Stably extendible vector bundles over the quaternionic projective spaces (Q1306511)

Let \(\mathbb HP^n\) denote quaternionic projective \(n\)-space. Let a quaternionic \(k\)-dimensional \((k\leq n)\) vector bundle \(\gamma\) over \(\mathbb HP^n\) be stably extendible (that is, for each \(m\geq n\), the bundle \(\gamma\) is stably equivalent to the \(\mathbb HP^n\)-restriction of a quaternionic \(k\)-dimensional vector bundle \(\widetilde\gamma_m\) over \(\mathbb HP^m\)). Supposing that the top nonzero symplectic Pontryagin class of \(\gamma\) does not vanish mod \(2\), the authors prove that \(\gamma\) is stably equivalent to a sum of \(k\) quaternionic line bundles. They give an example showing that the Pontryagin class condition cannot be removed.