On the fundamental group of real toric varieties

Let X(∆) be the real toric variety associated to a smooth fan ∆. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of X(∆), (ii) to give necessary and sufficient conditions on ∆ under which π1(X(∆)) is abelian, (iii) to give necessary and sufficient conditions on ∆ under which X(∆) is aspherical, and when ∆ is complete, (iv) to give necessary and sufficient conditions for C∆ to be a K(π,1) space where C∆ is the complement of a real subspace arrangement associated to ∆.