Relative family Gromov–Witten invariants and symplectomorphisms

Olguţa Buşe

Vol. 218 (2005), No. 2, 315–341

DOI: 10.2140/pjm.2005.218.315

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 2000

Primary: 57R17

Secondary: 53D35, 53D45, 57S05

Milestones

Received: 29 January 2003

Published: 1 February 2005