Classification Theory of 4-Manifolds

本研究项目是关于拓扑4-流形的几何分类理论。已知的分类技术适用于一类好的基本群，并且通常被猜想是失败的。这个猜想在四维拓扑中仍然是一个中心开放的问题。这个项目的目的之一是证明圆盘嵌入定理，以及随后的外科手术和S-余边定理，对于服从的基本群是成立的。在该项目中探索的另一种手术方法是由圆盘嵌入猜想提供的，直到S-柯博德猜想。4-流形的拓扑与链路切片问题密切相关，本项目的第三部分旨在将有关切片的结果推广到同伦平凡链路的怀特黑德对偶。4维可能的大尺度结构的分类，在局部上看起来像4维时空，在过去的20年里一直是深入研究的焦点。这一研究领域是拓扑学、几何学、分析和物理学的交叉点。这个项目的目的是对具有可服从的基本群的四维物体进行分类，并证明某些具有大的基本群的四维形状的不存在结果，这些形状包含许多不可收缩的回路。

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.