发表机构
Ben Gurion University（本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对所有次数下的变换群有界上同调问题，通过分析特定变换群在闭曲面或圆盘等流形上的相关性质，证明了不同情形下变换群的约化有界上同调及约化正合有界上同调的相关结论，尤其适用于任意维数的负曲率闭定向可微流形。
AI 中文摘要
设$M$为维数是$n\ge2$的紧致连通定向光滑流形。设$\mathcal T_MG$为以下变换群之一：$Homeo_0(M,\mu)$、$Diff_0(M,\mu)$、$Symp_0(M, \omega)$（当$M$为辛流形时）以及$Ham(M, \omega )$（当$M$为辛流形时）。记$\overline{H}_b^d(\mathcal T_M)$为$\mathcal T_M$在次数$d$下的约化有界上同调，$\overline{EH}^d(\mathcal T_M)$为$\mathcal T_M$在次数$d$下的约化正合有界上同调。本文证明，若$n=2$且$M$为闭曲面$\Sigma_g$或圆盘$\mathbb D$，则对每个$d\ge 2$，$$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$。此外，若$\mathcal T_M$为$Diff_0(\Sigma_g,\mu)$、$Ham(\Sigma_g, \omega)$或$Ham(\mathbb{D}, \omega)$，或$g>1$，则$\dim\overline{EH}^d(\mathcal T_M)=\infty$。当$n>2$时，在$\pi_1(M)$满足一定条件下，证明对每个次数$d\ge 2$，$$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$。特别地，当$M$为任意维数下具有严格负截面曲率黎曼度量的闭定向光滑流形时，这些结果对$\mathcal T_M$成立。
英文摘要
Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge2$. Let $\mathcal T_MG$ be one of the following transformation groups: $Homeo_0(M,μ)$, $Diff_0(M,μ)$, $Symp_0(M, ω)$ (in case $M$ is symplectic) and $Ham(M, ω)$ (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 $Σ_g$ 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 $Diff_0(Σ_g,μ)$ or $Ham(Σ_g, ω)$, or $Ham(\mathbb{D}, ω)$; or $g>1$. In case $n>2$, under certain conditions on $π_1(M)$, 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.
Comments74 pages, 6 figures
