Let $E$ be the total space of the sphere bundle $S^k\to E\to M$, is it true that there exists a disk bundle $D^{k+1}\to N\to M$ such that $E=\partial N$? (where $D^{k+1}$ is the unit disk in $\mathbb R^n$)



Is it true that all sphere bundles are boundaries of disk bundles?
If you don't specify what structure group you want the disc bundle $D^{k+1} \to N \to M$ to have, then it is always true: you just take the fibrewise cone on the original family.


If you want to know whether smooth $S^k$-bundles always bound smooth disc bundles, this is true iff
$$O(k+1) \to \mathrm{Diff}(S^k)$$
is a homotopy equivalence (known as the Smale conjecture). This is known to be true if $k=0, 1$ (classical), $2$ (Smale) or $3$ (Hatcher). I don't think it is known in any other dimension, and is definitely false in general. In fact, it is false on $\pi\_0$ in general, due to the existence of exotic spheres and hence also exotic self-diffeomoprhisms of spheres.


