To what extent is a vector bundle on a manifold with boundary determined by its restriction to the interior? Let $M$ be a manifold with boundary $\partial M$ and interior $M_0$. Let $E\rightarrow M_0$ be a fixed vector bundle. How many extensions of $E$ to a vector bundle $E'\rightarrow M$ are there, up to isomorphism? In terms of the bundles monoid: the restriction of $E'$ to $M_0$ gives a monoid morphism $\mathrm{Vec}_k(M)\rightarrow \mathrm{Vec}_k(M_0)$. Is it surjective/injective?
Intuitively, the bundle $E'|_{\partial M}$ is "the limit" of $E$ at $\partial M$, and therefore should be fixed up to isomorphism.
And perhaps in the same vein, the inclusion $\iota : M_0 \rightarrow M$ induces $\iota_*:\pi_1(M_0)\rightarrow \pi_1(M)$. Is this map surjective/injective? Can the bijectivity be deduced from a tubular neighborhood of $\partial M$ in $M$?
Counter-examples are appreciated.
 A: As I indicated in my comment, the inclusion $\iota : M_0 \to M$ is a homotopy equivalence. This can be shown using the fact that the boundary $\partial M$ has a collar neighbourhood; it then boils down to showing the inclusion $(0, 1) \hookrightarrow [0, 1)$ is a homotopy equivalence. Actually, one needs to show that there is a homotopy inverse $j : [0, 1) \to (0, 1)$ to $i$ such that $i\circ j$ and $j\circ i$ are homotopic to identity maps relative to $[\frac{1}{2}, 1)$. This is not difficult, see this answer for some details.
On any paracompact space $X$, there is a natural bijection between isomorphism classes of real vector bundles on $X$ of rank $r$ and $[X, BO(r)]$, the set of homotopy classes of maps $X \to BO(r)$; see section $1.2$ of Hatcher's Vector Bundles and K-Theory for example. In particular, given a map $f : X \to Y$, we get an induced map $f^* : [Y, BO(r)] \to [X, BO(r)]$ which corresponds to pulling back a vector bundle by $f$. The analogous statement is true for complex vector bundles too, one just replaces $BO(r)$ with $BU(r)$.
In the case that $f$ is a homotopy equivalence, then $f^*$ is a bijection: if $g$ is the homotopy inverse of $f$, then $g^*$ is the inverse of $f^*$. In particular, for the homotopy equivalence $\iota : M_0 \to M$, we see that there is a bijection between isomorphism classes of real/complex rank $r$ bundles on $M$ and $M_0$ given by $E \mapsto \iota^*E = E|_{M_0}$.
Finally, as $\iota : M_0 \to M$ is a homotopy equivalence, the induced map $\iota_* : \pi_1(M_0) \to \pi_1(M)$ is an isomorphism.

As Ben McKay indicates in the comment below, the above does not deal with smooth bundles but topological bundles. The statement for smooth bundles is also true, but requires a bit more work. The key is that every real rank $r$ vector bundle on a smooth manifold $M$ has a classifying map $M \to \operatorname{Gr}_r(\mathbb{R}^N)$ which is unique up to homotopy where $N = r + \dim M + 1$; this is Theorem 3.3.4 of Hirsch's Differential Topology. It follows that isomorphism classes of topological real rank $r$ vector bundles on $M$ are in bijection with $[M, \operatorname{Gr}_r(\mathbb{R}^N)]$; that is, the inclusion $\operatorname{Gr}_r(\mathbb{R}^N) \hookrightarrow \operatorname{Gr}_r(\mathbb{R}^{\infty})$ induces a bijection $[M, \operatorname{Gr}_r(\mathbb{R}^N)] \to [M, \operatorname{Gr}_r(\mathbb{R}^{\infty})] = [M, BO(r)]$.
If the classifying map of a bundle is smooth, then the bundle itself is smooth (the pullback of a smooth bundle by a smooth map is smooth). As every continuous map between smooth manifolds is homotopic to a smooth one, every topological vector bundle on $M$ is isomorphic to a smooth one. Moreover, two smooth maps are homotopic if and only if they are smoothly homotopic which implies that every topological vector bundle is isomorphic to a unique smooth vector bundle up to smooth isomorphism. It follows that isomorphism classes of smooth real rank $r$ vector bundles on $M$ are in bijection with $[M, \operatorname{Gr}_r(\mathbb{R}^N)]$.
Now we can argue as before to deduce that $\iota^*$ induces a bijection between the set of isomorphism classes of smooth real rank $r$ bundles on $M$ and $M_0$. Again, the statement is also true for smooth complex bundles.
