Topological regularity theorems for Alexandrov spaces
Usually, Gromov-Hausdorff limits of Riemannian manifolds with curvature bounded uniformly from below are almost never Riemannian manifolds, but they always are Alexandrov spaces, i.e., finite dimensional inner metric spaces with a lower curvature bound in the sense of distance comparison [see, for example, \textit{K. Grove} and \textit{P. Petersen} [J. Differ. Geom. 33, 379-394 (1991; Zbl 0729.53045)] for more details). The main purpose of the paper under review is to give some topological characterization of topological manifolds without boundary for Alexandrov spaces. The main result is as follows: Theorem 1.1. An \(n\)-dimensional Alexandrov space \(X\) is a topological manifold without boundary if and only if it satisfies the following two properties: (1) \(X\) is a homology manifold, that is, \(H_*(X,X-p)\cong H_*(\mathbb R^n, \mathbb R^n -\{0\})\) for any \(p\in X\), and (2) in the case that \(n\geq 4\), the space of directions \(\Sigma_p\) at \(p\) is simply connected for any \(p\in X\). To prove Theorem 1.1, the author first proves that for an Alexandrov space \(X\) the following three statements are equivalent: (i) \(X\) is a homology manifold, (ii) \(X\times\mathbb R^k\) is a topological manifold without boundary for some \(k\geq 1\), and (iii) \(X\times\mathbb R\) is a topological manifold without boundary. Some consequences of Theorem 1.1 are given. Special emphasis is given to the structure of compact nonnegatively curved Alexandrov spaces.
compact nonnegatively curved Alexandrov space
scientific article; zbMATH DE number 1185527
scientific article