# Tangent bundles of exotic manifolds

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?

• 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). – Moishe Kohan Aug 26 '13 at 13:20

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.

1. 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.

1. 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.
• When you say isomorphic as vector bundles, do you mean in the category of continuous vector bundles or smooth vector bundles? (Assuming that terminology is correct, it probably isn't.) I.e. does there exist a bundle map which is a homeomorphism but not a diffeomorphism? Or is the result even stronger, saying that the two tangent bundles aren't even homeomorphic? – Chill2Macht Aug 1 '17 at 17:52
• @Chill2Macht The fact that they are not smoothly isomorphic is easier since manifolds are not diffeomorphic. But the actual result is that they are not isomorphic as topological vector bundles. – Moishe Kohan Aug 1 '17 at 19:47
• Just to confirm then, that means that this is an example of two homeomorphic smooth manifolds with non-homeomorphic tangent bundles? (I was going to ask a question asking for examples of such objects, but this seems like it might qualify. Isomorphic as topological vector bundles implies homeomorphic, but not isomorphic as topological vector bundles doesn't imply not homeomorphic in general, without further assumptions, right? Is a homeomorphism of the underlying spaces sufficient for that?) – Chill2Macht Aug 1 '17 at 20:59
• @Chill2Macht You are using some nonstandard terminology so I do not understand your question. What do you mean by homeomorphic vector bundles? A vector bundle is a triple satisfying some properties. – Moishe Kohan Aug 1 '17 at 21:05
• @Chill2Macht I see. In the future, try to stick to the standard terminology, which distinguished bundles from their total spaces. The answer is that total spaces of tangent bundles are indeed homeomorphic . The paper to read is Microbundles-I by Milnor, as OP suggested. – Moishe Kohan Aug 1 '17 at 21:37