Integral cohomology of rational projection method patterns

Gähler, Franz¹ ; Hunton, John² ; Kellendonk, Johannes³

¹Faculty of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany
²The Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
³Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne cedex, France

Algebraic and Geometric Topology, Tome 13 (2013) no. 3, pp. 1661-1708

Résumé

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 ℝ3 – 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.

Détail

DOI : 10.2140/agt.2013.13.1661

Keywords: aperiodic patterns, cut and project, model sets, cohomology, tilings

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