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

规范群和电流群在数学物理中有着重要的意义，规范群在几何中有着重要的应用，而电流群在可积系统中有着重要的应用。它们的拓扑学关注的是在连续变形时保持不变的特性。在这两种情况下，群都可以通过映射空间来描述，映射空间更适合使用代数拓扑的方法进行分析。映射空间包括基空间X和李群g。最近有大量的工作是研究当X是曲面或四流形时规范群的拓扑结构，而不是三流形。很少有人研究当前群的拓扑结构。这个项目的目的是确定在3流形的背景下每个重要的性质，包括它们的同调和同伦群的性质。

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.