Algebraic & Geometric Topology Volume 13, issue 3 (2013)

Download this article
	For screen
	For printing

Recent 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

DOI: 10.2140/agt.2013.13.1661

Bibliography

1	R Ammann, B Grünbaum, G C Shephard, Aperiodic tiles, Discrete Comput. Geom. 8 (1992) 1 MR1156132
2	J E Anderson, I F Putnam, Topological invariants for substitution tilings and their associated $C^$–algebras*, Ergodic Theory Dynam. Systems 18 (1998) 509 MR1631708
3	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 MR1086074
4	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 MR1132337
5	M Barge, H Bruin, L Jones, L Sadun, Homological Pisot substitutions and exact regularity, Isr. J. Math. 188 (2012) 281
6	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)
7	J Bellissard, $K$–theory of $C^{}$–algebras in solid state physics*, from: "Statistical mechanics and field theory: mathematical aspects (Groningen, 1985)" (editor T C Dorlas), Lecture Notes in Phys. 257, Springer (1986) 99 MR862832
8	J Bellissard, R Benedetti, J M Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys. 261 (2006) 1 MR2193205
9	J Bellissard, D J L Herrmann, M Zarrouati, Hulls of aperiodic solids and gap labeling theorems, from: "Directions in mathematical quasicrystals" (editors M B Baake, R V Moody), CRM Monogr. Ser. 13, Amer. Math. Soc. (2000) 207 MR1798994
10	J Bellissard, J Kellendonk, A Legrand, Gap-labelling for three-dimensional aperiodic solids, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 521 MR1834062
11	M T Benameur, H Oyono-Oyono, Index theory for quasi-crystals I: Computation of the gap-label group, J. Funct. Anal. 252 (2007) 137 MR2357353
12	N G de Bruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane I, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 39 MR609465
13	N G de Bruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 53 MR609465
14	A Clark, L Sadun, When shape matters: deformations of tiling spaces, Ergodic Theory Dynam. Systems 26 (2006) 69 MR2201938
15	H Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer (1993) MR1228206
16	L Danzer, Three-dimensional analogs of the planar Penrose tilings and quasicrystals, Discrete Math. 76 (1989) 1 MR1002233
17	B Eick, F Gähler, W Nickel, Computing maximal subgroups and Wyckoff positions of space groups, Acta Cryst. Sect. A 53 (1997) 467 MR1461658
18	B Eick, F Gähler, W Nickel, Cryst – Computing with crystallographic groups, version 4.1.6 (2008)
19	A H Forrest, J Hunton, The cohomology and $K$–theory of commuting homeomorphisms of the Cantor set, Ergodic Theory Dynam. Systems 19 (1999) 611 MR1695911
20	A H Forrest, J R Hunton, J Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys. 226 (2002) 289 MR1892456
21	A Forrest, J Hunton, J Kellendonk, Topological invariants for projection method patterns, Mem. Amer. Math. Soc. 159 (2002) MR1922206
22	F Gähler, Matching rules for quasicrystals: The composition-decomposition method, J. Non-Cryst. Solids 153 (1993) 160
23	F Gähler, Torsion in the homology of the Tübingen triangle tiling, lectures and unpublished notes, Banff and elsewhere (2004)
24	F Gähler, J R Hunton, J Kellendonk, Integer Cech cohomology of icosahedral projection tilings, Z. Krystallogr. 223 (2008) 801
25	F Gähler, J Kellendonk, Cohomology groups for projection tilings of codimension 2, Mat. Sci. Eng. A 294–296 (2000) 438
26	C Irving, Euler characteristics and cohomology for quasiperiodic projection patterns, PhD thesis, University of Leicester (2006)
27	A Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010) 489 MR2599890
28	P Kalugin, Cohomology of quasiperiodic patterns and matching rules, J. Phys. A 38 (2005) 3115 MR2132522
29	J Kaminker, I Putnam, A proof of the gap labeling conjecture, Michigan Math. J. 51 (2003) 537 MR2021006
30	J Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems 28 (2008) 1153 MR2437225
31	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 MR1215344
32	P Kramer, R Neri, On periodic and nonperiodic space fillings of $\mathbb{E}^m$ obtained by projection, Acta Cryst. Sect. A 40 (1984) 580 MR768042
33	P Kramer, Z Papadopolos, Models of icosahedral quasicrystals from 6d lattices, from: "Proceedings of the International Conference on Aperiodic Crystals, Aperiodic '94" (editor G Chapuis), World Scientific (1995) 70
34	R V Moody, Model sets: a survey, from: "From Quasicrystals to More Complex Systems" (editors F Axel, F Dénoyer, J P Gazeau), Centre de physique Les Houches, Springer (2000)
35	A Pavlovitch, M Kléman, Generalised 2D Penrose tilings: structural properties, J. Phys. A 20 (1987) 687 MR880817
36	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 MR2585984
37	L Sadun, Topology of tiling spaces, University Lecture Series 46, American Mathematical Society (2008) MR2446623
38	L Sadun, Exact regularity and the cohomology of tiling spaces, Ergodic Theory Dyn. Syst. 31 (2011) 1819
39	L Sadun, R F Williams, Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems 23 (2003) 307 MR1971208
40	J E S Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B 39 (1989) 519 MR998533
41	W Steurer, S Deloudi, Crystallography of Quasicrystals, Springer Series in Materials Science 126, Springer (2009)
42	The GAP Group, GAP – Groups, Algorithms and Programming, version 4.4.12 (2008)
43	C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994) MR1269324