Bounded cohomology of transformation groups in all degrees

Abstract

Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge 2$. Let ${\mathcal{T}}_{M}G$ be one of the following transformation groups: $Home{o}_0(M,\mu )$, $Dif{f}_0(M,\mu )$, $Sym{p}_0(M,\omega )$ (in case $M$ is symplectic) and $Ham(M,\omega )$ (in case $M$ is symplectic). Denote by ${\overline{H}}_b^d({\mathcal{T}}_M)$ reduced bounded cohomology of ${\mathcal{T}}_M$ in degree $d$, and by ${\overline{EH}}^d({\mathcal{T}}_M)$ reduced exact bounded cohomology of ${\mathcal{T}}_M$ in degree $d$. In this paper we prove that if $n=2$ and $M$ is a closed surface ${\Sigma }_g$ or a disc $\mathbb{D}$, then for every $d\ge 2$ $$\dim (\overline{H}{b}^d({\mathcal{T}}_M))=\infty .$$ Moreover, $\dim {\overline{EH}}^d({\mathcal{T}}_M)=\infty$ if ${\mathcal{T}}_M$ is either $Dif{f}_0({\Sigma }_g,\mu )$ or $Ham({\Sigma }_g,\omega )$, or $Ham(\mathbb{D},\omega )$; or $g>1$. In case $n>2$, under certain conditions on ${\pi }_1(M)$, we prove that for every degree $d\ge 2$ $$\dim {\overline{EH}}^d({\mathcal{T}}_M)=\infty .$$ In particular, these results hold for ${\mathcal{T}}_M$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

arXiv comment74 pages, 6 figures