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

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

2008-05-09

arxiv.org/abs/0805.1375v1

Mathematics

Symplectic Geometry

13 pages

Scientific paper

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 $\pi_2(N)=0$.

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