I saw in the answer of this post:


[Is it true that all sphere bundles are boundaries of disk bundles?](https://mathoverflow.net/questions/74756/is-it-true-that-all-sphere-bundles-are-boundaries-of-disk-bundles)


that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is a homotopy equivalence. 


Can anyone direct me to a reference that proves it?


I'm studying Milnor's paper "On manifolds homeomorphic to the 7-sphere" and trying to avoid Thom's paper to see that a closed 7-manifold is the boundary of an 8-manifold. More specifically, I have a $SO(4)$-bundle 


$$S^3 \rightarrow M^7 \rightarrow S^4$$


and I want to see this bundle as the boundary of some bundle:


$$D^4 \rightarrow N^8 \rightarrow S^4$$


with $\partial N^8 = M^7$


Thanks in advance.



3-sphere bundles over 4-sphere bound smooth disc bundles
The map $O(4) \to \text{Diff}(S^3)$ being homotopy equivalence is *not* equivalent to the assertion that every smooth $S^3$-bundle bounds a disk bundle, but the much stronger statement that every smooth $S^3$-bundle can be linearized, i.e. arises as the unit bundle of some four-dimensional vector bundle (and this clearly implies that it is the boundary of a disk bundle).


You can see this by observing that smooth $S^3$-bundles over a space $X$ are in bijective correspondence with homotopy classes $[X,B \text{Diff}(S^3)]$ and the same holds for vector bundles and $[X,B O(4)]$ .


But I think you do not need all this stuff to understand what Milnor did: There we already start with a $S^3$-bundle with structure group $SO(4)$, i.e. a linear sphere bundle.


