Relative family Gromov-Witten invariants and symplectomorphisms

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We study the symplectomorphism groups Gλ=Symp₀(M,ωλ) of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms ωλ 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 exiarxiv.org stence of families of nontrivial elements in π*₋ᵢGλ, 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,ωλ)= (S² × S² × X,σF ⊕ λσB ⊕ ωst), with (X,ωst) an arbitrary symplectic manifold. We find families of nontrivial elements in πₖ(Gλˣ), for countably many k and different values of λ. In particular we show that the fragile elements wℓ found by Abreu-McDuff in π₄ ℓ(Gℓ₊₁pt) do not disappear when we consider them in S² × S² × X.