Combinatorial and geometric methods in topology

TL;DR

This paper explores the role of hyperbolic geometry in three-dimensional topology, focusing on how different 3D spaces can be formed by face gluings of an octahedron, and reviews recent combinatorial and computational advances.

Contribution

It highlights the significance of hyperbolic geometry in 3D topology and reviews recent combinatorial and computational methods in the study of three-manifolds.

Findings

Hyperbolic geometry is central to 3D topology.

Distinct topological spaces can be obtained by face gluings of an octahedron.

Recent combinatorial and computational approaches have advanced understanding of three-manifolds.

Abstract

Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two-dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years.