# Gluing vector bundles with isomorphic restrictions

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!

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, ....

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.

• Thanks for answering! So, after all, everything comes down to this isomorphism over $U \cap V$. In that case, we have a natural, well-defined projection map on $\tilde{G}$ and the argument in your analogy yields that the projection is smooth as well as that we also have a natural, well-defined local trivialisation atlas for $\tilde{G}$. Jun 27, 2021 at 13:53
• That's right. Perhaps surprisingly, it's the overlap maps or changes of coordinates that encode invariant geometric structure, not the coordinate systems/trivializations themselves. Jun 27, 2021 at 16:53
• My mind goes to how transition functions determine a vector bundle structure or how does a Haelfiger cocycle describes a foliation of a manifold. If you mean so, I also find this intrestingly surprising. Jun 28, 2021 at 10:28
• The fact that geometric structure resides in the overlaps maps does raise the prospect of using the powerful bookkeeping machinery of sheaf cohomology to study bundles, and deformations of structure. I wasn't aware of Haefliger cocycles, but they do appear to be an example of this "gluing" viewpoint. Jun 28, 2021 at 13:43
• The story with sheaf cohomology isn't brief, but I recall Complex Manifolds by Kodaira and Morrow being an accessible account for deformations of holomorphic structures. Jun 28, 2021 at 17:54