Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Cohomology of quotients in real symplectic geometry

### Thomas John Baird and Nasser Heydari

Algebraic & Geometric Topology 22 (2022) 3249–3276
##### Abstract

Given a Hamiltonian system $\left(M,\omega ,G,\mu \right)$, where $\left(M,\omega \right)$ is a symplectic manifold and $G$ is a compact connected Lie group acting on $\left(M,\omega \right)$ with moment map $\mu :M\to {\mathfrak{𝔤}}^{\ast }$, one may construct the symplectic quotient $\left(M∕∕G,{\omega }_{\mathrm{red}}\right)$, where $M∕∕G:={\mu }^{-1}\left(0\right)∕G$. 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 $\left(M,\omega ,G,\mu ,\sigma ,\varphi \right)$ is a Hamiltonian system along with a pair of involutions $\left(\sigma :M\to M,\varphi :G\to G\right)$ satisfying certain compatibility conditions. These imply that the fixed-point set ${M}^{\sigma }$ is a Lagrangian submanifold of $\left(M,\omega \right)$ and that ${M}^{\sigma }∕∕{G}^{\varphi }:=\left({\mu }^{-1}\left(0\right)\cap {M}^{\sigma }\right)∕{G}^{\varphi }$ is a Lagrangian submanifold of $\left(M∕∕G,{\omega }_{\mathrm{red}}\right)$. We prove analogues of Kirwan’s theorems that can be used to calculate the ${ℤ}_{2}$–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 ${ℤ}_{2}$ 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 ${ℤ}_{2}$–Betti numbers of ${M}^{\sigma }∕∕{G}^{\varphi }$.

##### Keywords
Kirwan surjectivity, Lagrangian submanifolds, Lagrangian quotients, real symplectic geometry, Hamiltonian actions, equivariant cohomology
Primary: 53D12