Whitehead products in symplectomorphism groups and Gromov-Witten invariants

Abstract

Consider any symplectic ruled surface $(M^g_λ,ω_λ)$ given by $(Σ_g \times S^2, λσ_{Σ_g} \oplus σ_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_λ;H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_λ$, where $A=[Σ_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_λ^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a new interpretation of the multiplicative structure in the ring $H^*(BG^0_λ ;\Q)$ found by Abreu-McDuff.