Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:36:14Z http://mathoverflow.net/feeds/question/106147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106147/is-partial-x-a-sphere-for-x-a-complete-cat0-space Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space? Joseph O'Rourke 2012-09-02T01:56:50Z 2012-09-02T14:05:04Z <p>Let $X$ be a <a href="http://en.wikipedia.org/wiki/CAT%28k%29_space" rel="nofollow">complete CAT$(0)$ metric space</a>, and $\partial X$ its boundary. One way to define $\partial X$ is as the equivalence class of geodesic rays $\gamma(t), \gamma'(t)$ that remain within a constant distance of one another for large $t$.</p> <blockquote> <p>Under what conditions and for which $n$ is it known that the boundary of a complete CAT$(0)$ $n$-manifold is homeomorphic to the $(n{-}1)$-sphere $\mathbb{S}^{n-1}$ ?</p> </blockquote> <p>I believe this is known if $X$ is a complete $n$-dimensional Riemannian manifold of nonpositive sectional curvature, but I have not found clear counterexamples otherwise. I am especially interested in $n{=}3$. Pointers would be appreciated, as this area is relatively new to me. Thanks! </p> <p><b>Answered</b>. Here is a snippet from the Davis-Januszkiewicz paper Igor cites, describing an $n{=}5$ example where $\partial X \neq \mathbb{S}^4$: <br />&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://cs.smith.edu/~orourke/MathOverflow/DavisJanuTop.png" alt="Davis-Janu."><br /> I would still be interested to learn if a similar example is known for $n &lt; 5$.</p> http://mathoverflow.net/questions/106147/is-partial-x-a-sphere-for-x-a-complete-cat0-space/106149#106149 Answer by Igor Rivin for Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space? Igor Rivin 2012-09-02T02:28:07Z 2012-09-02T02:28:07Z <p>For a PL manifold, the answer is YES. This is proved by M. Davis and T. Januszkiewicz in Davis, Michael W.(1-OHS); Januszkiewicz, Tadeusz(PL-WROC) Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347–388. </p> <p>For a topological manifold the answer is NO, as shown in the same paper.</p>