Integral cohomology of rational projection method patterns

TL;DR

This paper develops a comprehensive theoretical and computational framework for calculating the integral cohomology of aperiodic tilings generated by the cut and project method, which are important in quasicrystal physics.

Contribution

It extends previous work by providing a unified approach and practical toolkit for computing integral cohomology of these patterns, including explicit calculations for key examples.

Findings

Successfully computed cohomology for several icosahedral patterns in R^3

Extended rational cohomology results to integral cohomology

Unified various methods into a comprehensive framework

Abstract

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in $R^3$ -- the Danzer tiling, the Ammann-Kramer tiling and the Canonical and Dual Canonical $D_6$ tilings, including complete computations for the first of these, as well as results for many of the better known 2 dimensional examples.