{G}V\to U\times V$ for $p$, and the vector space structure on the fibres of $p$ is uniquely determined by the requirement that the bundle charts are fibrewise linear.

We now show that vector bundles are associated to principal bundles. Let $p:X\to B$ be an $n$-dimensional real vector bundle. Let $E_b=\mathrm{Iso}({\mathbb{R}}^n,X_b)$ be the space of linear isomorphisms. The group $G={\mathrm{GL}}_n(\mathbb{R})$ acts freely and transitively on $E_b$ from the right by composition of linear maps. We have the set map$$q:E(\xi )=E=\underset{b\in B}{⨆}{E}_b\to B,E_b\to b$$with fibrewise ${\mathrm{GL}}_n(\mathbb{R})$-action. If $\phi :p^{-1}(U)\to U\times {\mathbb{R}}^n,{\phi }_b:X_b\to {\mathbb{R}}^n$ is a bundle chart of $p$, we define$$\tilde{\phi }:q^{-1}(U)=\underset{b\in U}{⨆}{E}_b\to U\times G{L}_n(\mathbb{R}),\alpha \in E_b\mapsto (b,{\phi }_b\circ \alpha )$$to be a bundle chart for $q$. The transition function for two such that charts has the form$$(U\cap V)\times {\mathrm{GL}}_n(\mathbb{R})\to (U\cap V)\times {\mathrm{GL}}_n(\mathbb{R}),(b,\gamma )\mapsto (b,{\psi }_b{\phi }_b^{-1}\gamma )$$This map is continuous, because $b\mapsto {\psi }_b{\phi }_b^{-1}$ is continuous. Therefore there exists a unique topology on $E$ in which the sets $q^{-1}(U)$ are open and $\tilde{\phi }$ is equivariant. This shows $q:E\to B$ to be a ${\mathrm{GL}}_n(\mathbb{R})$-principal bundle. The evaluation $\mathrm{Iso}({\mathbb{R}}^n,X_b)\times {\mathbb{R}}^n\to X_b,(f,u)\mapsto f(u)$ induces an isomorphism $E(\xi ){\times }_{{\mathrm{GL}}_n(\mathbb{R})}{\mathbb{R}}^n\cong X(\xi )$.

命题 3. The assignment which associated to a ${\mathrm{GL}}_n(\mathbb{R})$-principal bundle $E\to B$ the vector bundle $E{\times }_{{\mathrm{GL}}_n(\mathbb{R})}{\mathbb{R}}^n\to B$ is an equivalence of the category of ${\mathrm{GL}}_n(\mathbb{R})$-principal bundles with the category of $n$-dimensional real vector bundles; the morphisms are in both cases the bundle maps.

证明.

证明. 略.

$\square$

[TG]	N.Bourbaki, Topologie Générale.