Topological invariants for projection method patterns
About this Title
Alan Forrest, John Hunton and Johannes Kellendonk
Publication: Memoirs of the American Mathematical Society
Publication Year 2002: Volume 159, Number 758
ISBNs: 978-0-8218-2965-3 (print); 978-1-4704-0351-5 (online)
MathSciNet review: 1922206
MSC: Primary 37B50; Secondary 19E20, 37B05, 46L55, 52C23, 55T05
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, formulæ 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.
Graduate students and research mathematicians interested in convex and discrete geometry.
Table of Contents
- General introduction
- I. Topological spaces and dynamical systems
- II. Groupoids, $C$*-algebras, and their invariants
- III. Approaches to calculation I: Cohomology for codimension one
- IV. Approaches to calculation II: Infinitely generated cohomology
- V. Approaches to calculation III: Cohomology for small codimension