|
|||||
|
|
|
|
|
|
|
|
|
Integral cohomology of
rational projection method patterns
Franz Gähler, John Hunton and Johannes Kellendonk |
|
Algebraic & Geometric Topology 13 (2013) 1661–1708
|
Abstract |
|
We study the cohomology and hence –theory of the aperiodic tilings formed by the so called “cut and project” method, that is, patterns in –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 – the Danzer tiling, the Ammann–Kramer tiling and the Canonical and Dual Canonical tilings, including complete computations for the first of these, as well as results for many of the better known 2–dimensional examples. |
Keywords
aperiodic patterns, cut and project, model sets,
cohomology, tilings
|
Mathematical Subject Classification 2000
Primary: 52C23
Secondary: 52C22, 55R20
|
References |
Publication
Received: 10 February 2012
Accepted: 4 December 2012
Published: 16 May 2013
|
Authors |
| Franz Gähler | |
| Faculty of Mathematics University of Bielefeld D-33615 Bielefeld Germany |
|
| http://www.math.uni-bielefeld.de/~gaehler/ | |
| John Hunton | |
| The Department of Mathematics University of Leicester Leicester LE1 7RH UK |
|
| http://www.le.ac.uk/people/jrh7/ | |
| Johannes Kellendonk | |
| Université de Lyon Université Claude Bernard Lyon 1 Institut Camille Jordan CNRS UMR 5208 43 Boulevard du 11 Novembre 1918 F-69622 Villeurbanne cedex France |
|
| http://math.univ-lyon1.fr/~kellendonk/ | |
|
