Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

Abstract

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of ${\rm SU}(2),\,{\rm U}(2),\,{\rm SO}(3),\,{\rm SO}(3) \times S^1$ and ${{\rm Spin}}^{\mathbb{C}}(3)$. We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism $f:L(p, -1)\times S^1\rightarrow L(p, -1)\times S^1$, the induced isomorphism $(\pi\circ f\circ i)_*$ maps each element in the fundamental group of $L(p, -1)$ to itself or its inverse, where $i:L(p,-1)\rightarrow L(p, -1)\times S^1$ is the natural inclusion and $\pi:L(p, -1)\times S^1\rightarrow L(p, -1)$ is the projection.