Singular Cohomology of Symplectic Quotients by Circle Actions and Kirwan Surjectivity

Let M be a compact symplectic manifold carrying a Hamiltonian S¹ -action with momentum map J:M → R and consider the corresponding symplectic quotient M₀:= J⁻¹(0)/S¹ . We extend Sjamaar’s complex of differential forms on M₀ , whose cohomology is isomorphic to the singular or Čech cohomology H*(M₀) of M₀ with real coefficients, to a complex of differential forms on M₀ associated with a partial desingularization M₀ of M₀ , which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology H*(M₀) of M₀ . Based on this, we derive a long exact sequence involving both H*(M₀) and H*(M₀) and give conditions for its splitting. We then define a Kirwan map K:HS¹*(M) → H*(M₀) from the equivariant cohomology HS¹*(M) of M to H*(M₀) and show that its image contains the image of H*(M₀) in H*(M₀) 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 κ :HᵉvS¹(M) → Hᵉv(M₀) 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.