Topological regularity theorems for Alexandrov spaces
type: Item
instance of: scholarly article
title: Topological regularity theorems for Alexandrov spaces
zbMATH Open document ID: 0915.53021
author: Jyh-Yang Wu
published in: Journal of the Mathematical Society of Japan
publication date: 1999-06-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 1185527
zbMATH Keywords: Gromov-Hausdorff convergence theory
zbMATH Keywords: topological manifold
zbMATH Keywords: compact nonnegatively curved Alexandrov space
zbMATH Keywords: sectional curvature
zbMATH Keywords: diameter
zbMATH Keywords: contractibility radius
zbMATH Keywords: homology manifold
reviewed by: Victor Alexandrov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04940741
OpenAlex ID: W2027914259
DOI: 10.2969/JMSJ/04940741
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description: scientific article; zbMATH DE number 1185527