Cohomology of matching rules

Details Cohomology of matching rules Cohomology of matching rules

2004-07-29

arxiv.org/abs/math-ph/0407076v1

Physics

Mathematical Physics

Scientific paper

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.

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