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

Abstract

Let $(M,\omega )$ be a closed symplectic manifold and $\text{Ham}(M,\omega )$ the group of Hamiltonian diffeomorphisms of $(M,\omega )$. Then the Seidel homomorphism is a map from the fundamental group of $\text{Ham}(M,\omega )$ to the quantum homology ring $Q{H}_{\ast }(M;\Lambda )$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $\psi$ in $\text{Ham}(M,\omega )$ determines a nontrivial loop $\psi \times {\text{id}}_N$ in $\text{Ham}(M\times N,\omega \oplus \eta )$, where $(N,\eta )$ is a closed symplectic manifold such that ${\pi }_2(N)=0$.

13 pages