Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

John M lee, Introduction to smooth manifolds, Exercise 5-4:

Let $M$ be a smooth manifold and let $\{U_α\}_{α∈A}$ be an open cover of $M$. Suppose for each $α, β ∈ A$ we are given a smooth map $τ_{αβ} : U_α∩U_β → GL(k,\mathbb{R})$ such that $τ_{αβ}(p)τ_{βγ}(p) = τ_{αγ}(p)$, $p∈ U_α ∩ U_β ∩ U_γ$ is satisfied for all $α, β, γ ∈ A$. Show that there is a smooth rank $k$ vector bundle $E → M$ with smooth local trivializations $Φ_α : π^{−1}(U_α) → U_α×\mathbb{R}^k$ whose transition functions are the given maps $τ_{αβ}$. [Hint: Define an appropriate equivalence relation on $\coprod_{α∈A}(U_α × \mathbb{R}^k)$, and use the bundle construction lemma.]

But the bundle construction lemma has a different hypothesis. It uses $E= \coprod _{p \in M} E_p$ where $E_p$ is a real vector space. How can I use the lemma?

share|cite|improve this question
The obvious equivalence relation should turn the union of trivializations into a union of vector spaces indexed by $M$ (transition isomorphisms will make sure of the vector space structure) so that you can use the bundle construction lemma. – Neal Dec 2 '12 at 5:43
up vote 2 down vote accepted

Let $X=\coprod_{\alpha\in A}(U_\alpha\times\mathbf R^k)$. You want to find a relation ${\sim}$ on $X$ such that $X/{\sim}$ and $E$ are in bijection. The obvious candidate for a bijection is the one defined on each $U_\alpha\times\mathbf R^k$ by $\phi:(p,v)\in U_\alpha\times\mathbf R^k \mapsto (p,v)\in E_p$. This $\phi$ is surjective, but not injective: if $p$ is in $U_\alpha\cap U_\beta$, then both $(p,v)\in U_\alpha\times\mathbf R^k$ and $(p,v)\in U_\beta\times\mathbf R^k$ are sent to $(p,v)\in E_p$. The obvious solution is to define the relation ${\sim}$ by: for $(p,v)\in U_\alpha\times\mathbf R^k$ and $(q,w)\in U_\beta\times\mathbf R^k$, say that $(p,v)\sim(q,w)$ when $p=q$ and $w=\tau_{\alpha\beta}(p)v$. Then $\phi$ gives a bijection $X/{\sim}\to E$, and you can use it to build the bijections $\Phi_\alpha$ in the construction lemma.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.