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.

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 has nonzero second Pontryagin class. See remark 1.3.

Such examples do not exist in dimensions $\le 8$ by Corollary 2.6.

In dimensions $\ge 18$ there 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].

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.

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].

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.

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 has nonzero second Pontryagin class. See remark 1.3.

Such examples do not exist in dimensions $\le 8$ by Corollary 2.6.

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.

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 has nonzero second Pontryagin class. See remark 1.3.

Such examples do not exist in dimensions $\le 8$ by Corollary 2.6.

In dimensions $\ge 18$ there 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].