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Integral cohomology of rational projection method patterns

Franz Gähler, John Hunton and Johannes Kellendonk

Algebraic & Geometric Topology 13 (2013) 1661–1708

We study the cohomology and hence K–theory of the aperiodic tilings formed by the so called “cut and project” method, that is, patterns in d–dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in 3 – the Danzer tiling, the Ammann–Kramer tiling and the Canonical and Dual Canonical D6 tilings, including complete computations for the first of these, as well as results for many of the better known 2–dimensional examples.

aperiodic patterns, cut and project, model sets, cohomology, tilings
Mathematical Subject Classification 2000
Primary: 52C23
Secondary: 52C22, 55R20
Received: 10 February 2012
Accepted: 4 December 2012
Published: 16 May 2013
Franz Gähler
Faculty of Mathematics
University of Bielefeld
D-33615 Bielefeld
John Hunton
The Department of Mathematics
University of Leicester
Leicester LE1 7RH
Johannes Kellendonk
Université de Lyon
Université Claude Bernard Lyon 1
Institut Camille Jordan
43 Boulevard du 11 Novembre 1918
F-69622 Villeurbanne cedex