Cohomology of Quotients in Real Symplectic Geometry

Abstract

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