Fundamental Groups of Small Covers Revisited

We study the topology of small covers from their fundamental groups. We find a way to obtain explicit presentations of the fundamental group of a small cover. Then we use these presentations to study the relations between the fundamental groups of a small cover and its facial submanifolds. In particular, we can determine exactly when a facial submanifold of a small cover is $\pi_1$-injective in terms of some purely combinatorial condition of the underlying simple polytope. In addition, our study reveals some connections between several topological notions for 3-dimensional small covers. This allows us to determine when a 3-dimensional small cover and its regular $Z_2^k$-covering spaces are Haken manifolds.