On the rank of $\pi_1(\text{Ham})$

TL;DR

This paper demonstrates that in symplectic 4-manifolds, the fundamental group of the Hamiltonian diffeomorphism group can have arbitrarily large rank, revealing complex topological structures in symplectic geometry.

Contribution

It constructs examples of symplectic 4-manifolds with Hamiltonian diffeomorphism groups of arbitrarily large fundamental group rank.

Findings

Existence of symplectic 4-manifolds with arbitrarily large fundamental group rank of Hamiltonian diffeomorphisms

Shows the fundamental group of Hamiltonian diffeomorphisms can be arbitrarily large in rank

Provides new insights into the topology of Hamiltonian diffeomorphism groups

Abstract

We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold, such that the rank of the fundamental group of the group of Hamiltonian diffeomorphisms is at least $k.$