The question is a follow-up to [this one](https://mathoverflow.net/questions/363360/to-what-extent-is-a-vector-bundle-on-a-manifold-with-boundary-determined-by-its).


* Let $N\subset M$ be a closed smooth submanifold of codimension $k\geq 1$ of a smooth manifold $M$. Let $E\_1\rightarrow M$ and $E\_2\rightarrow M$ be two vector bundles over $M$. Suppose $E\_1|\_{M-N}\sim E\_2|\_{M-N}$. Is it true that $E\_1|\_N\sim E\_2|\_N$?
* Can we impose some conditions on $k$ and $n=\dim M$ such that under the condition $E\_1|\_{M-N}\sim E\_2|\_{M-N}$, we can conclude $E\_1\sim E\_2$? (a counter example for this statement in general is provided by $E\_1=T\mathbb{S}^1\rightarrow\mathbb{S}^1,E\_2=\text{mobius}\rightarrow\mathbb{S}^1$ and $N=\{\cdot\}$) (the difference with the [previous question](https://mathoverflow.net/questions/363360/to-what-extent-is-a-vector-bundle-on-a-manifold-with-boundary-determined-by-its) is that $M$ and $M-N$ are generally not homotopic).



To what extent is a vector bundle on a smooth manifold determined by its restriction to the complement of a closed smooth submanifold?
Here's an example to show that the answer to the first question is no.


Let $M = \mathbb{CP}^2$ and $N = \mathbb{CP}^1$. Note that $M - N$ is diffeomorphic to $\mathbb{C}^2$ and is therefore contractible. So for any two vector bundles $E\_1, E\_2 \to \mathbb{CP}^2$ of the same rank, we have $E\_1|\_{M - N} \cong E\_2|\_{M-N}$. However, we need not have $E\_1|\_N \cong E\_2|\_N$. For example, if $E\_1 = \mathcal{O}\_{\mathbb{CP}^2}(1)$ and $E\_2 = \mathcal{O}\_{\mathbb{CP}^2}$, then $E\_1|\_N \cong \mathcal{O}\_{\mathbb{CP}^1}(1)$ and $E\_2|\_N \cong \mathcal{O}\_{\mathbb{CP}^1}$ which are distinguished by the second Stiefel-Whitney class.


