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Is there an example of a topological manifold in which different smooth structures give rise to tangent bundles which are not isomorphic as topological vector bundles?

My (our) own attempts or remarks, mostly obtained from discussing this question with other people:

  • By the Wu formula, the two tangent bundles would have in any case the same Stiefel--Whitney classes. This is in fact what originally motivated my question.

  • Exotic 7-spheres are no good to produce such examples because they all have trivial tangent bundles.

  • "Different smooth structures correspond to different (stable) linear structures on the tangent microbundle."

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Yes, there are. In fact, there are examples where the tangent bundles aren't even stably equivalent.

In section 9 of Microbundles: Part I, Milnor constructs an open set $U \subset \mathbb{R}^m$. With its standard smooth structure, the (stable) tangent bundle of $U\times\mathbb{R}^k \subset \mathbb{R}^{m+k}$ is trivial, while in Corollary 9.3, Milnor shows that it admits a smooth structure for which the tangent bundle has a non-zero Pontryagin class. As Pontryagin classes are stable, the stable tangent bundle of the latter manifold is not trivial, and hence not isomorphic to the stable tangent bundle of $U\times\mathbb{R}^k$ with its standard smooth structure.

Milnor, John W., Microbundles: Part I, Topology 3, Suppl. 1, 53-80 (1964). ZBL0124.38404.

A closed example is constructed in this answer.

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    $\begingroup$ My answer is essentially the same as the one I gave here. $\endgroup$ – Michael Albanese Dec 1 '20 at 12:36
  • $\begingroup$ Thank you for your answer! I'm sorry I made you write it down twice. I did try to look for similar questions on MSE and MO but somehow the wording I chose didn't allow me to find that MO post $\endgroup$ – Pedro Dec 1 '20 at 13:04
  • $\begingroup$ No problem at all. It's a very natural question to ask, so I'm not surprised it has appeared more than once. $\endgroup$ – Michael Albanese Dec 1 '20 at 13:14

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