TITLE: Metric spheres in CAT(0) manifolds
QUESTION [9 upvotes]: Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are sufficiently small metric spheres in $X$ homeomorphic to metric spheres in Euclidean space $\mathbb{E}^n$?
[In "Ideal boundary of CAT(0) spaces" (1998) by Myung-Jin Jeon, this was unclear to the author; see the bottom of Page 104. That paper examined geodesic completeness for CAT(0) manifolds.]
EDIT: It is well-known that sufficiently small metric balls in any CAT(0) space are contractible.  Using the manifold property, I believe that my question reduces to a very basic one:
If $U$ is a contractible open subset of $\mathbb{R}^n$, then is $U$ homeomorphic to $\mathbb{R}^n$?

REPLY [16 votes]: The answer is no. By the double suspension theorem of Cannon and Edwards, if $X^n$ is a  homology $n$-sphere, then the double suspension $S^2X$ is a sphere. In particular, the cone $CSX$ on $SX$ will be a simply connected manifold homeomorphic to $R^{n+2}$, since it is obtained from the double suspension by deleting a point. If there is a simplicial complex structure on $SX$ which is flag (e.g. by barycentric subdivision), then the cone $CSX$ will have a CAT(0) metric. The point is that one may make each of the simplices of the triangulation of $SX$ into a right-angled $n+1$ spherical simplex, then take the metric cone on this to get a quadrant of $R^{n+2}$, and glue these quadrants together in the manner prescribed by the simplicial complex. The metric ball about the cone point of $CSX$ will not be homeomorphic to a ball, since its boundary will be $SX$ which is not a manifold. Check out the papers of Davis-Januskiewicz for constructions of CAT(0) complexes.