Whitehead products in symplectomorphism groups and Gromov-Witten invariants

Abstract

Consider any symplectic ruled surface $(M_{\lambda }^g,{\omega }_{\lambda })$ given by $({\Sigma }_g\times S^2,\lambda {\sigma }_{{\Sigma }_g}\oplus {\sigma }_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EG{W}_{g,0}(M_{\lambda }^g;H_k,A-kF)$ for all hamiltonian circle actions $H_k$ on $M_{\lambda }^g$, where $A=[{\Sigma }_g\times pt]$ 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 $G_{\lambda }^g$, $g\ge 0$. 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.

22 pages