This is a preliminary report on work in progress with Graeme Wilkin. Let $G$ be a compact Lie group. The well-known Kirwan surjectivity theorem in equivariant symplectic geometry states that the $G$-equivariant rational cohomology of a Hamiltonian $G$-space $(M,\omega)$ surjects onto the ordinary rational cohomology of the symplectic quotient of $M$ by $G$. This surjective ring homomorphism ("the Kirwan map") has been a key tool in computations of the topology of symplectic quotients. I will discuss our recent progress on the analogous hyperk\"ahler question, namely: if $(M,\omega_1, \omega_2, \omega_3)$ is a hyperk\"ahler hyperhamiltonian $G$-space, then does the $G$-equivariant cohomology of $M$ surject onto the ordinary rational cohomology of the hyperk\"ahler quotient of $M$ by $G$? We restrict to the case of Nakajima quiver varieties, which are spaces built from the combinatorics of a quiver and an accompanying dimension vector, and which arise in geometric representation theory. We develop a Morse theory for the hyperk\"ahler moment map analogous to the case of the moduli space of Higgs bundles. In particular, we show that the Harder-Narasimhan stratification of spaces of representations of quivers coincide with the Morse-theoretic stratification associated to the norm-square of the real moment map. Our approach also provides insight into the topology of specific examples of small-rank quiver varieties, including hyperpolygon spaces and some ADHM quivers.