Cohomology in one-dimensional substitution tiling spaces
HTML articles powered by AMS MathViewer
- by Marcy Barge and Beverly Diamond PDF
- Proc. Amer. Math. Soc. 136 (2008), 2183-2191 Request permission
Abstract:
Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which âforces its border.â 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. For one-dimensional substitution tiling spaces, we describe a modification of the Anderson-Putnam complex on collared tiles that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology.References
- Jared E. Anderson and Ian F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 509â537. MR 1631708, DOI 10.1017/S0143385798100457
- Marcy Barge and Beverly Diamond, A complete invariant for the topology of one-dimensional substitution tiling spaces, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1333â1358. MR 1855835, DOI 10.1017/S0143385701001638
- Marcy Barge, James Jacklitch, and Gioia Vago, Homeomorphisms of one-dimensional inverse limits with applications to substitution tilings, unstable manifolds, and tent maps, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999) Contemp. Math., vol. 246, Amer. Math. Soc., Providence, RI, 1999, pp. 1â15. MR 1732368, DOI 10.1090/conm/246/03771
- Ola Bratteli, Palle E. T. Jorgensen, Ki Hang Kim, and Fred Roush, Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1625â1655. MR 1869063, DOI 10.1017/S014338570100178X
- Fabien Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), no. 1-3, 89â101. MR 1489074, DOI 10.1016/S0012-365X(97)00029-0
- Brigitte MossĂ©, Puissances de mots et reconnaissabilitĂ© des points fixes dâune substitution, Theoret. Comput. Sci. 99 (1992), no. 2, 327â334 (French). MR 1168468, DOI 10.1016/0304-3975(92)90357-L
- B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom. 20 (1998), no. 2, 265â279. MR 1637896, DOI 10.1007/PL00009386
Additional Information
- Marcy Barge
- Affiliation: Department of Mathematics, Montana State University, Bozeman, Montana 59717
- Email: barge@math.montana.edu
- Beverly Diamond
- Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
- Email: diamondb@cofc.edu
- Received by editor(s): February 14, 2007
- Received by editor(s) in revised form: May 4, 2007
- Published electronically: February 19, 2008
- Communicated by: Jane M. Hawkins
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2183-2191
- MSC (2000): Primary 37B05; Secondary 37A30, 37B50, 54H20
- DOI: https://doi.org/10.1090/S0002-9939-08-09225-3
- MathSciNet review: 2383524