Classification Theory of 4-Manifolds

本研究课题是关于拓扑4-流形的几何分类理论。已知分类技术适用于一类好的基本群，但一般都失败了。这个猜想在四维拓扑学中仍然是一个中心的开放问题。这个项目的一个目标是证明圆盘嵌入定理，以及随后的手术和s-协边定理，对顺从的基本群成立。在该项目中探索的另一种手术方法是由直到s-配边的盘嵌入猜想提供的。4-流形的拓扑与链切片问题密切相关，本项目的第三部分旨在将切片结果推广到同伦平凡链的Whitehead双。在过去的20年里，四维空间中可能存在的大尺度结构的分类一直是人们研究的热点。这一领域的研究在于拓扑学，几何学，分析和物理学的交叉点。这个项目的目的是分类的四维物体与顺从的基本群体，并在证明不存在的结果，某些四维形状与大的基本群体，其中包含许多循环，不能收缩。

This research project concerns the geometric classification theory of topological 4-manifolds. The classification techniquesare known to hold for a class of good fundamental groups, and are conjectured to fail in general. This conjecture remains a centralopen problem in 4-dimensional topology.One goal of this project is to prove that the disk embedding theorem, and consequently the surgery and s-cobordism theorems, hold for amenable fundamental groups. Another approach to surgery, explored in the project, is provided by the disk embedding conjecture up to s-cobordism. The topology of 4-manifolds is closely related to link-slicing problems, and the third part of the project aims at extending the results about slicing to Whitehead doubles of homotopically trivial links. The classification of possible large-scale structures in dimension 4, which locally look like the 4-dimensional space-time, has beena focus of intensive research over the past 20 years. This area of research lies at the intersection of topology, geometry, analysis and physics. This project is aimed at classification of 4-dimensional objects with amenable fundamental groups, andat proving the non-existence result of certain 4-dimensional shapes with large fundamental groups, which contain many loops that cannot be contracted.