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Cohomology of substitution tiling spaces

Published online by Cambridge University Press:  04 November 2009

MARCY BARGE ,
BEVERLY DIAMOND ,
JOHN HUNTON  and
LORENZO SADUN
MARCY BARGE
Affiliation:
Department of Mathematics, Montana State University, Bozeman, MT 59717, USA (email: [email protected])
BEVERLY DIAMOND
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA (email: [email protected])
JOHN HUNTON
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK (email: [email protected])
LORENZO SADUN
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: [email protected])
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Abstract

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1607 - 1627
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C *-algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509–537.CrossRefGoogle Scholar
[2]Bellissard, J.. Gap labeling theorems for Schrödinger operators. From Number Theory to Physics (Les Houches, 1989). Springer, Berlin, 1992, pp. 538–630.CrossRefGoogle Scholar
[3]Bellissard, J., Benedetti, R. and Gambaudo, J.-M.. Spaces of tilings finite telescopic approximations and gap-labelling. Comm. Math. Phys. 261 (2006), 1–41.CrossRefGoogle Scholar
[4]Barge, M. and Diamond, B.. Cohomology in one-dimensional substitution tiling spaces. Proc. Amer. Math. Soc. 136(6) (2008), 2183–2191.CrossRefGoogle Scholar
[5]Forrest, A., Hunton, J. and Kellendonk, J.. Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 758 (2002).Google Scholar
[6]Frank, N. P. and Sadun, L.. Topology of some tiling spaces without finite local complexity. Discrete Contin. Dyn. Syst. 23 (2009), 847–865.CrossRefGoogle Scholar
[7]Gaehler, F.. Lectures given at workshops Applications of Topology to Physics and Biology, Max-Planck-Institut for Physik komplexer Systeme, Dresden, June 2002, and Aperiodic Order, Dynamical Systems, Operator Algebras and Topology, Victoria, British Columbia, August, 2002.Google Scholar
[8]Gaehler, F., Hunton, J. and Kellendonk, J.. Integer Cech cohomology of projection tilings. Z. Krist. 223 (2008), 801–804.CrossRefGoogle Scholar
[9]Holton, C., Radin, C. and Sadun, L.. Conjugacies for tiling dynamical systems. Comm. Math. Phys. 254 (2005), 343–359.CrossRefGoogle Scholar
[10]Kellendonk, J.. The local structure of tilings and their integer group of coinvariants. Comm. Math. Phys. 187 (1997), 115–157.CrossRefGoogle Scholar
[11]Kellendonk, J. and Putnam, I.. Tilings, C *-algebras and K-theory. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). Eds. Baake, M. and Moody, R. V.. American Mathematical Society, Providence, RI, 2000.Google Scholar
[12]Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99 (1992), 327–334.CrossRefGoogle Scholar
[13]Ormes, N., Radin, C. and Sadun, L.. A homeomorphism invariant for substitution tiling spaces. Geom. Dedicata 90 (2002), 153–182.CrossRefGoogle Scholar
[14]Radin, C.. The pinwheel tilings of the plane. Ann. of Math. (2) 139 (1994), 661–702.CrossRefGoogle Scholar
[15]Sadun, L.. Topology of Tiling Spaces (University Lecture Series, 46). American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
[16]Shechtman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1953.CrossRefGoogle Scholar
[17]Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265–279.CrossRefGoogle Scholar
[18]Wang, H.. Proving theorems by pattern recognition—II. Bell Syst. Tech. J. 40 (1961), 1–41.CrossRefGoogle Scholar