Whitehead products in symplectomorphism groups and Gromov-Witten invariants

TL;DR

This paper computes equivariant Gromov-Witten invariants for symplectic ruled surfaces and uses these to demonstrate the nontriviality of certain Whitehead products in the homotopy groups of symplectomorphism groups, providing new insights into their structure.

Contribution

It introduces explicit calculations of equivariant Gromov-Witten invariants for all Hamiltonian circle actions on symplectic ruled surfaces and applies these to study Whitehead products in symplectomorphism groups.

Findings

Computed all equivariant Gromov-Witten invariants for symplectic ruled surfaces.

Demonstrated nontrivial higher order Whitehead products in symplectomorphism groups.

Provided sharper results for genus 0 and 1 cases, answering a question by D. McDuff.

Abstract

Consider any symplectic ruled surface $(M^g_{\lambda},\omega_{\lambda})$ given by $(\Sigma_g \times S^2, \lambda \sigma_{\Sigma_g} \oplus \sigma_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_{\lambda};H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_{\lambda}$, 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 $G_{\lambda}^g$, $g \geq 0$. 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 $H^*(BG^0_{\lambda} ;\Q)$ found by Abreu-McDuff.