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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 (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.

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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