Integer Cech Cohomology of Icosahedral Projection Tilings

TL;DR

This paper computes the integer Cech cohomology of 3D icosahedral projection tilings, revealing that all such tilings have torsion in their cohomology, including the simplest Danzer tiling, contrasting with 2D cases.

Contribution

It provides explicit cohomology formulae for 3D icosahedral tilings and demonstrates the presence of torsion in their cohomology groups, a novel finding.

Findings

All studied 3D icosahedral tilings have torsion in their cohomology.

The Danzer tiling, the simplest icosahedral tiling, also exhibits torsion.

Contrasts with 2D tilings where torsion is often absent.

Abstract

The integer Cech cohomology of canonical projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional canonical projection tilings, where many examples without torsion are known.