Integer Cech cohomology of icosahedral projection tilings

Cohomology of Canonical Projection Tilings

We define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in Schlottmann’s formulation. We prove the cohomology to be equivalent to a certain cohomology of the lattice. We discuss one of its qualitative features, namely tha...

Cohomology groups for projection tilings of codimension 2

The gap-labelling group, which provides the set of possible values of the integrated density of states on gaps in the spectrum of a Hamiltonian describing particles in a tiling, is frequently related to the cohomology of the tiling. We present explicit results for the cohomology of many well-known tilings obtained from the cut and projection method with codimension 2, including the (generalized...

Generalized Integer Partitions , Tilings of

Quasiperiodic Tilings: A Generalized Grid–Projection Method

We generalize the grid–projection method for the construction of quasiperiodic tilings. A rather general fundamental domain of the associated higher dimensional lattice is used for the construction of the acceptance region. The arbitrariness of the fundamental domain allows for a choice which obeys all the symmetries of the lattice, which is important for the construction of tilings with a give...

The integer cohomology of toric Weyl arrangements

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if T W̃ is the toric arrangement defined by the cocharacters lattice of a Weyl group W̃ , then the integer cohomology of its complement is torsion free.