Cohomology of Quotients in Real Symplectic Geometry

Abstract

Given a Hamiltonian system $ (M,\omega, G,\mu) $ where $(M,\omega)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,\omega)$ with moment map $ \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, \omega_{red})$ where $M//G := \mu^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|\mu|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,\omega, G,\mu, \sigma, \phi) $ is a Hamiltonian system along with a pair of involutions $(\sigma:M \rightarrow M, \phi:G \rightarrow G) $ satisfying certain compatibility conditions. These imply that the fixed point set $M^{\sigma}$ is a Lagrangian submanifold of $(M,\omega)$ and that $M^{\sigma}//G^{\phi} := (\mu^{-1}(0) \cap M^{\sigma})/G^{\phi}$ is a Lagrangian submanifold of $(M//G, \omega_{red})$. In this paper we prove analogues of Kirwan's Theorems that can be used to calculate the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi} $. In particular, we prove (under appropriate hypotheses) that $|\mu|^2$ restricts to a $G^{\phi}$-equivariantly perfect Morse-Kirwan function on $M^{\sigma}$ over $\mathbb{Z}_2$ coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for $G^{\phi}$ acting on $M^{\sigma}$, and combine these results to produce formulas for the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi}$.