TITLE: Tangent bundles of exotic manifolds
QUESTION [12 upvotes]: 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?

REPLY [7 votes]: 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):

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.

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.

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.