A note on the homotopy invariance of Pontrjagin classes

We prove that the classical result of Wu that, for every n ⩾ 1, the integral Pontrjagin class pn modulo 3 is homotopy invariant is the only and best possible result, ‘only’ in the sense that no other Pontrjagin classes of stable vector bundles—rational, integral, multiple of integral or integral mod p, p ≠ 3, are homotopy invariant and ‘best’ in the sense that pn mod 3r is not homotopy invariant if r > 1.