Noncontractible Hamiltonian loops in the kernel of Seidel's representation

Sílvia Anjos and Rémi Leclercq

Vol. 290 (2017), No. 2, 257–272

DOI: 10.2140/pjm.2017.290.257

Abstract

The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of a 1-parameter family of 2-point blow-ups of $S^2\times S^2$, exactly one of which is 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.

Keywords

symplectic geometry, Seidel morphism, toric symplectic manifolds, Hirzebruch surfaces

Mathematical Subject Classification 2010

Primary: 53D45

Secondary: 53D05, 57S05

Milestones

Received: 6 April 2016

Revised: 9 March 2017

Accepted: 10 March 2017

Published: 25 July 2017