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.



Can a topological manifold have different tangent bundles?
This is answered in [Crowley, Diarmuid J.; Zvengrowski, Peter D, On the non-invariance of span and immersion co-dimension for manifolds, Arch. Math. (Brno) 44 (2008), no. 5, 353–365], see [here](http://www.dcrowley.net/Span.pdf). 


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].


