I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", *Annals of Mathematics*, and I am trying to prove (or disprove) the following result. The more elementary the proof, the better.



> 
> If two smooth manifolds are homeomorphic, then their stable tangent
> bundles (i.e. the Whitney sum of the tangent bundle with the trivial
> line bundle) are vector bundle isomorphic.
> 
> 
> 


I am trying to prove this as an intermediate step to give an alternative proof for KM's Theorem 3.1: Every homotopy sphere is $s$-parallelizable.



If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic
The result you are hoping for is in fact false.


In section 9 of *[Microbundles: Part I](http://dx.doi.org/10.1016/0040-9383(64)90005-9)*, 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**](http://dx.doi.org/10.1016/0040-9383(64)90005-9), Topology 3, Suppl. 1, 53-80 (1964). [ZBL0124.38404](https://zbmath.org/?q=an:0124.38404).


