Pathologies in Alexandrov spaces with curvature bounded above
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 \(\mathbb R^4\).
scientific article; zbMATH DE number 2097032