Gluing vector bundles with isomorphic restrictions

Question

I am dealing with the following situation. We have two vector bundles $E \xrightarrow{\pi_1} U$ and $F \xrightarrow{\pi_2} V$ over some open subsets of a smooth manifold whose restrictions on the intersection $U \cap V$ are isomoprhic, i.e. we have a vector bundle isomorphism $\phi : E|_{U\cap V} \rightarrow F|_{U\cap V}$. I am trying to glue $E$ and $F$ to a vector bundle over $U\cup V$ and so, here is my idea.

First, I consider the disjoint union $G=E_p \displaystyle\bigsqcup_{p \in U\cup V} F_p$ equipped with the disjoint union topology and then define an equivalence relation: $u \sim v \Longleftrightarrow$ u and v belong to to $E_p$ or $F_p$ for $p\in U \cap V$. Then, we pass to the quotient space \begin{equation*} \tilde{G}=\big (E_p \displaystyle\bigsqcup_{p \in U\cup V} F_p\big)mod\sim \end{equation*} and consider the projection map \begin{equation*} \pi_(v)= \begin{cases} \pi_1(v), \text{if}~ v\in E_p~ \text{for}~ p\in U\setminus U\cap V\\ \pi_2(v), \text{if}~ v\in F_p~ \text{for}~ p\in V\setminus U\cap V\\ \pi_1(v)=\pi_2(v), \text{if}~ v\in E_p\simeq F_p~ \text{for}~ p\in U\cap V \end{cases} \end{equation*} As I see it, there is a well-defined vector space structure on the fibres of $\pi$ so my problems begin with the local trivialization of $\tilde{G}$. There are some straightforward local trivializations charts over a point p, coming from $E$ and $F$ , when $p \in U \setminus \overline{U \cap V}$ or $p \in V \setminus \overline{U \cap V}$ respectively. I could even define such charts over points in $U \cap V$.

However, I just cannot see how things work out in the boundary $\partial(U \cap V)$ and more generally, I do not feel comfortable with the simplicity of this approach.

I would be thankful, If someone could shed some light to this question!

Answer 1

It may help to consider a simple analogy: Suppose we have two open sets $U$ and $V$ of real numbers, and two continuous functions, $f$ on $U$ and $g$ on $V$, such that $f = g$ in $U \cap V$. The "patched" formula $$ (f \cup g)(x) = \begin{cases} f(x) & x \in U, \\ g(x) & x \in V, \end{cases} $$ is well-defined in the intersection $U \cap V$, and therefore represents a function on the union $U \cup V$. Moreover (in relation to your question), $f \cup g$ is continuous: If $p$ is an arbitrary point of $U \cup V$, there exists a neighborhood $W$ of $p$ such that either

• $W \subset U$, so $f \cup g = f$ throughout $W$, or
• $W \subset V$, so $f \cup g = g$ throughout $W$.

In either case, $f \cup g$ is continuous. The same argument works for any local property a function might satisfy: smoothness, real-analyticity, ....

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In the situation at hand we have two vector bundles, and we know there exists a vector bundle isomorphism $\phi:E|_{U \cap V} \to F|_{U \cap V}$. The total space of the vector bundle over $U \cup V$ may be constructed as the disjoint union of fibres just as you say, but it may be desirable to write the fibre equivalence over $U \cap V$ explicitly: $$ \bigl[(p, v) \in E|_{U \cap V} \bigr] \sim \bigl[(p, \phi(v)) \in F|_{U \cap V} \bigr]. $$ The point is now to see that the quotient space $\widetilde{G} = E \cup F$ is locally trivial over $U \cup V$. Conceptually, this is precisely the simple analogy above. At risk of taking a joke past its punch line, for each point $p$ of $U \cup V$, there exists a neighborhood $W$ of $p$ such that either $W \subset U$ trivializes $E$, or $W \subset V$ trivializes $F$, or both (and the notions of trivialization agree).

Whether or not $p$ is a boundary point of the intersection $U \cap V$ turns out to be immaterial, though it's also perfectly understandable why that contingency might seem to require attention.