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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |