On the fundamental group of real toric varieties

Let $X(\Delta)$ be the real toric variety associated to a smooth fan $\Delta$. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of $X(\Delta)$, (ii) to give necessary and sufficient conditions on $\Delta$ under which $\pi_1(X(\Delta))$ is abelian, (iii) to give necessary and sufficient conditions on $\Delta$ under which $X(\Delta)$ is aspherical, and when $\Delta$ is complete, (iv) to give necessary and sufficient conditions for $\cc_{\Delta}$ to be a $K(\pi,1)$ space where $\cc_{\Delta}$ is the complement of a real subspace arrangement associated to $\Delta$.