Fundamental groups of small covers revisited

TL;DR

This paper investigates the fundamental groups of small covers, providing explicit presentations and exploring their relations with facial submanifolds, leading to classifications of 3D small covers with certain geometric properties.

Contribution

It introduces explicit fundamental group presentations for small covers and characterizes when facial submanifolds are π₁-injective using combinatorial data.

Findings

Explicit fundamental group presentations for small covers

Characterization of π₁-injective facial submanifolds

Classification of 3D small covers with nonnegative scalar curvature

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.