# Can a topological manifold have different tangent bundles?

We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of smooth structures and these tangent bundles.

• You might want to look up the Novikov conjecture. – Mikhail Katz Apr 18 '16 at 15:25
• It seems like a negative answer to the (open) Novikov conjecture would give an example here where the tangent bundles are different. That doesn't directly imply that the original question is open, however. – Dylan Thurston Apr 18 '16 at 16:21
• Thanks Mckay for editing them to make them more aware and Belegradek for giving an enlightening answer. – Jialong Deng Apr 23 '16 at 14:29

Specifically, in each dimension $>8$ there is a closed PL manifold admitting two smooth structures whose tangent bundles are non-isomorphic. One tangent bundle is trivial and the other one has nonzero second Pontryagin class. See remark 1.3.
Such examples do not exist in dimensions $\le 8$ by Corollary 2.6.
In dimensions $\ge 18$ this was known since 1969 and due to Roitberg in [On the PL noninvariance of the span of a smooth manifold, Proceedings of the American Mathematical Society Vol. 20, No. 2 (Feb., 1969), pp. 575-579].