Alexandrov Geometry in the light of symmetry and topology

亚历山德罗夫空间是具有低截面曲率界的完全黎曼流形的一种推广。然而，由于它们所携带的拓扑和度量奇点，它们可能表现出与黎曼流形不同的行为。因此，研究是否可以将黎曼环境中的给定性质推广到亚历山德罗夫环境中是至关重要的。在这个项目中，我们解决了其中的一些性质，并探讨了亚历山德罗夫空间在每个性质方面的行为。在一个方向上，我们将注意力集中在拓扑特征上。这里的主要目标是了解亚历山德罗夫空间的拓扑与光滑流形的拓扑有多远。更具体地说，一方面，我们的目的是研究Alexandrov空间是否允许三角化的基本和重要问题，另一方面，我们想用有理同伦理论分析Alexandrov空间在对称存在下的上同调性质，如poincar<s:1>对偶性。在另一个方向上，我们研究具有正曲率的Alexandrov空间的同质性，目的是对它们进行分类。为此，我们需要找到障碍和识别工具，特别是依赖于我们对在前一个方向上研究的亚历山德罗夫空间的同质性拓扑行为的理解。

Alexandrov spaces are a generalization of complete Riemannian manifolds with a lower sectional curvature bound. However, they may exhibit behaviors different from Riemannian manifolds due to the topological and metric singularities that they carry. It is then of fundamental importance to investigate whether one can extend a given property in the Riemannian setting to the Alexandrov setting.In this project we address some of these properties and explore how Alexandrov spaces behave with respect to each property. In the one direction, we focus our attention on the topological features. The primary objective here is to understand how far the topology of Alexandrov spaces are from that of the smooth manifolds. More specifically, on the one hand we aim to investigate the basic and important question of whether Alexandrov spaces admit a triangulation and on the other hand we want to analyze, using rational homotopy theory, the cohomological properties of Alexandrov spaces, such as Poincaré duality, in the presence of symmetry. In the other direction, we examine cohomogeneity one Alexandrov spaces with positive curvature and the goal is to classify them. To this end, we need to find obstructions and recognition tools, which, in particular, rely upon our understanding of topological behaviors of cohomogeneity one Alexandrov spaces investigated in the former direction.