Existence of CAT(0)-metrics

Question

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?

Answer 1

Here are some thoughts about your question:

1. Every topological space X which admits a complete CAT(0) metric is contractible, locally contractible and completely metrizable. In the locally compact case, such $X$ also admits a Z-compactification. I cannot think of any further restrictions on topology of such a space. One may conjecture that these are the only restrictions. However, I think, such a conjecture is completely beyond the realm of, currently, provable.

2. One may ask a similar question in the context of simplicial complexes, in which case local contractibility is automatic. The only result I know is in

F. Ancel, C. Guilbault, Interiors of compact contractible n-manifolds are hyperbolic ($n\ge 5$), J. Differential Geom. 45 (1997), no. 1, 1–32.

where the positive answer is given in the case of all contractible tame PL manifolds of dimensions $\ne 4$. (In the case $n=3$ this is a corollary of the Poincare conjecture.)

3. It is proven in

V. Berestovskii, Borsuk's problem on metrization of a polyhedron, Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 273–277

that each simplicial complex admits a (complete) CAT(1) metric. Along the same lines, one may conjecture that each completely metrizable locally contractible topological space admits a CAT(1) metric.

4. Another useful reference is

M. Davis, T. Januszkiewicz, Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347–388