Non-contractible Hamiltonian loops in the kernel of Seidel's representation

Abstract

The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone. As side remarks, we show that Seidel's morphism is injective on all Hirzebruch surfaces and discuss how to adapt the monotone example to the Lagrangian setting.Comment: 13 pages. In the second version the title is changed to emphasize the actual point of the paper and a "background" section is added so the paper is more self-containe