Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-27T03:45:28.366Z Has data issue: false hasContentIssue false
  • English
  • Français

A spectral sequence for the K-theory of tiling spaces

Published online by Cambridge University Press:  01 June 2009

JEAN SAVINIEN  and
JEAN BELLISSARD
JEAN SAVINIEN
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA (email: savinien@math.gatech.edu, jeanbel@math.gatech.edu)
JEAN BELLISSARD
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA (email: savinien@math.gatech.edu, jeanbel@math.gatech.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let 𝒯 be an aperiodic and repetitive tiling of ℝd with finite local complexity. We present a spectral sequence that converges to the K-theory of 𝒯 with page-2 given by a new cohomology that will be called PV in reference to the Pimsner–Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of 𝒯 generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to the Čech cohomology of the hull of 𝒯 (a compactification of the family of its translates).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 997 - 1031
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associatedC *-algebra. Ergod. Th. Dynam. Sys. 18 (1998), 509–537.CrossRefGoogle Scholar
[2]Atiyah, M. F. and Hirzebruch, F.. Vector Bundles and Homogeneous Spaces (Proceedings of Symposia in Pure Mathematics, 3). American Mathematical Society, Providence, RI, 1961, pp. 7–38.Google Scholar
[3]Atiyah, M. F. and Singer, I. M.. The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), 422–433.CrossRefGoogle Scholar
[4]Atiyah, M. F., Segal, G. B. and Singer, I. M.. The index of elliptic operators. I, II, III. Ann. of Math. (2) 87 (1968), 484–530, 531–545, 546–604.CrossRefGoogle Scholar
[5]Bellissard, J.. Schrödinger’s Operators with an Almost Periodic Potential: An Overview (Lecture Notes in Physics, 153). Springer, Berlin, 1982.Google Scholar
[6]Bellissard, J.. K-theory of C *-algebras in solid state physics. Statistical Mechanics and Field Theory, Mathematical Aspects (Lecture Notes in Physics, 257). Eds. T. C. Dorlas, M. N. Hugenholtz and M. Winnink. Springer, Berlin, 1986, pp. 99–156.CrossRefGoogle Scholar
[7]Bellissard, J., Iochum, B., Scoppola, E. and Testard, D.. Spectral properties of one dimensional quasi-crystals. Comm. Math. Phys. 125 (1989), 527–543.CrossRefGoogle Scholar
[8]Bellissard, J., Bovier, A. and Ghez, J. M.. Gap labelling theorem for one dimensional Schrödinger operators. Rev. Math. Phys. 4(1) (1992), 1–37.CrossRefGoogle Scholar
[9]Bellissard, J.. Gap labelling theorems for Schrödinger’s operators. From Number Theory to Physics (Les Houches March 89). Eds. J. M. Luck, P. Moussa and M. Waldschmidt. Springer, Berlin, 1993,pp. 538–630.Google Scholar
[10]Bellissard, J., Contensou, E. and Legrand, A.. K-théorie des quasi-cristaux, image par la trace: le cas du réseau octogonal. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 197–200.CrossRefGoogle Scholar
[11]Bellissard, J., Hermmann, D. and Zarrouati, M.. Hull of aperiodic solids and gap labeling theorems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). Eds. M.B. Baake and R.V. Moody. American Mathematical Society, Providence, RI, 2000, pp. 207–259.Google Scholar
[12]Bellissard, J., Kellendonk, J. and Legrand, A.. Gap-labelling for three-dimensional aperiodic solids. C. R. Acad. Sci. Paris, Sér. I Math. 332 (2001), 521–525.CrossRefGoogle Scholar
[13]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
[14]Benedetti, R. and Gambaudo, J.-M.. On the dynamics of 𝔾-solenoids. Applications to Delone sets. Ergod. Th. Dynam. Sys. 23 (2003), 673–691.CrossRefGoogle Scholar
[15]Blackadar, B.. K-Theory for Operator Algebras, 2nd edn. Cambridge University Press, Cambridge, 1998.Google Scholar
[16]Borel, A. and Serre, J.-P.. Impossibilité de fibrer un espace euclidien par des fibres compactes. C. R. Acad. Sci. Paris 230 (1950), 2258–2260.Google Scholar
[17]Bratelli, O.. Inductive limits of finite dimensional C *-algebras. Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
[18]Claro, F. H. and Wannier, G. H.. Closure of bands for bloch electrons in a magnetic field. Phys. Status Solidi b 88 (1978), K147–K151.CrossRefGoogle Scholar
[19]Coburn, L. A., Moyer, R. D. and Singer, I. M.. C *-algebra of almost periodic pseudo-differential operators. Acta Math. 130 (1973), 279–307.CrossRefGoogle Scholar
[20]Connes, A.. Sur la théorie non commutative de l’intégration. Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) (Lecture Notes in Mathematics, 725). Springer, Berlin, 1979, pp. 19–143.CrossRefGoogle Scholar
[21]Connes, A.. An analogue of the Thom isomorphism for crossed products of a C *-algebra by an action of R. Adv. Math. 39(1) (1981), 31–55.CrossRefGoogle Scholar
[22]Connes, A.. A survey of foliations and operator algebras. Operator Algebras and Applications, Part I, (Kingston, Ontario, 1980) (Proceedings of Symposia in Pure Mathematics, 38). American Mathematical Society, Providence, RI, 1982, pp. 521–628.CrossRefGoogle Scholar
[23]Connes, A.. Géométrie Non Commutative. InterEditions, Paris, 1990.Google Scholar
[24]Connes, A.. Noncommutative Geometry. Academic Press, San Diego, 1994.Google Scholar
[25]Davis, J. F. and Kirk, P.. Lecture Notes in Algebraic Topology (AMS Graduate Studies in Mathematics, 35). American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
[26]Van Elst, A.. Gap-labelling theorems for Schrödinger operators on the square and cubic lattice. Rev. Math. Phys. 6 (1994), 319–342.CrossRefGoogle Scholar
[27]Forrest, A. H. and Hunton, J. R.. The cohomology and K-theory of commuting homeomorphisms of the Cantor set. Ergod. Th. & Dynam. Sys. 19 (1999), 611–625.CrossRefGoogle Scholar
[28]Forrest, A. H., Hunton, J. R. and Kellendonk, J.. Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(758) (2002).Google Scholar
[29]Gähler, F., Hunton, J. and Kellendonk, J.. Torsion in tiling homology and cohomology. arXiv.com math-phys/0505048, May 2005.Google Scholar
[30]Gambaudo, J.-M. and Martens, M.. Algebraic topology for minimal Cantor sets. Ann. Henri Poincaré 7 (2006), 423–446.CrossRefGoogle Scholar
[31]Ghys, E.. Laminations par surfaces de Riemann. Dynamique et géométrie complexes, Panoramas et Synthèses 8 (1999), 49–95.Google Scholar
[32]Grothendieck, A.. La théorie des classes de Chern. Bull. Soc. Math. France 86 (1958), 137–154.CrossRefGoogle Scholar
[33]Grünbaum, B. and Shephard, G. C.. Tilings and Patterns, 1st edn. W.H. Freemand and Co, New York, 1987.Google Scholar
[34]Hatcher, A.. Algebraic Topology, 1st edn. Cambridge University Press, Cambridge, 2002 (and available online at http://www.math.cornell.edu/∼hatcher/#ATI).Google Scholar
[35]Hirzebruch, F.. Arithmetic genera and the theorem of Riemann–Roch for algebraic varieties. Proc. Natl. Acad. Sci. USA 40 (1954), 110–114.CrossRefGoogle ScholarPubMed
[36]Hirzebruch, F.. Neue topologische Methoden in der algebraischen Geometrie (Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9). Springer, Berlin, 1956 (Topological Methods in Algebraic Geometry, Reprint of the 3rd edn. Springer, Berlin, 1978).Google Scholar
[37]Hofstadter, D. R.. Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B 14 (1976), 2239–2249.CrossRefGoogle Scholar
[38]Johnson, R. and Moser, J. The rotation number for almost periodic potentials. Comm. Math. Phys. 84 (1982), 403–438.CrossRefGoogle Scholar
[39]Kasparov, G. G.. The operator K-functor and extensions of C*-algebras. Izv. Akad. Nauk SSSR 44 (1980), 571–636.Google Scholar
[40]Kasparov, G. G.. Equivariant KK-theory and the Novikov conjecture. Invent Math. 91 (1988), 147–201.CrossRefGoogle Scholar
[41]Kellendonk, J.. Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7 (1995), 1133–1180.CrossRefGoogle Scholar
[42]Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A: Math. Gen. 36 (2003), 5765–5772.CrossRefGoogle Scholar
[43]Kellendonk, J. and Putnam, I. F.. The Ruelle–Sullivan map for actions of ℝn. Math. Ann. 334 (2006), 693–711.CrossRefGoogle Scholar
[44]Koszul, J.-L.. Sur les opérateurs de dérivation dans un anneau. C. R. Acad. Sci., Paris 225 (1947), 217–219.Google Scholar
[45]Lagarias, J. C.. Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21 (1999), 161–191.CrossRefGoogle Scholar
[46]Lagarias, J. C.. Geometric models for quasicrystals. II. Local rules under isometries. Discrete Comput. Geom. 21 (1999), 345–372.CrossRefGoogle Scholar
[47]Lagarias, J. C. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. Dynam. Sys. 23 (2003), 831–867.CrossRefGoogle Scholar
[48]Leray, J.. Structure de l’anneau d’homologie d’une représentation. C. R. Acad. Sci., Paris 222 (1946), 1419–1422.Google Scholar
[49]Massey, W. S.. Exact couples in algebraic topology (Parts I & II). Ann. Math. 56 (1952), 363–396; Exact couples in algebraic topology (Parts III, IV & V). Ann. Math. 57 (1953), 248–286.CrossRefGoogle Scholar
[50]McCleary, J.. A User’s Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics, 58), 2nd edn. Cambridge University Press, Cambridge, 2001.Google Scholar
[51]Meyer, Y.. Algebraic Number and Harmonic Analysis. North-Holland, Amsterdam, 1972.Google Scholar
[52]Moore, C. C. and Schochet, C.. Global Analysis on Foliated Spaces (Mathematical Sciences Research Institute Publications, 9). Springer, New York, 1988.CrossRefGoogle Scholar
[53]Moser, J.. An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56 (1981), 198–224.CrossRefGoogle Scholar
[54]Palais, R.. Seminar on the Atiyah–Singer Index Theorem (Annals of Mathematics Studies, 57). Princeton University Press, Princeton, NJ, 1965.Google Scholar
[55]Pimsner, M. and Voiculescu, D.. Exact sequences for K-groups and Ext groups of certain cross-product C *-algebras. J. Operator Theory 4 (1980), 93–118.Google Scholar
[56]Pimsner, M. V.. Ranges of traces on K 0 of reduced crossed products by free groups. Operator Algebras and their Connections with Topology and Ergodic Theory (Buşteni, 1983) (Lecture Notes in Mathematics, 1132). Springer, Berlin, 1985, pp. 374–408.CrossRefGoogle Scholar
[57]Penrose, R.. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10 (1974), 55–65.Google Scholar
[58]Queffelec, M.. Substitution Dynamical Systems-Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.CrossRefGoogle Scholar
[59]Radin, C.. The pinwheel tiling of the plane. Ann. of Math. (2) 139 (1994), 257–264.Google Scholar
[60]Radin, C.. Miles of Tiles. Vol. 1. American Mathematical Society, Student Mathematical Library, Providence, RI, 1999.Google Scholar
[61]Renault, J.. A Groupoid Approach to C * (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.CrossRefGoogle Scholar
[62]Rieffel, M. A.. Morita equivalence for operator algebras. Operator Algebras and Applications, Part 1 (Proceedings of Symposia in Pure Mathematics, 38). American Mathematical Society, Providence, RI, 1982, pp. 285–298.CrossRefGoogle Scholar
[63]Sadun, L.. Tiling spaces are inverse limits. J. Math. Phys. 44 (2003), 5410–5414.CrossRefGoogle Scholar
[64]Sadun, L. and Williams, R. F.. Tiling spaces are Cantor set fiber bundles. Ergod. Th. & Dynam. Sys. 23 (2003), 307–316.CrossRefGoogle Scholar
[65]Sadun, L.. Explicit computation of the cohomology for the chair tiling, talk delivered at Banff Research Station, July 2005.Google Scholar
[66]Sadun, L.. Pattern-equivariant cohomology with integer coefficients, arXiv.com math.DS/0602066, February 2006. Available at ftp://ftp.ma.utexas.edu/pub/papers/sadun/2006/integer4.pdf.Google Scholar
[67]Serre, J.-P.. Homologie singulière des espaces fibrés. Applications. Ann. of Math. 54 (1951), 425–505.CrossRefGoogle Scholar
[68]Shechtman, D., Blech, I., Gratias, D. and Cahn, J. V.. Metallic phase with long range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1953.CrossRefGoogle Scholar
[69]Schochet, C.. Topological methods for C*-algebra I: spectral sequences. Pacific J. Math. 96(1) (1981), 193–211.CrossRefGoogle Scholar
[70]Singer, I. M.. Future extensions of index theory and elliptic operators. Prospects in Mathematics (Proc. Symp., Princeton University, NJ, 1970) (Annals of Mathematical Studies, 70). Princeton University Press, Princeton, NJ, 1971, pp. 171–185.Google Scholar
[71]Spanier, E.. Algebraic Topology. McGraw-Hill, New York, 1966 (reprinted by Springer, Berlin).Google Scholar
[72]Subin, M. A.. Spectral theory and the index of elliptic operators with almost-periodic coefficients (Russian). Uspekhi Mat. Nauk 34(2(206)) (1979), 95–135.Google Scholar
[73]Süto, A.. The spectrum of a quasi-periodic Schrödinger operator. Comm. Math. Phys. 111 (1987), 409–415.CrossRefGoogle Scholar
[74]Thouless, D. J., Kohmoto, M., Nightingale, M. and den Nijs, M.. Quantum hall conductance in two dimensional periodic potential. Phys. Rev. Lett. 49 (1982), 405–408.CrossRefGoogle Scholar
[75]Williams, R. F.. Expanding attractors. Publ. Math. IHES 43 (1974), 169–203.CrossRefGoogle Scholar