In this paper we consider the topological side of a problem which is the
analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of
rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann
surface. We prove that all intersection numbers in the compactly supported
cohomology vanish, i.e. "there are no topological L^2 harmonic forms on
Hitchin's space". This result generalizes the well known vanishing of the Euler
characteristic of the moduli space of rank 2 stable bundles of fixed
determinant of odd degree over the given Riemann surface. Our proof shows that
the vanishing of all intersection numbers in the compactly supported cohomology
of Hitchin's space is given by relations analogous to Mumford's relations in
the cohomology ring of the moduli space of stable bundles.Comment: 30 pages (published version