Cohomology of matching rules

TL;DR

This paper investigates the cohomology of quasiperiodic patterns with matching rules, showing it is isomorphic to the cohomology of a complement of thickened tori, and computes Betti numbers for various tilings.

Contribution

It establishes a new isomorphism between the cohomology of pattern hulls and complements of torus arrangements, with explicit calculations for several tilings.

Findings

Cohomology ring is isomorphic to that of a torus complement.

Betti numbers computed for multiple tilings match previous results.

Cohomology groups have a module structure over the symmetry group.

Abstract

Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus $T^N$ to an arrangement $A$ of thickened affine tori of codimension two. 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 $T^N \backslash A$ 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.