# How to judge a manifold generated by Coxeter system smooth?

For the definition of Coxeter System, you can see: http://en.wikipedia.org/wiki/Coxeter_group

Given a chamber Q, given a Coxeter System $(\Gamma,V)$, we can defined a set M by the following way: defined $M=\Gamma\times X/$~, the ~ is defined by $(g,x)$~$(h,y)$ if and only if x=y and $g^{-1}h\in \Gamma_{V(x)}$

V(x) is the element in V that the reflection generated by the element fixed x. $\Gamma_{(V(x))}$ is the group generated by $V(x)$

if the Coxeter system satisfies that all the $\Gamma_{V(x)}$ is finite, then Davis[1] has a theorem that: the manifold M generated by Q and $(\Gamma,V)$ is a topological manifold.

My question is in what condition can we judge that M is smooth?

If we assume the coxeter system is right angular, can we claim that M is smooth?

[1] Groups Generated by reflections and aspherical manifolds not covered by Euclidean space. Annals of Math. pp.293-324

-