TITLE: Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
QUESTION [16 upvotes]: 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 $n < 5$.

REPLY [13 votes]: For a piecewise Euclidean or piecewise hyperbolic metric on a PL manifold, the answer is YES. This is proved (p348 in published version) by M. Davis and T. Januszkiewicz in 
M. Davis, T.Januszkiewicz, Hyperbolization of polyhedra. 
J. Differential Geom. 34 (1991), no. 2, 347–388. Link: projecteuclid or (non-paywalled) Davis' page.
For a topological manifold with a piecewise Euclidean or piecewise hyperbolic metric, the answer is NO, as shown in the same paper (Section 5).