The Pontryagin Rings of Moduli Spaces of Arbitrary Rank Holomorphic Bundles Over a Riemann Surface

The cohomology of M ( n , d ), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g ⩾ 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [ 17 ] originally showed that the topology of M ( n , d ) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of M ( n , d ) was first given by Harder and Narasimhan [ 7 ] and subsequently by Atiyah and Bott [ 1 ]. The integral cohomology of M ( n , d ) is known to have no torsion [ 1 ] and a set of generators was found by Newstead [ 19 ] for n = 2, and by Atiyah and Bott [ 1 ] for arbitrary n . Much progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of M (2, 1) is now known to be complete [ 14 ]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of M (2, 1) [ 2 , 13 , 20 , 25 ]. Unless otherwise stated all cohomology in this paper will have rational coefficients.