高维流形拓扑学若干问题的研究

在本项研究计划中将考虑以下高维流形拓扑学的问题。.1. 推广的L^2-rho-不变量的研究. rho-不变量是光滑流形的重要微分不变量, L^2-rho-不变量是对流形间的无限复叠,基于von Neumann代数的rho-不变量的推广,具有诸多应用.本项目的申请人进一步将L^2-rho-不变量推广为取值在复叠变换群的von Neumann代数的代数K群中的不变量.此推广的L^2-rho-不变量蕴含了前人定义的数值L^2-rho-不变量,且包含有更多的信息.我们将系统研究此推广的L^2-rho-不变量,包括其性质,与其他不变量间的关系,具体计算,应用等..2. 单连通5维流形上自由Z/k作用的研究.此项研究包括具有给定代数拓扑数据的自由Z/p作用的存在性,以及在代数拓扑性质较简单时自由Z/k作用的完全分类.后者等价于基本群为Z/k的一类5维流形的分类.

This research program consists of the study of the following problems on the topology of high dimensional manifolds..1. Generalized L^2-rho-invariants. rho-invariants are important diffeomorphism invariants of smooth manifolds. L^2-rho-invariant is an L^2-version of the rho-invariants for infinite covers of manifolds, based on ideas and techniques from von Neumann algebra. A generalized L^2-rho-invariant, taking value in the algebraic K-group of the group von Neumann algebra of the deck transformation group, has been obtained by the applicant, which contains more information than the ordinary L^2-rho-invariant. We will systematically study the properties, computations and applications of the generalized L^2-rho-invariant..2. Free Z/k-actions on 1-connected 5-manifolds. This project consists of the study of the existence of free Z/p-actions with given algebraic data, and the classification of free Z/k-actions when the algebro-topological data is simple. The latter is equivalent to the classification of certain 5-manifolds with fundamental group Z/k.