Pathologies in Alexandrov spaces with curvature bounded above
type: Item
instance of: scholarly article
title: Pathologies in Alexandrov spaces with curvature bounded above
zbMATH Open document ID: 1051.57029
published in: Siberian Advances in Mathematics
publication date: 2004-09-06T00:00:00Z
review text: The author constructs two examples of Alexandrov spaces \(A\) with curvature bounded above, with certain pathological properties. The first is a CAT\((-1)\)-space \(A\) homeomorphic to \(\mathbb R^n\), \(n\geq 5\), whose hyperbolic boundary in the sense of M. Gromov is not a topological manifold. The construction proposed is much simpler than that of the corresponding example by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [J. Differ. Geom. 34, No. 2, 347--386 (1991; Zbl 0723.57017)]. The second example yields \(A\) of nonpositive curvature and entropy dimension (around some point) greater than the (coincident) topological and Hausdorff dimensions. The author also proves that every closed simply connected 3-manifold \(\Sigma^3\) is homeomorphic to the space of directions at a point in \(S^4\) for a CAT(1)-metric on \(S^4\) or to Gromov's hyperbolic boundary for a CAT\((-1)\)-metric on \(\mathbb R^4\).
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 2097032
zbMATH Keywords: topological dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: entropy dimension
author: V. N. Berestovskij
reviewed by: Victor Alexandrov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2097032