Topology of 4-manifolds, links and Engel groups

流形的拓扑是一个旨在分类形状（流形）的主题，这些形状在给定维度的欧几里得空间上局部建模。四维流形的拓扑是特别感兴趣的，因为这个维度在物理学中是相关的，而研究时空，它是拓扑学中一些突出的开放问题的主题。这方面也是特殊的意义上说，它是边界之间的“低维”，许多分类结果已经解决和“高维”的几何分类问题是已知的承认代数重新。这个项目的目的是分类到具有特殊特征的4-流形的连续变形，称为大基本群。这个项目的主要部分是关于拓扑4-流形的几何分类，特别是拓扑手术猜想和“大”基本群的s-配边猜想。提出了1/2-null手术问题的概念，作为通用手术问题背景下的中间步骤。一个关键的组成部分是群论恩格尔关系，这是密切相关的高阶奇异曲面在4-流形。AB切片问题的框架也被认为是。该项目的另一部分涉及正在进行的工作应用（2+1）维拓扑量子场论的组合和表示的映射类组。PI还计划考虑4维嵌入的几何和拓扑复杂性问题。

Topology of manifolds is a subject aimed at classification of shapes (manifolds) which are locally modeled on the Euclidean space of a given dimension. The topology of 4-dimensional manifolds is of particular interest since this dimension is relevant in physics while studying space-time, and it is the subject of some of the outstanding open problems in Topology. This dimension is also special in the sense that it is borderline between "low dimensions" where many classification results have been settled and "higher dimensions" where geometric classification problems are known to admit an algebraic reformulation. This project is aimed at classification up to continuous deformations of 4-manifolds with special characteristics, known as large fundamental groups. The main part of this project concerns geometric classification of topological 4-manifolds, specifically the topological surgery conjecture and the s-cobordism conjecture for "large" fundamental groups. The notion of a 1/2-null surgery problem is proposed, as an intermediate step in the context of universal surgery problems. A key ingredient is the group-theoretic Engel relation, which is closely related to higher-order singularities of surfaces in 4-manifolds. The framework of the AB-slice problem is also considered. Another part of the project concerns ongoing work on applications of (2+1)-dimensional topological quantum field theories to combinatorics and to representations of mapping class groups. The PI also plans to consider questions in geometric and topological complexity of embeddings in dimension 4.