Seidel's Representation on the Hamiltonian Group of a Cartesian Product

Abstract

Let $(M,ω)$ be a closed symplectic manifold and $\textup{Ham}(M,ω)$ the group of Hamiltonian diffeomorphisms of $(M,ω)$. Then the Seidel homomorphism is a map from the fundamental group of $\textup{Ham}(M,ω)$ to the quantum homology ring $QH_*(M;Λ)$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $ψ$ in $\textup{Ham}(M,ω)$ determines a nontrivial loop $ψ\times\textup{id}_N$ in $\textup{Ham}(M\times N,ω\oplusη)$, where $(N,η)$ is a closed symplectic manifold such that $π_2(N)=0$.