Topological Invariants for Projection Method Patterns

This memoir develops, discusses and compares a range of commutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general - any compact set which is the closure of its interior - while in the second half we concentrate on the so-called canonical patterns. The topological invariants used are various forms of $K$-theory and cohomology applied to a variety of both $C^*$-algebras and dynamical systems derived from such a pattern. The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or self-similarity, but one of the main motivations is also to provide an accessible approach to the the $K_0$ group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern. The main results provide complete descriptions of the (unordered) $K$-theory and cohomology of codimension 1 projection patterns, formule for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canonical projection patterns. Computations are made for the Ammann-Kramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One corollary is that, generically, projection patterns cannot be derived via substitution systems.