On the rank of ${\pi }_1({\mathrm{Symp}}_0)$

Abstract

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

Error in the section 2, in the construction of a loop of sympletic diffeomorphisms