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Topological Invariants for Projection Method Patterns

Alan Forrest , Glasgow, Scotland
John Hunton University of Leicester, Leicester, England
Johannes Kellendonk Cardiff University, Cardiff, Wales
Available Formats:
Electronic ISBN: 978-1-4704-0351-5
Product Code: MEMO/159/758.E
List Price: $62.00 MAA Member Price:$55.80
AMS Member Price: $37.20 Click above image for expanded view Topological Invariants for Projection Method Patterns Alan Forrest , Glasgow, Scotland John Hunton University of Leicester, Leicester, England Johannes Kellendonk Cardiff University, Cardiff, Wales Available Formats:  Electronic ISBN: 978-1-4704-0351-5 Product Code: MEMO/159/758.E  List Price:$62.00 MAA Member Price: $55.80 AMS Member Price:$37.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1592002; 120 pp
MSC: Primary 52; Secondary 19; 37; 46; 55; 82;

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.

• Chapters
• 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
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Volume: 1592002; 120 pp
MSC: Primary 52; Secondary 19; 37; 46; 55; 82;

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.

• II. Groupoids, $C$*-algebras, and their invariants