Substitutions with Vanishing Rotationally Invariant First Cohomology
The cohomology groups of tiling spaces with three-fold and nine-fold symmetries are obtained. The substitution tilings are characterized by the fact that they have vanishing first cohomology group in the space of tilings modulo a rotation. The rank of the rational first cohomology, in the tiling spa...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2012/818549 |
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| _version_ | 1832554881421934592 |
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| author | Juan García Escudero |
| author_facet | Juan García Escudero |
| author_sort | Juan García Escudero |
| collection | DOAJ |
| description | The cohomology groups of tiling spaces with three-fold and nine-fold symmetries
are obtained. The substitution tilings are characterized by the fact that they have vanishing
first cohomology group in the space of tilings modulo a rotation. The rank of the rational first
cohomology, in the tiling space formed by the closure of a translational orbit, equals the Euler
totient function evaluated at 𝑁 if the underlying rotation group is 𝐙𝑁. When the symmetries
are of crystallographic type, the cohomologies are infinitely generated. |
| format | Article |
| id | doaj-art-a98a5ac3d5f94452a4651624dcf41d12 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a98a5ac3d5f94452a4651624dcf41d122025-02-03T05:50:16ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2012-01-01201210.1155/2012/818549818549Substitutions with Vanishing Rotationally Invariant First CohomologyJuan García Escudero0Facultad de Ciencias Matemáticas y Físicas, Universidad de Oviedo, 33007 Oviedo, SpainThe cohomology groups of tiling spaces with three-fold and nine-fold symmetries are obtained. The substitution tilings are characterized by the fact that they have vanishing first cohomology group in the space of tilings modulo a rotation. The rank of the rational first cohomology, in the tiling space formed by the closure of a translational orbit, equals the Euler totient function evaluated at 𝑁 if the underlying rotation group is 𝐙𝑁. When the symmetries are of crystallographic type, the cohomologies are infinitely generated.http://dx.doi.org/10.1155/2012/818549 |
| spellingShingle | Juan García Escudero Substitutions with Vanishing Rotationally Invariant First Cohomology Discrete Dynamics in Nature and Society |
| title | Substitutions with Vanishing Rotationally Invariant First Cohomology |
| title_full | Substitutions with Vanishing Rotationally Invariant First Cohomology |
| title_fullStr | Substitutions with Vanishing Rotationally Invariant First Cohomology |
| title_full_unstemmed | Substitutions with Vanishing Rotationally Invariant First Cohomology |
| title_short | Substitutions with Vanishing Rotationally Invariant First Cohomology |
| title_sort | substitutions with vanishing rotationally invariant first cohomology |
| url | http://dx.doi.org/10.1155/2012/818549 |
| work_keys_str_mv | AT juangarciaescudero substitutionswithvanishingrotationallyinvariantfirstcohomology |