Volume 13, issue 3 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 6, 2971–3570
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
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 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.

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/