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Cohomology of quotients in real symplectic geometry

Thomas John Baird and Nasser Heydari

Algebraic & Geometric Topology 22 (2022) 3249–3276

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 𝔤, one may construct the symplectic quotient (MG,ωred ), where MG := μ1(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 MG.

A real Hamiltonian system (M,ω,G,μ,σ,ϕ) is a Hamiltonian system along with a pair of involutions (σ: M M,ϕ: G G) satisfying certain compatibility conditions. These imply that the fixed-point set Mσ is a Lagrangian submanifold of (M,ω) and that MσGϕ := (μ1(0) Mσ)Gϕ is a Lagrangian submanifold of (MG,ωred ). We prove analogues of Kirwan’s theorems that can be used to calculate the 2–Betti numbers of MσGϕ. In particular, we prove (under appropriate hypotheses) that |μ|2 restricts to a Gϕ–equivariantly perfect Morse–Kirwan function on Mσ over 2 coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for Gϕ acting on Mσ, and combine these results to produce formulas for the 2–Betti numbers of MσGϕ.

Kirwan surjectivity, Lagrangian submanifolds, Lagrangian quotients, real symplectic geometry, Hamiltonian actions, equivariant cohomology
Mathematical Subject Classification 2010
Primary: 53D12
Received: 20 September 2018
Revised: 22 June 2021
Accepted: 28 August 2021
Published: 30 January 2023
Thomas John Baird
Department of Mathematics & Statistics
Memorial University of Newfoundland
St. John’s, NL
Nasser Heydari
Department of Mathematics & Statistics
Memorial University of Newfoundland
St. John’s, NL