Integral cohomology of rational projection method patterns

Abstract

We study the cohomology and hence $K$–theory of the aperiodic tilings formed by the so called “cut and project” method, that is, 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.