TITLE: Is it true that all sphere bundles are boundaries of disk bundles?
QUESTION [16 upvotes]: 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 $\mathbb R^n$)

REPLY [6 votes]: The question of whether or not a smooth sphere bundle fibrewise extends to a smooth disc bundle over a space $X$ boils down to whether or not the classifying map
$$ X \to BDiff(S^n) $$
lifts up
$$ BDiff(D^{n+1}) \to BDiff(S^n)$$
Where $Diff(D^{n+1})$ is the group of diffeomorphisms of the disc. 
The map $Diff(D^{n+1}) \to Diff(S^n)$ splits as a product:
$$O_{n+1} \times PDiff(D^{n}) \to O_{n+1} \times Diff(D^n rel \partial)$$
where $PDiff(D^{n})$ is the group of pseudo-isotopy diffeomorphisms of $D^{n}$.  These are diffeomorphisms of $D^{n} \times [0,1]$ that are the identity on $(D^{n} \times \{0\}) \cup (S^{n-1} \times [0,1])$. 
There is a fibre-bundle:
$$Diff(D^{n+1} rel \partial) \to PDiff(D^{n}) \to Diff(D^n rel \partial)$$
so basically this is asking whether or not this bundle has a section.   I think it can't have a section, in particular the map $\pi_1 Diff(D^n rel \partial) \to \pi_0 Diff(D^{n+1} rel \partial)$ is epic by Cerf's Pseudoisotopy theorem. 
Okay, so this is now an answer. So this is saying that there are sphere bundles over $S^2$ which do not extend to smooth disc bundles over $S^2$.