Disk bundles as unit disk bundles Let $M^n$ be a closed connected manifold and let $\pi : X \to M$ be a $D^k$-disk bundle.  Under what circumstances is $X$ the unit disk bundle of a rank $k$ vector bundle over $M$?  I know for example that this is true for $n = k =1$ and $n=k=2$ - though I am not sure of why.  
An explicit example of a disk bundle that can't be extended to a vector bundle would be great.  
 A: Given a principal $O(k)$-bundle $P \to M$, there is an associated real rank $k$ vector bundle $E \to M$. Conversely, given any real rank $k$ vector bundle, we can associate to it a principal $O(k)$-bundle, namely the bundle of orthonormal frames. These two constructions are inverses of one another. Likewise, there is a correspondence between $D^k$ bundles and principal $\operatorname{Diff}(D^k)$-bundles.
If $G$ is a topological group, the collection of isomorphism classes of principal $G$-bundles on $M$ is given by $[M, BG]$. If $\varphi : G \to H$ is a topological group homomorphism, there is an induced map $BG \to BH$ and hence a map $[M, BG] \to [M, BH]$. If $\varphi$ is a homotopy equivalence, then $BG$ and $BH$ are homotopy equivalent and the induced map $[M, BG] \to [M, BH]$ is a bijection.
Because orthogonal transformations preserve length, there is an inclusion $O(k) \to \operatorname{Diff}(D^k)$. The induced map $[M, BO(k)] \to [M, B\operatorname{Diff}(D^k)]$ corresponds to taking the disk bundle of a vector bundle. Therefore, asking whether a given disk bundle can be obtained 
from a vector bundle is equivalent to asking whether it is in the image of this map.
For any $k$ such that the inclusion $O(k) \to \operatorname{Diff}(D^k)$ is a homotopy equivalence (i.e. $O(k)$ is a deformation retract of $\operatorname{Diff}(D^k)$), the map $[M, BO(k)] \to [M, B\operatorname{Diff}(D^k)]$ would be a bijection, and hence every $D^k$ bundle would come from a vector bundle. This is known to be true for $k = 1$ (exercise), $k = 2$ (Smale), and $k = 3$ (Hatcher). It is unknown in dimension four, but in higher dimensions it is known to be false and is related to the existence of exotic spheres. For more details, see this and this.
