高维流形的对称性与结构

在本项研究计划中我们将对高维流形的对称性和几何结构中的若干问题展开研究.. 1. 流形的对称性. (1)流形的映射类群和微分自同胚群的拓扑.我们将研究一类6维流形的映射类群.这类6维流形包含包含了一批卡-丘流形.这一类流形的映射类群的结果将应用于相应空间上的几何和模空间的研究.进一步我们计划对高维流形映射类群其相关子群，如Torelli群的有限/无限性做出探讨.. (2) 流形上的群作用是流形对称性的一种体现.作为一个重要的例子,我们将研究高连通流形上的有限循环群自由作用的分类问题.. 2. 流形的几何结构拓扑研究中一类重要的问题是,在给定的代数拓扑条件下,流形是否具有某种几何上的结构,如连通和,纤维丛等.判断流形是否是圆周上的纤维丛是拓扑学的经典结果.我们将进一步研究流形成为曲面上的纤维丛的充要条件.

In this research program we are going to study certain problems on the symmetries and geometric structure of high dimensional manifolds.. 1. Symmetries of manifolds. (1) The mapping class group and diffeomorphism group of manifolds. We are going to study the mapping class group of a class of 6 manifolds, which include certain Calabi-Yau 3-folds. The result will have applications to the study of the geometry and moduli problem of these spaces. Further we will study the finitess/infinitess properties of the mapping class groups and the Torelli groups of high dimensional manifolds.. (2) The symmetries of a space are reflected by group actions. As a basic case, we are going to study free actions of cyclic groups on highly-connected manifolds.. 2. Geometric structure of manifolds . It's a fundamental question in geometric topology to study the geometric structure of manifolds with given algebraic topological data. A classical result of Browder-Levine gives the sufficient and necessary condition for a manifold being a fiber bundle over the circle. We are going to study the sufficient and necessary condition for manifolds being a fiber bundle over surfaces