Flat 2-orbifolds and Seifert fibred 4-manifolds

TL;DR

This paper explores the properties of flat 2-orbifold groups and their relation to Seifert fibred 4-manifolds, focusing on geometric, topological, and covering aspects of these structures.

Contribution

It provides a detailed analysis of flat 2-orbifold groups and their connection to Seifert fibrations in 4-manifolds, including decomposition and covering relations.

Findings

Relation between geometric and topological presentations of orbifold groups

Descriptions of orbifold decompositions as fibrations or unions

Classification of Seifert fibrations in solvable Lie type geometries

Abstract

This is a summary of some of the basic facts about flat 2-orbifold groups, otherwise known as 2-dimensional crystallographic groups. We relate the geometric and topological presentations of these groups, and consider structures corresponding to decompositions of the orbifolds as fibrations or as unions. We also consider covering relations, and record the bases of Seifert fibrations of 4-manifolds with geometries of solvable Lie type.