## Algebraic & Geometric Topology

### Integral cohomology of rational projection method patterns

#### 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 $ℝ 3$ – the Danzer tiling, the Ammann–Kramer tiling and the Canonical and Dual Canonical $D 6$ tilings, including complete computations for the first of these, as well as results for many of the better known 2–dimensional examples.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1661-1708.

Dates
Accepted: 4 December 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715595

Digital Object Identifier
doi:10.2140/agt.2013.13.1661

Mathematical Reviews number (MathSciNet)
MR3071138

Zentralblatt MATH identifier
1270.52025

#### Citation

Gähler, Franz; Hunton, John; Kellendonk, Johannes. Integral cohomology of rational projection method patterns. Algebr. Geom. Topol. 13 (2013), no. 3, 1661--1708. doi:10.2140/agt.2013.13.1661. https://projecteuclid.org/euclid.agt/1513715595

#### References

• R Ammann, B Grünbaum, G C Shephard, Aperiodic tiles, Discrete Comput. Geom. 8 (1992) 1–25
• J E Anderson, I F Putnam, Topological invariants for substitution tilings and their associated $C^*$–algebras, Ergodic Theory Dynam. Systems 18 (1998) 509–537
• M Baake, P Kramer, M Schlottmann, D Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in 4–space, Internat. J. Modern Phys. B 4 (1990) 2217–2268
• M Baake, M Schlottmann, P D Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A 24 (1991) 4637–4654
• M Barge, H Bruin, L Jones, L Sadun, Homological Pisot substitutions and exact regularity, Isr. J. Math. 188 (2012) 281–300
• F P M Beenker, Algebraic theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus, Eindhoven University of Technology Report 82-WSK-04 (1982)
• J Bellissard, $K$–theory of $C^{*}$–algebras in solid state physics, from: “Statistical mechanics and field theory: mathematical aspects (Groningen, 1985)”, (T C Dorlas, editor), Lecture Notes in Phys. 257, Springer, Berlin (1986) 99–156
• J Bellissard, R Benedetti, J-M Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys. 261 (2006) 1–41
• J Bellissard, D J L Herrmann, M Zarrouati, Hulls of aperiodic solids and gap labeling theorems, from: “Directions in mathematical quasicrystals”, (M B Baake, R V Moody, editors), CRM Monogr. Ser. 13, Amer. Math. Soc., Providence, RI (2000) 207–258
• J Bellissard, J Kellendonk, A Legrand, Gap-labelling for three-dimensional aperiodic solids, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 521–525
• M-T Benameur, H Oyono-Oyono, Index theory for quasi-crystals I: Computation of the gap-label group, J. Funct. Anal. 252 (2007) 137–170
• N G de Bruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane I, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 39–52
• N G de Bruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 53–66
• A Clark, L Sadun, When shape matters: deformations of tiling spaces, Ergodic Theory Dynam. Systems 26 (2006) 69–86
• H Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer, Berlin (1993)
• L Danzer, Three-dimensional analogs of the planar Penrose tilings and quasicrystals, Discrete Math. 76 (1989) 1–7
• B Eick, F Gähler, W Nickel, Computing maximal subgroups and Wyckoff positions of space groups, Acta Cryst. Sect. A 53 (1997) 467–474
• B Eick, F Gähler, W Nickel, Cryst – Computing with crystallographic groups, version 4.1.6 (2008) Available at \setbox0\makeatletter\@url http://www.gap-system.org/Packages/cryst.html {\unhbox0
• A H Forrest, J Hunton, The cohomology and $K$–theory of commuting homeomorphisms of the Cantor set, Ergodic Theory Dynam. Systems 19 (1999) 611–625
• A H Forrest, J R Hunton, J Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys. 226 (2002) 289–322
• A Forrest, J Hunton, J Kellendonk, Topological invariants for projection method patterns, Mem. Amer. Math. Soc. 159 (2002) x+120
• F Gähler, Matching rules for quasicrystals: The composition-decomposition method, J. Non-Cryst. Solids 153 (1993) 160–164
• F Gähler, Torsion in the homology of the Tübingen triangle tiling, lectures and unpublished notes, Banff and elsewhere (2004)
• F Gähler, J R Hunton, J Kellendonk, Integer Cech cohomology of icosahedral projection tilings, Z. Krystallogr. 223 (2008) 801–804
• F Gähler, J Kellendonk, Cohomology groups for projection tilings of codimension 2, Mat. Sci. Eng. A 294–296 (2000) 438–440
• C Irving, Euler characteristics and cohomology for quasiperiodic projection patterns, PhD thesis, University of Leicester (2006)
• A Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010) 489–523
• P Kalugin, Cohomology of quasiperiodic patterns and matching rules, J. Phys. A 38 (2005) 3115–3132
• J Kaminker, I Putnam, A proof of the gap labeling conjecture, Michigan Math. J. 51 (2003) 537–546
• J Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems 28 (2008) 1153–1176
• R Klitzing, M Schlottmann, M Baake, Perfect matching rules for undecorated triangular tilings with 10-, 12-, and 8–fold symmetry, Internat. J. Modern Phys. B 7 (1993) 1455–1473
• P Kramer, R Neri, On periodic and nonperiodic space fillings of $\mathbb{E}^m$ obtained by projection, Acta Cryst. Sect. A 40 (1984) 580–587
• P Kramer, Z Papadopolos, Models of icosahedral quasicrystals from 6d lattices, from: “Proceedings of the International Conference on Aperiodic Crystals, Aperiodic '94”, (G Chapuis, editor), World Scientific, Singapore (1995) 70–76
• R V Moody, Model sets: a survey, from: “From Quasicrystals to More Complex Systems”, (F Axel, F Dénoyer, J P Gazeau, editors), Centre de physique Les Houches, Springer (2000)
• A Pavlovitch, M Kléman, Generalised 2D Penrose tilings: structural properties, J. Phys. A 20 (1987) 687–702
• I F Putnam, Non-commutative methods for the $K$–theory of $C^*$–algebras of aperiodic patterns from cut-and-project systems, Comm. Math. Phys. 294 (2010) 703–729
• L Sadun, Topology of tiling spaces, University Lecture Series 46, American Mathematical Society (2008)
• L Sadun, Exact regularity and the cohomology of tiling spaces, Ergodic Theory Dyn. Syst. 31 (2011) 1819–1834
• L Sadun, R F Williams, Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems 23 (2003) 307–316
• J E S Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B 39 (1989) 519–551
• W Steurer, S Deloudi, Crystallography of Quasicrystals, Springer Series in Materials Science 126, Springer (2009)
• The GAP Group, GAP – Groups, Algorithms and Programming, version 4.4.12 (2008) Available at \setbox0\makeatletter\@url http://www.gap-system.org/ {\unhbox0
• C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)