  The cohomology of the moduli spaces of stable bundles M(n,d), of coprime rank
n and degree d, over a Riemann surface (of genus g > 1) have been intensely
studied over the past three decades. We prove in this paper that the Pontryagin
ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is
strict (i.e. there exists a non-zero element of degree 2n(n-1)(g-1) in
Pont(M(n,d)).) This result is a generalisation of a 1967 Newstead conjecture
that Pont(M(2,1)) vanished above 4(g-1) (or equivalently that \beta^g =0.)
These results have been independently proved by Lisa Jeffrey and Jonathan
Weitsman.
