#### Volume 9, issue 4 (2009)

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The Seidel morphism of Cartesian products

### Rémi Leclercq

Algebraic & Geometric Topology 9 (2009) 1951–1969
##### Abstract

We prove that the Seidel morphism of $\left(M×{M}^{\prime },\omega \oplus {\omega }^{\prime }\right)$ is naturally related to the Seidel morphisms of $\left(M,\omega \right)$ and $\left({M}^{\prime },{\omega }^{\prime }\right)$, when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with nontrivial 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 [Int. Math. Res. Not. (2008) Art. ID rnn049].

##### Keywords
symplectic manifolds, Hamiltonian diffeomorphisms, Seidelś morphism
##### Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57R58, 57S05