On the rank of $\pi_1(\mathrm{Symp}_0)$

TL;DR

This paper constructs specific symplectic 4-manifolds with prescribed first de Rham cohomology dimension and Flux group, exploring the algebraic structure of their symplectomorphism groups.

Contribution

It demonstrates the existence of symplectic 4-manifolds with arbitrary first Betti number and a Flux group equal to a particular integral lattice.

Findings

Existence of symplectic 4-manifolds with arbitrary first Betti number.

Construction of manifolds with Flux group equal to a specified lattice.

Insights into the rank of the fundamental group of symplectomorphism groups.

Abstract

We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold $(M,\omega)$, such that $H^1_\mathrm{dR}(M;\mathbb{R})$ is a $k$-dimensional real vector space and its Flux group is equal to $H^1_\mathrm{dR}(M_1;\mathbb{Z})$.