Fundamental groups of small covers revisited

Abstract

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 when a facial submanifold of a small cover is $\pi_1$-injective in terms of some purely combinatorial data on the underlying simple polytope. In addition, we find that any 3-dimensional small cover has an embedded non-simply-connected $\pi_1$-injective surface. Using this result and some results of Schoen and Yau, we characterize all the 3-dimensional small covers that admit Riemannian metrics with nonnegative scalar curvature.