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

TL;DR

This paper extends PV cohomology to pinwheel tilings, establishing its isomorphism with Čech cohomology, and computes the gap-labelling group explicitly, linking topological invariants to spectral properties.

Contribution

It adapts PV cohomology to pinwheel tilings and proves its isomorphism with Čech cohomology, providing a new approach to gap-labelling in aperiodic tilings.

Findings

PV cohomology for pinwheel tilings is isomorphic to Čech cohomology.

The top integer Čech cohomology equals the integer group of coinvariants.

Explicit gap-labelling group computed as (1/264) Z [1/5].

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 \v{C}ech cohomology of the quotient of the hull by $S^1$ which let us prove that the top integer \v{C}ech 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 $\mu^t \big(C(\Xi,\ZZ) \big) = \dfrac{1}{264} \ZZ [\dfrac{1}{5}]$.