2012/02/29 by Franz Gähler, Franz Gaehler, John Hunton +1 · 19 citations
Engineering · Materials Science · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Algebra over a field #Aperiodic graph #Cohomology #Dual (grammatical number) #Euclidean geometry #Euclidean space #Manifold (fluid mechanics) #Metamaterials and Metasurfaces Applications #Projection (relational algebra) #Quasicrystal #Quasicrystal Structures and Properties #Topological space #math-ph #math.AT #math.KT #math.MP #msc:52C22 #msc:52C23 #msc:55R20
paper · pdf · doi:10.2140/agt.2013.13.1661
published in Algebraic & Geometric Topology 13(3), 1661-1708 (Mathematical Sciences Publishers) · Extends, corrects and replaces 2005 preprint math-ph/0505048 'Torsion in Tiling Homology and Cohomology'. V2 corrects some calculations in math.KT/1202.2240v1
arxiv created 2012/10/26 · openalex publication_date 2013/05/16 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.