A topological proof of the modified Euler characteristic based on the orbifold concept.

Bartosz Naskręcki,Zbigniew Dauter,Mariusz Jaskolski

doi:10.1107/s2053273321004320

Abstract

The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.

Highlights

The famous Euler formula V À E + F = 2 applies to any single 3D solid with V vertices, E edges (1-cells) and F faces (2-cells) that is completely bounded by those ‘surface’ elements
The way to prove this theorem is to reinterpret first the formula m as the orbifold Euler characteristic ðOÞ (Section 3) of the orbifold O 1⁄4 RN=À (Definition 7.2 and Theorem 7.1) for the space group À acting properly discontinuously (Definition 7.1) on the Euclidean space E 1⁄4 RN, and to prove ðOÞ 1⁄4 0 in Theorem 4.2
Since each crystallographic space group À acting on the Euclidean space E 1⁄4 Rn is acting properly discontinuously, the quotient space of orbits E/À has a natural structure of an orbifold (Theorem 7.1)

Summary

Introduction

We provide a completely general proof (applicable to any periodic tessellation of space of any dimension by identical polytopes) rooted entirely in the topological notion of the orbifold and its properties. We recall the fundamental properties of the Euler characteristic, which are well known to topologists, i.e. invariance under the change in cell decomposition, and the multiplicativity property for finite covers of spaces. These ideas are extensively discussed in the excellent textbook by Thurston (2002). The proof is brief and uniform, without going into detailed combinatorics of particular examples, and our argument applies to the action of crystallographic groups in arbitrary dimension

Manifolds and orbifolds

Orbifolds associated with space-filling polyhedra

Proof of the main theorem

Examples of orbifolds which arise from crystallographic space groups

Coxeter’s Nj summation revisited

Applicability