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.