Cohomology of quasiperiodic patterns and matching rules

Abstract

Quasiperiodic patterns described by polyhedral 'atomic surfaces' are considered. It is shown that under certain rationality conditions (which coincide with the necessary conditions for the existence of matching rules), the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus TN to an arrangement A of thickened affine tori of codimension 2. Explicit computation of Betti numbers for several two-dimensional tilings and for the icosahedral Ammann–Kramer tiling confirms in most cases the results obtained previously by different methods. The cohomology groups of TN\A have a natural structure of a right module over the group ring of the space symmetry group of the pattern and can be decomposed in a direct sum of its irreducible representations. An example of such decomposition is shown for the Ammann–Kramer tiling.