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

TL;DR

This paper investigates how Seidel's representation affects the Hamiltonian group of a product symplectic manifold, providing conditions under which nontrivial loops in the Hamiltonian group of one factor induce nontrivial loops in the product.

Contribution

It establishes a sufficient condition linking Seidel's homomorphism on a symplectic manifold to the nontriviality of loops in the Hamiltonian group of its Cartesian product.

Findings

Nontrivial loops in $ extup{Ham}(M, extomega)$ can induce nontrivial loops in $ extup{Ham}(M imes N, extomega igoplus extomega')$ under certain conditions.

The condition involves the Seidel homomorphism and the property $ extpi_2(N)=0$.

Provides a criterion for detecting nontrivial Hamiltonian loops in product manifolds.

Abstract

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 $\pi_2(N)=0$.