# Topological Invariants for Projection Method Patterns

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*Alan Forrest; John Hunton; Johannes Kellendonk*

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.

#### Table of Contents

# Table of Contents

## Topological Invariants for Projection Method Patterns

- Table of Contents vii8 free
- General Introduction 112 free
- I: Topological Spaces and Dynamical Systems 1021 free
- 1 Introduction 1021
- 2 The projection method and associated geometric constructions 1122
- 3 Topological spaces for point patterns 1627
- 4 Tilings and point patterns 2031
- 5 Comparing II[sub(u)] and II[sub(u)] 2435
- 6 Calculating M P[sub(u)] and M P[sup(u)] 2637
- 7 Comparing M P[sup(u)] with M P[sup(u)] 3041
- 8 Examples and counter-examples 3344
- 9 The topology of the continuous hull 3748
- 10 A Cantor Z[sup(d)] dynamical system 4051

- II: Groupoids, C*-algebras, and their Invariants 4657
- III: Approaches to Calculation I: Cohomology for Codimension One 6475
- IV: Approaches to Calculation II: Infinitely Generated Cohomology 6980
- V: Approaches to Calculation III: Cohomology for Small Codimension 94105
- Bibliography 116127