Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

2012-09-02T01:56:50Z

Let $X$ be a complete CAT$(0)$ metric space, 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 $t$.

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 $\mathbb{S}^{n-1}$ ?

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!

Answered. Here is a snippet from the Davis-Januszkiewicz paper Igor cites, describing an $n{=}5$ example where $\partial X \neq \mathbb{S}^4$:

I would still be interested to learn if a similar example is known for $n < 5$.

Answer by Igor Rivin for Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

2012-09-02T02:28:07Z

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.

For a topological manifold the answer is NO, as shown in the same paper.

Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space? - MathOverflow

Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

Answer by Igor Rivin for Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?