I saw in the answer of this post:
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.