  This paper uses Morse-theoretic techniques to compute the equivariant Betti
numbers of the space of semistable rank two degree zero Higgs bundles over a
compact Riemann surface, a method in the spirit of Atiyah and Bott's original
approach for semistable holomorphic bundles. This leads to a natural proof that
the hyperk\"ahler Kirwan map is surjective for the non-fixed determinant case.
