Tangent bundles of exotic manifolds

Question

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and $\phi^*TM_2$ are always isomorphic as topological fiber bundles (that's because they are both isomorphic to a subbundle of the tangent microbundle).

Are they always isomorphic as vector bundles?

Answer 1

John Milnor gives an example of two homeomorphic smooth manifolds whose tangent bundles are not isomorphic as topological vector bundles, see his ICM-1962 address, Corollary 1. I think, this was the first such example.

Edit. Few more things (motivated by questions in comments below):

1. The total space of the tangent bundle to any exotic $n$-sphere is diffeomorphic to $TS^n$, see

R. De Sapio, Disc and sphere bundles over homotopy spheres, Math. Z. 107 (1968) 232-236.

2. If $M_1, M_2$ are two homeomorphic manifolds, then the total spaces of their tangent bundles $TM_1, TM_2$ are always homeomorphic (the tangent bundles are even topologically isomorphic as microbundles), this follows from

J.Milnor, Microbundles-I, Topology, 3 (1964) 53-80.

3. I do not know of any examples where tangent bundles of two smooth homeomorphic manifolds $M_1, M_2$ such that the total spaces of $TM_1, TM_2$ are not diffeomorphic, but I did not spend much time thinking about this either.