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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?

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In the case of oriented 4-manifolds the equality follows from the Dold-Whitney Theorem: "If two oriented 4-plane bundles over an oriented 4-manifold have the same second Stiefel-Whitney class $w_2,$ Pontryagin class $p_1$ and Euler class $e,$ then they must be isomorphic". In lower dimensions, of course, the phenomena doesn't appear. So, what happens in higher dimensions and non-oriented 4-dimensional case? –  Misha Aug 19 '13 at 10:44
    
If they would be always isomorphic as vector bundles, it would force topological invariance of integer Pontryagin classes of smooth manifolds, but they (unlike rational ones) are not topologically invariant (I think, non-invariance is due to Milnor). –  studiosus Aug 26 '13 at 13:20

1 Answer 1

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

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