# Short Exact Sequence of Vector Bundles

Just wish to clarify, is it true that in order to show some vector bundles (over the same space) fit into a short exact sequence we just need to check that their fibers fit into a short exact sequence of vector spaces?

Thank you very much for your attention!

For a counterexample, let $X$ be any space that has a nontrivial vector bundle $E$ of rank $n$. Let $F$ denote the trivial vector bundle over $X$ of rank $n$, and let $0$ denote the vector bundle of rank $0$ over $X$. Then for each $x\in X$, there is a short exact sequence $0\to 0_x\to E_x\to F_x\to 0$ of fibers at $x$, since $E_x$ and $F_x$ are both $n$-dimensional vector spaces and $0_x$ is the trivial vector space. But there is no short exact sequence of vector bundles $0\to 0\to E\to F\to 0$, since this would imply $E\cong F$, but $E$ is nontrivial.
However, as Mariano Suárez-Alvarez commented, if you already have maps of vector bundles $E\to F$ and $F\to G$, then it is true that the sequence $0\to E\to F\to G\to 0$ is exact iff $0\to E_x\to F_x\to G_x\to 0$ is exact for each $x\in X$. I can't prove this without knowing what your definition of "exact sequence of vector bundles" is (indeed, one such definition is that a sequence is exact iff it is exact on each fiber).
• It is true, though, that if one has maps $0\to E\to F\to G\to0$ such that on each fiber give a short exact sequence, the maps form a short exact sequence. Commented May 9, 2016 at 2:27
• Thanks a lot for the answer Eric. But Mariano seems to suggest that if $0 \to E_x \to F_x \to G_x \to 0$ is a short exact sequence of vector spaces and this is true for any $x \in X$ then it is sufficient? Do you agree? Commented May 9, 2016 at 2:33
• @PhysicsMath: The point is that you have to already have maps of vector bundles $E\to F$ and $F\to G$ which give these short exact sequences on the fibers. If all you have is a linear map $E_x\to F_x$ and a linear map $F_x\to G_x$ for each $x$, these linear maps might not combine to give a map of vector bundles (because the map you get by combining them might not be continuous). Commented May 9, 2016 at 2:36