The Seidel morphism of cartesian products

TL;DR

This paper investigates how the Seidel morphism behaves under Cartesian products of monotone symplectic manifolds, revealing a relationship that enables understanding the fundamental groups of Hamiltonian diffeomorphism groups of product manifolds.

Contribution

It establishes a natural relation between the Seidel morphisms of product manifolds and their factors, extending previous results and providing new insights into the topology of Hamiltonian diffeomorphism groups.

Findings

Seidel morphism of product relates to factors' Seidel morphisms

Homotopy classes with non-trivial Seidel image induce injections between fundamental groups

Extends and generalizes previous results by Pedroza

Abstract

We prove that the Seidel morphism of $(M \times M', \omega \oplus \omega')$ is naturally related to the Seidel morphisms of $(M,\omega)$ and $(M',\omega')$, when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with non-trivial image via Seidel's morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [P08].