The Mumford relations and the moduli of rank three stable bundles

Abstract

We find a complete set of relations for the rational cohomology ring of the moduli space of rank three stable bundles over a Riemann surface of genus g and also show that the Pontryagin ring vanishes in degree 12g-8 and greater. The results are obtained by introducing some 'dual' Mumford relations and generalising Kirwan's proofs of the Mumford and Newstead conjectures in the rank two case. (In this revised version of the paper the vanishing degree of the Pontryagin ring of the moduli space has been improved from `in and above degree 12g-4' to `in and above degree 12g-8'. This degree is now known to be sharp.)