Topology of 4-manifolds, links and Engel groups

流形的拓扑学是一门研究在给定维数欧几里德空间上局部建模的形状（流形）分类的学科。四维流形的拓扑是特别有趣的，因为这个维度与研究时空的物理相关，并且它是拓扑学中一些突出的开放问题的主题。这个维度的特殊之处在于，它介于“低维度”和“高维度”之间，在“低维度”中，许多分类结果已经得到了解决，而在“高维度”中，已知几何分类问题允许代数重新表述。该项目旨在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“大”基群的拓扑外科猜想和s协猜想。提出了1/2-零手术问题的概念，作为通用手术问题背景下的中间步骤。一个关键的因素是群论恩格尔关系，它与4流形中曲面的高阶奇异性密切相关。本文还考虑了ab片问题的框架。该项目的另一部分涉及正在进行的（2+1）维拓扑量子场论在组合学和映射类群表示中的应用。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.