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 $\mu :M\to {\mathfrak{g}}^{\ast }$, 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 $(M,\omega ,G,\mu ,\sigma ,\varphi )$ is a Hamiltonian system along with a pair of involutions $(\sigma :M\to M,\varphi :G\to G)$ satisfying certain compatibility conditions. These imply that the fixed point set $M^{\sigma }$ is a Lagrangian submanifold of $(M,\omega )$ and that $M^{\sigma }//G^{\varphi }:=({\mu }^{-1}(0)\cap M^{\sigma })/G^{\varphi }$ 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 $M^{\sigma }//G^{\varphi }$. In particular, we prove (under appropriate hypotheses) that $|\mu {|}^2$ restricts to a $G^{\varphi }$-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 $G^{\varphi }$ acting on $M^{\sigma }$, and combine these results to produce formulas for the ${\mathbb{Z}}_2$-Betti numbers of $M^{\sigma }//G^{\varphi }$.

29 pages

References in corpus (2)

• The Yang-Mills equations over Klein surfaces
• Moduli spaces of vector bundles over a real curve: Z/2-Betti numbers

Cited by in corpus (1)

• The equivariant cohomology for semidirect product actions