Integral cohomology of rational projection method patterns

Abstract

Abstract. We study the cohomology and hence K-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in d dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in R 3 -the Danzer tiling, the Ammann-Kramer tiling and the Canonical and Dual Canonical D6 tilings, including complete computations for the first of these, as well as results for many of the better known 2 dimensional examples.

Self Affine Delone Sets and Deviation Phenomena

“…Cohomology for cut and project sets. In this section we summarize the results of [FHK02,GHK13] which are relevant to our work. Namely, we review sufficient conditions under which the cohomology spaces defined in (16) are finite dimensional for cut and project constructions.…”

“…We call the CAPS Λ(Γ, K) almost canonical if for each face f i of the window K, the set f i + Γ ⊥ contains the affine space spanned by f i . This is equivalent [GHK13] to having a finite family of (n − d − 1)-dimensional affine subspaces…”

“…The space A is a union of (n − ν)tori, the tori themselves being D i /Γ D i , where the D i , i ∈ I n−d−1 , are the elements of D; furthermore, the intersection of finitely many of these tori is empty, or a subtorus itself (see the comment following [GHK13, Corollary 4.13]). Thus, as pointed out in [GHK13], computation of H * (A) and the maps j * are possible, in principle, using a Meyer-Vietoris spectral sequence (as in [Kal05]). For our purposes, it is enough to compute these cohomology groups rationally, since we are interested mainly in the spectrum of the map induced by Φ A on cohomology.…”

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

“…The duality between E and E ⊥ carries through to the singular subspaces of E ⊥ , which are dual to the hyperplanes of E corresponding to the "worms". In the case of the structures described by the so called model sets ( [29], see also [30] for a survey), this requirement leads to a constraint on the boundary of the acceptance domain (or the "window"), known as the rationality condition [31]. This condition is quite a strong one, since, as it is shown in [32], it implies that Ω is homeomorphic to the inverse limit of a sequence of CW-spaces X n with cellular maps ι m : X m+1 → X m .…”

Ellis enveloping semigroup for almost canonical model sets of an Euclidean space

“…In this work we propose a qualitative description of the Ellis semigroup of dynamical systems associated with particular point patterns, the almost canonical model sets. These particular patterns are relevant in the crystallographic sense, as well as very accessible mathematically: One can get a complete picture of the hull X Λ 0 of such patterns (Le [19]), as well as their associated C * -Algebras (a recent source is Putnam [27], see references therein), and also compute their cohomology and K-theory groups (Forrest, Hunton and Kellendonk [8], Gähler, Hunton and Kellendonk [9] and Putnam [27]) as well as the asymptotic exponent of their complexity function (Julien [16]). We show that in our situation it is possible to completely describe elements of the Ellis semigroup, their action onto the underlying space, as well as the algebraic and topological structure of this semigroup.…”