On the fundamental group of real toric varieties

Abstract

Let $X(Δ)$ be the real toric variety associated to a smooth fan $Δ$. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of $X(Δ)$, (ii) to give necessary and sufficient conditions on $Δ$ under which $π_1(X(Δ))$ is abelian, (iii) to give necessary and sufficient conditions on $Δ$ under which $X(Δ)$ is aspherical, and when $Δ$ is complete, (iv) to give necessary and sufficient conditions for $\cc_Δ$ to be a $K(π,1)$ space where $\cc_Δ$ is the complement of a real subspace arrangement associated to $Δ$.