Abstract

We study the symplectomorphism groups G_λ = Symp₀(M,ω_λ) of a closed manifold M equipped with a one-parameter family of symplectic forms ω_λ with variable cohomology class. We show that the existence of nontrivial elements in π_*(A,A′), where (A,A′) is a suitable pair of spaces of almost complex structures, implies the existence of nontrivial elements in π_*−i(G_λ), for i = 1 or 2. Suitable parametric Gromov–Witten invariants detect nontrivial elements in π_*(A,A′). By looking at certain resolutions of quotient singularities we investigate the situation (M,ω_λ) = (S² × S² × X,σ_F ⊕ λσ_B ⊕ ω_arb), with (X,ω_arb) an arbitrary symplectic manifold. We find nontrivial elements in higher homotopy groups of G_λ^X, for various values of λ. In particular we show that the fragile elements w_ℓ found by Abreu and McDuff in π_4ℓ(G_ℓ+1^pt) do not disappear when we consider them in S² × S² × X.

Keywords

symplectomorphism group, Gromov–Witten invariant, almost complex structure

Mathematical Subject Classification

Primary: 57R17

Secondary: 53D35, 53D45, 57S05

Authors