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arXiv · 0906.2107

PV cohomology of pinwheel tilings, their integer group of coinvariants and gap-labelling

Haïja Moustafa
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Abstract

In this paper, we first remind how we can see the "hull" of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam) and we then adapt the PV cohomology introduced in a paper of Bellissard and Savinien to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by $S^1$ which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants on some transversal of the hull. The gap-labelling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labelling, showing that $μ^t \big(C(Ξ,\ZZ) \big) = \dfrac{1}{264} \ZZ [\dfrac{1}{5}]$.

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BibTeXRIS

Haïja Moustafa. 2009-06-12. PV cohomology of pinwheel tilings, their integer group of coinvariants and gap-labelling. https://doi.org/10.1007/s00220-010-1070-3

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