Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity

Abstract

Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with momentum map $J:M\to \mathbb{R}$ and consider the corresponding symplectic quotient ${\mathcal{M}}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex of differential forms on ${\mathcal{M}}_0$, whose cohomology is isomorphic to the singular cohomology $H({\mathcal{M}}_0;\mathbb{R})$ of ${\mathcal{M}}_0$ with real coefficients, to a complex of differential forms on ${\mathcal{M}}_0$ associated with a partial desingularization ${\tilde{\mathcal{M}}}_0$, which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology $H({\tilde{\mathcal{M}}}_0)$ of ${\tilde{\mathcal{M}}}_0$. Based on this, we derive a long exact sequence involving both $H({\mathcal{M}}_0;\mathbb{R})$ and $H({\tilde{\mathcal{M}}}_0)$ and give conditions for its splitting. We then define a Kirwan map $\mathcal{K}:H_{S^1}(M)\to H({\tilde{\mathcal{M}}}_0)$ from the equivariant cohomology $H_{S^1}(M)$ of $M$ to $H({\tilde{\mathcal{M}}}_0)$ and show that its image contains the image of $H({\mathcal{M}}_0;\mathbb{R})$ in $H({\tilde{\mathcal{M}}}_0)$ 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 $\check{\kappa }:H_{S^1}^{\text{ev}}(M)\to H^{\text{ev}}({\mathcal{M}}_0;\mathbb{R})$ 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.

46 Pages