Cohomology of quotients in real symplectic geometry

Baird, Thomas John; Heydari, Nasser

doi:10.2140/agt.2022.22.3249

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Cohomology of quotients in real symplectic geometry

Thomas John Baird and Nasser Heydari

Algebraic & Geometric Topology 22 (2022) 3249–3276

DOI: 10.2140/agt.2022.22.3249

Abstract

Given a Hamiltonian system $(M, ω, G, μ)$ , where $(M, ω)$ is a symplectic manifold and $G$ is a compact connected Lie group acting on $(M, ω)$ with moment map $μ : M \to 𝔤^{*}$ , 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 $(σ : M \to M, ϕ : G \to G)$ 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 $(M ∕ ∕ G, ω_{red})$ . We prove analogues of Kirwan’s theorems that can be used to calculate the $ℤ_{2}$ –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 $ℤ_{2}$ 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 $ℤ_{2}$ –Betti numbers of $M^{σ} ∕ ∕ G^{ϕ}$ .

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Keywords

Kirwan surjectivity, Lagrangian submanifolds, Lagrangian quotients, real symplectic geometry, Hamiltonian actions, equivariant cohomology

Mathematical Subject Classification 2010

Primary: 53D12

References

Publication

Received: 20 September 2018

Revised: 22 June 2021

Accepted: 28 August 2021

Published: 30 January 2023

Authors

Thomas John Baird
Department of Mathematics & Statistics Memorial University of Newfoundland St. John’s, NL Canada

Nasser Heydari
Department of Mathematics & Statistics Memorial University of Newfoundland St. John’s, NL Canada

Volume 22, issue 7 (2022)