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

TL;DR

This paper constructs a Hamiltonian diffeomorphism loop in specific symplectic manifolds that the Seidel morphism does not detect, revealing limitations of the Seidel representation in these contexts.

Contribution

It demonstrates the existence of non-contractible Hamiltonian loops in certain blow-ups of S^2 x S^2 that are invisible to the Seidel morphism, and shows injectivity on Hirzebruch surfaces.

Findings

Identifies Hamiltonian loops undetected by Seidel's morphism in specific blow-ups.

Proves Seidel's morphism is injective on all Hirzebruch surfaces.

Discusses adaptation of examples to the Lagrangian setting.

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.