Symplectic Manifolds with Vanishing Action-Maslov Homomorphism

TL;DR

This paper investigates conditions under which the action--Maslov homomorphism vanishes in monotone symplectic manifolds, proving it is zero for certain product spaces and Grassmannians.

Contribution

It establishes criteria for the vanishing of the action--Maslov homomorphism based on Seidel element order and homology properties, extending understanding of Hamiltonian group topology.

Findings

The homomorphism vanishes when the Seidel element has finite order and homology satisfies property D.

It is shown to be zero for products of projective spaces.

It is also zero for the Grassmannian of 2-planes in C^4.

Abstract

The action--Maslov homomorphism $I\co\pi_1(\text{Ham}(X,\omega))\to\R$ is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property $\mathcal{D}$ (a generalization of having homology generated by divisor classes). We use these results to show that $I=0$ for products of projective spaces and the Grassmannian of $2$ planes in $\C^4$.