Let M be a compact symplectic manifold carrying a Hamiltonian \(S^1\) -action with momentum map \({ \mathrm J}:M \rightarrow {\mathbb R}\) and consider the corresponding symplectic quotient \({\mathscr {M}}_0:={ \mathrm J}^{-1}(0)/S^1\) . We extend Sjamaar’s complex of differential forms on \({\mathscr {M}}_0\) , whose cohomology is isomorphic to the singular or Čech cohomology \(H^{*}({\mathscr {M}}_0)\) of \({\mathscr {M}}_0\) with real coefficients, to a complex of differential forms on \({\mathscr {M}}_0\) associated with a partial desingularization \(\widetilde{{\mathscr {M}}_0}\) of \({\mathscr {M}}_0\) , which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology \(H^{*}(\widetilde{{\mathscr {M}}_0})\) of \(\widetilde{{\mathscr {M}}_0}\) . Based on this, we derive a long exact sequence involving both \(H^{*}({\mathscr {M}}_0)\) and \(H^{*}(\widetilde{{\mathscr {M}}_0})\) and give conditions for its splitting. We then define a Kirwan map \(\mathcal {K}:H_{S^1}^{*}(M) \rightarrow H^{*}(\widetilde{{\mathscr {M}}_0})\) from the equivariant cohomology \(H_{S^1}^{*}(M)\) of M to \(H^{*}(\widetilde{{\mathscr {M}}_0})\) and show that its image contains the image of \(H^{*}({\mathscr {M}}_0)\) in \(H^{*}(\widetilde{{\mathscr {M}}_0})\) under the natural inclusion. Combining both results in the case that all fixed point components of M have vanishing odd cohomology we obtain a surjection \(\check{\kappa }:H^\text {ev}_{S^1}(M) \rightarrow H^\text {ev}({\mathscr {M}}_0)\) in even degrees, while already simple examples show that a similar surjection in odd degrees does not exist in general. As an interesting class of examples we study abelian polygon spaces.
