Let $(M,\omega)$ be a closed symplectic manifold and $\textup{ Ham}(M,\omega)$ the group of Hamiltonian diffeomorphisms of $(M,\omega)$. Then the Seidel homomorphism is a map from the fundamental group of $\textup{ Ham}(M,\omega)$ to the quantum homology ring $QH_*(M;\Lambda)$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $\psi$ in $\textup{ Ham}(M,\omega)$ determines a nontrivial loop $\psi\times\textup{ id}_N$ in $\textup{ Ham}(M\times N,\omega\oplus\eta)$, where $(N,\eta)$ is a closed symplectic manifold such that $\pi_2(N)=0$.
Categories: math.SG math.AT