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Topological Invariants for Projection Method Patterns
Alan Forrest, Glasgow, Scotland, John Hunton, University of Leicester, England, and Johannes Kellendonk, Cardiff University, Wales
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Memoirs of the American Mathematical Society
2002; 120 pp; softcover
Volume: 159
ISBN-10: 0-8218-2965-3
ISBN-13: 978-0-8218-2965-3
List Price: US$59
Individual Members: US$35.40
Institutional Members: US$47.20
Order Code: MEMO/159/758
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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.

Readership

Graduate students and research mathematicians interested in convex and discrete geometry.

Table of Contents

  • General introduction
  • Topological spaces and dynamical systems
  • Groupoids, \(C^\ast\)-algebras, and their invariants
  • Approaches to Calculation I: Cohomology for codimension one
  • Approaches to Calculation II: Infinitely generated cohomology
  • Approaches to Calculation III: Cohomology for small codimension
  • Bibliography
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