Fundamental groups of 3-dimensional small covers

TL;DR

This paper presents a Morse-theoretic method to explicitly compute the fundamental groups of 3-dimensional small covers, linking their algebraic invariants to minimal Heegaard splittings.

Contribution

It introduces a procedure to obtain explicit, balanced presentations of fundamental groups of orientable small covers with minimal generators.

Findings

Explicit fundamental group presentations for 3D small covers.

Connection between fundamental group presentations and minimal Heegaard splittings.

Method applicable to a broad class of orientable Haken manifolds.

Abstract

Small covers arising from 3-dimensional simple polytopes are an interesting class of 3-manifolds. The fundamental group is a rigid invariant for wide classes of 3-manifolds, particularly for orientable Haken manifolds, which include orientable small covers. By using Morse-theoretic approach we give a procedure to get an explicit, balanced presentation of the fundamental group of a closed, orientable 3-dimensional simple cover with minimal number of generators. Beside that the minimal Heegaard splitting is determined by this presentation.