The topology of gauge groups and current groups for 3-manifolds

规范群和流群在数学物理中是重要的，规范群在几何中有重要的应用，流群在可积系统中有重要的应用。它们的拓扑与连续变形时保持不变的属性有关。在这两种情况下，群都可以通过映射空间来描述，映射空间更适合于使用代数拓扑学的方法进行分析。映射空间涉及一个基空间X和一个Lie群G，当X是一个曲面或四维流形，而不是三维流形时，最近已经做了大量的工作来研究规范群的拓扑。对当前群体的拓扑结构进行调查的工作很少。这个项目的目的是在3-流形的背景下确定每个流形的重要性质，包括它们的同调和同伦群。

Gauge groups and current groups are important in mathematical physics, with gauge groups having important applications in geometry and current groups having important applications in integrable systems. Their topology is concerned with properties that remain unchanged when continuously deformed. In both cases, the group can be described via mapping spaces, which are better suited to analysis using methods from algebraic topology. The mapping spaces involve a base space X and a Lie group G. A great deal of recent work has been done investigating the topology of gauge groups when X is a surface or a four-manifold, but not a 3-manifold. Little has been done to investigate the topology of current groups. This project aims to determine important properties of each in the context of 3-manifolds, properties including their homology and homotopy groups.