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.