Substitutions with Vanishing Rotationally Invariant First Cohomology

Abstract

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 𝐙_𝑁. When the symmetries are of crystallographic type, the cohomologies are infinitely generated.

1. Introduction

The computation of topological invariants of tiling spaces Ω is a subject of increasing research activity, mainly since the work of Anderson and Putnam [1].

We use substitution rules to define collections of tilings. The expanding hierarchical structures that arise are essentially the same at each level, because they are described by a single substitution map. More general formalisms for handling general spaces of hierarchical tilings are studied in [2].

The simplest invariants of tiling spaces are the Čech cohomology groups. A relevant fact, also from the point of view of applications, is that the Čech cohomology is related to the gap distribution in the spectrum of the Schroedinger operator with a potential associated to a particular tiling. The Čech cohomology groups may be interpreted also in terms of certain tiling properties. For projection tilings, there is in the first cohomology at least a subgroup isomorphic to the reciprocal lattice of the tiling [3], and hence for a tiling with 𝑁-fold symmetry its minimal dimension is given by the Euler totient function Φ(𝑁). If the stretching factor of a one-dimensional substitution tiling is a Pisot number of algebraic degree 𝑘 and the rank of the first rational Čech cohomology is also 𝑘, then the number of appearances of a patch in a return word is determined by the Euclidean length of the return word [4]. Based on this results, it has been shown in [5] that the top-dimensional cohomology for a tiling in 𝐑^𝑑 governs patch frequencies and the number of appearances of a patch in a finite region.

A class of tiling spaces with fivefold symmetry has been analyzed in [6]. The rank of the first cohomology coincides with Φ(5) and its restriction in the space of tilings modulo a fivefold rotation is zero. In this paper we study several examples of tiling spaces with crystallographic and noncrystallographic symmetries with analogous properties. They have vanishing first cohomology in the space of tilings modulo a rotation Ω. When the integer first cohomology in Ω is finitely generated, as in the tilings with ninefold and pentagonal symmetries, its rank coincides with the Eulers totient function. In the cases with crystallographic symmetries, it is the rank of the rational first cohomology which equals the Eulers totient function.

2. Geometric Constructions of Substitution Tilings with 𝑑-Fold Symmetry and Algebraic Surfaces

An extension of the methods introduced in [7] for tilings with odd symmetries nondivisible by three to all the symmetries has been given in [8]. There are four types of geometric constructions consisting in straight lines in 𝑑 orientations: (A) for 𝑑=2𝑚, (B) for 𝑑=2𝑚+1, (C) for 𝑑=6𝑚, and (D) for 𝑑=6𝑚+3. They produce simplicial arrangements 𝑆_𝑋,𝑋=𝐴,𝐵,𝐶,𝐷, which are the basis for the generation of planar tilings with arbitrarily high symmetry. The systems with 𝑑 odd are included in the systems with 𝑑 even: (B) is included as a subsystem of (A), and (D) is included in (C). The fact that, in general, a system with 𝑑 lines is included in another with 2𝑑 lines allows to introduce in a systematic way arrows on the edges, which are necessary for the definition of the inflation rules generating face-to-face tilings [6]. We give an alternative formulation which simplifies the construction. Systems of lines, equivalent up to scaling, translation, and rotation, to those obtained with the constructions (A) and (B) are {Λ_𝑘,𝑑(𝑥,𝑦,0)=0},𝑘=1,2,…𝑑, whereΛ_𝑘,𝑑𝑥,𝑦,𝜏∶=𝑦−cos𝑘2𝜋𝑑−𝑥tan𝑘𝜋𝑑+𝜏−sin𝑘2𝜋𝑑 (2.1)

The simplicial arrangements 𝑆_𝐶,𝑆_𝐷 and their mirror images 𝑆_𝐶,𝑆_𝐷 can be obtained also by using Λ_𝑘,𝑑(𝑥,𝑦,𝜏) for certain values of 𝜏. For fixed 𝑑 we consider the subset formed by the lines Λ_{2𝑘−1,2𝑑}(𝑥,𝑦,𝜏) with 𝑘=1,2,…𝑑 and 𝜏=𝑛𝜋/4𝑑. When 𝑑=3𝑚, we get, for 𝑛=0,1,2,…8𝑑, successive rotations of two types of simple arrangements of lines (no more than two lines intersect) Σ₁,Σ₂ and the two types of simplicial arrangements 𝑆_𝐴,𝑆_𝐶 and 𝑆_𝐵,𝑆_𝐷 for 𝑑 even and odd, respectively. In addition to the nontriangular shapes, Σ₁,Σ₂, have two sets of triangular prototiles: Σ₁ has the prototiles of 𝑆_𝐴,𝑆_𝐵 while Σ₂ and its mirror image Σ₂ have the prototiles of 𝑆_𝐶,𝑆_𝐷 and 𝑆_𝐶,𝑆_𝐷. In Figure 1(a) we show the case 𝑑=5 with (from top to bottom) 𝑛=0,1,2,…6. Two rotated versions of 𝑆_𝐵 correspond to 𝑛=2,6, the others are simple arrangements of lines Σ₁ obtained when 𝑛=0,4. The case 𝑑=6 with 𝑛=0,1,2,…6 can be seen in Figure 1(b). Now we have two simplicial arrangements: 𝑆_𝐶 with 3 prototiles when 𝑛=2 and 𝑆_𝐴 with one prototile for 𝑛=6. When 𝑛=0, the simple arrangement Σ₁ has 6 copies of the prototile appearing in 𝑆_𝐴, and when 𝑛=4, Σ₂ has seven triangles with 3 different shapes (prototiles in 𝑆_𝐶). In both cases the prototiles have edges with three different lengths as in Section 4.

(a)

(b)

Figure 1 

Line configurations described by (2.1) for (from top to bottom) (a) 𝑑=5;𝑛=0(Σ1),1,2(𝑆𝐵),3,4(Σ1),5,6(𝑆𝐵) and (b) 𝑑=6;𝑛=0(Σ1),1,2(𝑆𝐶),3,4(Σ2),5,6(𝑆𝐴).

The line configurations for 𝑑=9 can be seen in Figure 2(a) when 𝑛=0,1,2,3 and Figure 2(b) for 𝑛=4,5,6. We get Σ₁ when 𝑛=0,12, while Σ₂ and Σ₂ are obtained with 𝑛=4,8. The simplicial arrangement 𝑆_𝐵 corresponds to 𝑛=6 and we get 𝑆_𝐷,𝑆_𝐷 when 𝑛=2,10. By using the notation in [9], the prototiles appearing in Σ₁ and 𝑆_𝐵 are 𝑎,𝑏,𝑐 (we use capital letters in Section 3 to describe the same tiles), while in Σ₁ and 𝑆_𝐷 there are seven prototiles 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔. All the prototiles now have edges with four different lengths as in Section 3.

(a)

(b)

Figure 2 

Simple and simplicial arrangements for 𝑑=9 (a) 𝑛=0(Σ1),1,2(𝑆𝐷),3, (b) 𝑛=4(Σ2),5,6(𝑆𝐵).

The simple arrangements Σ₁,Σ₂ are on the basis of a construction of algebraic surfaces with many real nodes proposed recently [10]. The surfaces have affine equations 𝑆(𝑥,𝑦,𝑧)∶=𝐹(𝑥,𝑦)+𝜆𝐹(𝑧,0)+𝜇=0,𝜆,𝜇∈𝐑, where the plane curve 𝐹(𝑥,𝑦) is a product of lines corresponding to a simple arrangement. Real variants of Chmutov surfaces [11] are based on Σ₁ as shown in [12]. On the other hand Σ₂, and its mirror image Σ₂, have one more triangle than Σ₁, and this property can be used to construct surfaces with more singularities. The dynamical formulation of the line configurations given by (2.1) can then be used to get deformations of the surfaces by varying 𝜏, where some singularities disappear. Other deformations of interest, giving smooth surfaces, are obtained by varying 𝜇.

3. Cohomology Groups of Tiling Spaces with Ninefold Symmetry

The tiling spaces Ω_𝑋 are formed by the closure of the translational orbit of one tiling 𝑋. A substitution 𝜎 is said to force the border [13] if there is a positive integer 𝑛 such that if 𝑡₁ and 𝑡₂ are two tiles of the same type, then the two level-𝑛 supertiles 𝜎^𝑛(𝑡₁) and 𝜎^𝑛(𝑡₂) have, up to translation, the same pattern of neighboring tiles. If a substitution forces the border, then the tiling space is the inverse limit of the Anderson-Putnam complex Γ₀, which contains one copy of every kind of tile that is allowed with some edges identified [1]. For 2D tiling spaces the cohomology classes are generated by the vertices, edges, and faces of Γ₀.

In addition to Ω_𝑋 we consider also another type of tiling spaces Ω_𝑋. The finite rotation group 𝐙_𝑁 associated to the tiling acts on Ω_𝑋, and the quotient Ω_𝑋/𝑍_𝑁 yields the space Ω_𝑋 of tilings modulo a rotation about the origin [14].

We study first the cohomology of a ninefold symmetry tiling space Ω_𝑁. The substitution rules for this and other composite patterns were studied in [9]. They can be obtained by using 𝑆_𝐵 in Figure 2(b).

Up to mirror reflection and rotation, the tiling has three triangular prototiles 𝒯 with inflation or substitution rules (level-1 supertiles) given in Figure 3. Iteration of the inflation rules shows that the tiling has 18 vertex configurations. After three inflation steps, they are transformed into just two vertex configurations that we call 𝑎 and 𝑏 representing a threefold and a ninefold star, respectively (Figure 4). The substitution forces the border in four steps. The analysis of the vertex sequences appearing in the level-4 superedges shows that there are four edge types 𝛼_𝑏,𝑎,𝛽_𝑎,𝑏,𝛾_𝑎,𝑎,𝛿_𝑏,𝑎, where the subindexes denote the vertices appearing on the edge borders. The lengths of 𝛼,𝛽,𝛾,𝛿 are 𝑠₁,𝑠₂,𝑠₃,𝑠₄, with 𝑠_𝜈≡sin(𝜈𝜋/9).

Figure 3 

Inflation rules for the ninefold symmetry tiling.

Figure 4 

Level-3 supertiles in the ninefold symmetry tiling.

The rotation group 𝐙₉ acts freely on edges and tiles. The two vertices satisfy 𝑎=𝑟³𝑎,𝑏=𝑟𝑏 with 𝑟⁹=1. We have six tile types 𝐴=𝒯(𝑟⁴𝛾,𝑟⁴𝛽,𝑟⁵𝛼),𝐴=𝒯(𝑟⁴𝛾,𝑟³𝛼,𝑟⁵𝛽),𝐵=𝒯(𝛿,𝑟⁸𝛾,𝑟𝛼),𝐵=𝒯(𝛿,𝑟⁷𝛼,𝛾),𝐶=𝒯(𝛿,𝑟³𝛾,𝑟⁶𝛽),𝐶=𝒯(𝛿,𝑟³𝛽,𝑟⁵𝛾), and each appears in 9 orientations. The Anderson-Putnam complex Γ₀ has Euler characteristic 𝜒=22.

The three irreducible representations of 𝐙₉ over the integers are the one-dimensional scalar (𝑟=1), a two-dimensional and a six-dimensional representation. The 2D and the 6D representations have 𝑟 acting by multiplication on the rings ℛ₁=𝐙[𝑟]/(𝑟²+𝑟+1) and ℛ₂=𝐙[𝑟]/(𝑟⁶+𝑟³+1), respectively. In this case the vertex 𝑏 appears only in the scalar representation, the vertex 𝑎 in the scalar and 2D representations, while the edges and faces appear in all representations. Vertices, edges, and faces form a basis for the spaces of chains 𝐶_𝑘 with 𝑘=0,1,2. The boundary maps 𝜕_𝑘∶𝐶_𝑘+1↦𝐶_𝑘 for 𝑘=0,1 are𝜕₁=𝑟⁵−𝑟³−𝑟𝑟⁷𝑟00₄−𝑟⁵00𝑟⁶−𝑟³−𝑟⁴𝑟⁴−𝑟⁸1𝑟³−𝑟⁵001−11−1 (3.1) in all representations and 𝜕₀=−𝑟²10−11−101 (3.2) in the scalar and 2D representations, but with appropriate modification having in mind that only the vertex 𝑎 appears in the 2D representation. If the cochain groups are denoted by 𝐶^𝑘, then, for 𝑘=0,1, the coboundary maps 𝛿_𝑘∶𝐶^𝑘↦𝐶^𝑘+1 are the transposes of the matrices 𝜕_𝑘 given above with 𝑟 replaced with 𝑟^𝑇=𝑟⁻¹.

Scalar Representation 𝑟=1
In the scalar representation, 𝐶⁰=𝐙²,𝐶¹=𝐙⁴,𝐶²=𝐙⁶. We have rank𝛿₀=1 and rank𝛿₁=3. The cohomologies of Γ₀ in this representation are ℋ⁰=Ker𝛿₀=𝐙,ℋ¹=Ker𝛿₁/Im𝛿₀=0, and ℋ²=𝐙⁶/Im𝛿₁=𝐙³.

2D Representation
There is one vertex in the 2D representation, and 𝐶⁰,𝐶¹,𝐶² are free modules of dimensions 1,4,6 over the ring ℛ₁. The rank of 𝛿₀ over ℛ₁ is 1, the matrix 𝛿₁ has rank 3 over ℛ₁, and, as abelian groups, we have ℋ⁰=ℋ¹=0,ℋ²=𝐙⁶.

6D Representation
There are no vertices in the 6D representation, and 𝐶¹,𝐶² are free modules of dimensions 1 and 3 over the ring ℛ₂. The map 𝛿₁ has rank 3 over ℛ₂ and ℋ⁰=0,ℋ¹=𝐙⁶,ℋ²=𝐙¹⁸.

We have obtained the cohomology of the complex Γ₀ which is enough because the tiling forces the border [1]. To get the cohomology of the tiling space Ω_𝑁, we need to compute the direct limit of the cohomologies under the substitution.

The substitution matrix on vertices is𝜎₀=𝑟²001, (3.3) while the matrix on 1-chains is𝜎₁=000100𝑟𝑟⁵01𝑟⁵1𝑟𝑟⁵𝑟𝑟⁵, (3.4) and the substitution on 2-chains𝜎₂=0𝑟0𝑟1𝑟10101𝑟0𝑟⁵0𝑟⁵+𝑟⁸𝑟⁴𝑟⁵+𝑟⁸𝑟⁵0𝑟²+𝑟⁵0𝑟²+𝑟⁵𝑟⁶𝑟⁶𝑟𝑟⁶𝑟+𝑟⁴1+𝑟³+𝑟⁶𝑟+𝑟⁴+𝑟⁷1𝑟⁴1+𝑟⁶𝑟⁴1+𝑟³+𝑟⁶𝑟+𝑟⁴+𝑟⁷. (3.5)

The induced matrices on cochains 𝜎^∗_𝑘 are obtained by transposition of the matrices 𝜎_𝑘 with 𝑟 replaced with 𝑟⁻¹. They are isomorphisms, and the direct limit of each ℋ^𝑘 is simply ℋ^𝑘. By taking into account all the irreducible representations, we get the followingℋ̌⁰Ω_𝑁=𝐙,ℋ̌¹Ω_𝑁=𝐙⁶,ℋ̌²Ω_𝑁=𝐙²⁷. (3.6)

The cohomology of Ω_𝑋 is the rotationally invariant part of the cohomology of Ω_𝑋, and we haveℋ̌⁰Ω_𝑁=𝐙,ℋ̌¹Ω_𝑁=0,ℋ̌²Ω_𝑁=𝐙³. (3.7)

Another tiling space with noncrystallographic symmetries and vanishing first cohomology in Ω_𝑋 was analyzed in [6]. The pentagonal tiling space Ω_Ξ+ has the same prototiles as the Robinson decomposition of the Penrose tilings: 𝐴,𝐵 with edges 𝛼,𝛽 and lengths 𝑠₁,𝑠₂, with 𝑠_𝜈≡sin(𝜈𝜋/5). The inflation rules can be obtained from 𝑆_𝐵 in Figure 1(a) with arrows as indicated in [6]. The cohomology groups areℋ̌⁰Ω_Ξ+=𝐙,ℋ̌¹Ω_Ξ+=𝐙⁴,ℋ̌²Ω_Ξ+=𝐙¹⁴, (3.8) and the rotationally invariant part is ℋ̌⁰(Ω_Ξ+)=𝐙,ℋ̌¹(Ω_Ξ+)=0,ℋ̌²(Ω_Ξ+)=𝐙².

Fractal tilings related with Penrose patterns have been studied in [15, 16]. In addition to the computation of the cohomologies, the analysis of the patterns of vertex configurations, appearing in Ξ₊ on the level-6 superedges, can be used also to construct tilings whose edges and tiles have irregular shapes. In Figure 5(a) we have shown the vertex sequences arising in the 𝐴-type level-6 supertile (see Figure  12 in [6]). The borders of the new prototile shapes are formed by concatenation of scaled copies of oriented 𝛼 and 𝛽. The substitution rules for the corresponding pattern can be seen in Figure 5(b).

(a)

(b)

Figure 5 

Pentagonal tiling with irregular shapes: (a) construction of the boundaries, (b) inflation rules.

Tiling spaces with crystallographic symmetries also may have zero first cohomology. For instance the chair tiling space Ω_CH has [14]ℋ̌⁰Ω_CH=𝐙,ℋ̌¹Ω_CH=𝐙12²,ℋ̌²Ω_CH1=𝐙41⊕𝐙2² (3.9) with ℋ̌⁰(Ω_CH)=𝐙,ℋ̌¹(Ω_CH)=0,ℋ̌²(Ω_CH)=𝐙[1/4]. In the next section we study other tiling spaces with crystallographic symmetries, which also have ℋ̌¹(Ω_𝑇)=0.

4. Tiling Spaces with Threefold Symmetry

4.1. Equithirds Tiling

The equithirds tiling was obtained independently by L. Danzer and B. Kalahurka ([5] and references therein). It is based on the substitution of Figure 6. The prototiles are an equilateral triangle of side length 1 and an isosceles triangle with sides of length 1,1, and √3.

Figure 6 

Inflation rules for the equithirds tiling.

The analysis of the five vertex configurations shows that, after one inflation step, they are transformed into one: vertex 𝑎. The substitution forces the border in two steps. We denote the edges by 𝛼,𝛽 with lengths 1,√3, and the rotation group 𝐙₃ acts freely on them.

We have four tile types 𝐴=𝒯(𝛼,𝑟𝛼,𝑟²𝛼),𝐴=𝒯(𝛼,𝑟𝛼,𝑟²𝛼),𝐵=𝒯(𝛼,𝛽,𝑟𝛼),𝐵=𝒯(𝛼,𝑟²𝛼,𝑟²𝛽), with 𝑟³=1. The tiles 𝐵,𝐵 appear in three orientations and 𝐴,𝐴 in one. The Anderson-Putnam complex Γ₀ has therefore Euler characteristic 𝜒=3.

The two irreducible representations of 𝐙₃ over the integers are the one-dimensional scalar (𝑟=1) and a two-dimensional vector representation. The vector representation has 𝑟 acting by multiplication on the ring ℛ₁=𝐙[𝑟]/(𝑟²+𝑟+1). The edges and the tiles 𝐵,𝐵,𝐷 appear in all representations, while the vertex and the tiles 𝐴,𝐴 appear only in the scalar representation. The boundary maps are 𝜕₀=0 and 𝜕₁=1−11−𝑟1−𝑟²00−1𝑟². (4.1)

Scalar Representation 𝑟=1
In the scalar representation, 𝐶⁰=𝐙,𝐶¹=𝐙²,𝐶²=𝐙⁴. In this case rank𝛿₁=2 and we have ℋ⁰=Ker𝛿₀=𝐙,ℋ¹=Ker𝛿₁/Im𝛿₀=0, and ℋ²=𝐙⁴/Im𝛿₁=𝐙².

Vector Representation
There are two tiles in the 2D representation, and 𝐶¹,𝐶² are free modules of dimensions 2 over the ring ℛ₁. The rank of 𝛿₁ over ℛ₁ is 1 and we have ℋ⁰=0,ℋ¹=𝐙² and ℋ²=𝐙².

The cohomology of the complex Γ₀ is therefore ℋ⁰(Γ₀)=𝐙,ℋ¹(Γ₀)=𝐙² and ℋ²(Γ₀)=𝐙⁴. Now we compute the direct limit of the cohomologies under the substitution.

The substitution on the vertex is the identity. The substitution on edges is represented by𝜎₁=3003, (4.2) and the substitution on 2-chains is𝜎₂=301103111+𝑟+𝑟²1+𝑟+𝑟²3+𝑟+𝑟²1+𝑟1+𝑟+𝑟²1+𝑟+𝑟²1+𝑟²3+𝑟+𝑟². (4.3)

In the scalar representation the direct limits of each ℋ^𝑘, under the action of the induced matrices on cochains 𝜎^∗_𝑘, are ℋ^𝑘 for 𝑘=0,1. We can take a basis for ℋ² in terms of the canonical basis of 𝐙⁴ formed by 𝑒₁,𝑒₁+𝑒₃ with the classes of 𝑣₃=𝑒₄−𝑒₃,𝑣₄=𝑒₂−𝑒₁ cohomologous to zero. The eigenvalues of 𝜎^∗₂ are 9,3,1 with eigenvectors 𝑣₁=(1,1,1,1)^𝑇,𝑣₂,𝑣₃,𝑣₄=(−3,−3,1,1)^𝑇. Its action on ℋ² gives the direct limit 𝐙⊕𝐙[1/3].

In the vector representation the direct limit of ℋ¹ is 𝐙[1/3]². A base for ℋ² in this representation is given by the vector (−1,0)^𝑇∼(𝑟,1)^𝑇 which is the eigenvector of 𝜎^∗₂ with eigenvalue 1. The direct limit is now 𝐙². The cohomology groups for the equithirds tiling space Ω_EQ are (see also [5])ℋ̌⁰Ω_EQ=𝐙,ℋ̌¹Ω_EQ=𝐙13²,ℋ̌²Ω_EQ=𝐙³1⊕𝐙3. (4.4)

The rotationally invariant part of the cohomology isℋ̌⁰Ω_EQ=𝐙,ℋ̌¹Ω_EQ=0,ℋ̌²Ω_EQ=𝐙⊕𝐙13. (4.5)

4.2. The Tiling Spaces Ω_𝑇

Although the paper [7] deals with tilings having odd symmetries nondivisible by three, the authors consider a case which can be derived with a set of six lines in the plane. The tiling spaces Ω_𝑇 that we treat now are based on such system of lines. In the construction discussed in Section 2, it corresponds to the simplicial arrangements 𝑆_𝐴; 𝑆_𝐶 in Figure 1(b).

The substitution rules for 𝑇 are represented in Figure 7, and its seven vertex configurations can be seen in Figure 8. After two inflation steps the vertex configurations are transformed into the configurations 1 and 2 which are denoted by 𝑎,𝑏. The substitution forces the border also in two steps: the vertex sequences in the level-2 superedges (Figure 9) are 𝛼∶53,𝛽∶676,𝛾∶2354 with lengths 𝑠₁,𝑠₂,𝑠₃, 𝑠_𝜈≡sin(𝜈𝜋/6), and the vertices in the edge borders are 𝛼∶𝑎𝑏,𝛽∶𝑎𝑎,𝛾∶𝑎𝑏. The rotation group 𝐙₃ acts freely on the edges and 𝑎=𝑟𝑎,𝑏=𝑟𝑏 with 𝑟³=1.

Figure 7 

Inflation rules for 𝑇.

Figure 8 

Vertex configurations in 𝑇.

Figure 9 

Level-2 supertiles in 𝑇.

The four tile types are 𝐴=𝒯(𝛼,𝑟²𝛼,𝛽),𝐷=𝒯(𝛽,𝑟𝛼,𝛾),𝐷=𝒯(𝛾,𝛼,𝑟²𝛽),and𝐸=𝒯(𝛽,𝑟𝛽,𝑟²𝛽), and each appears in 3 orientations, except 𝐸 which appears in only one. The Euler characteristic of Γ₀ is 𝜒=3.

In this case the edges and the tiles 𝐴,𝐷,𝐷 appear in all representations of 𝐙₃, while the vertices 𝑎,𝑏, and the tile 𝐸 appear only in the scalar representation. The boundary maps are 𝜕₀=,𝜕−10−1101₁=1−𝑟²𝑟−10−11𝑟²−1.0−110 (4.6)

Scalar Representation  𝑟=1
In this representation, 𝐶⁰=𝐙²,𝐶¹=𝐙³,𝐶²=𝐙⁴. Here rank𝛿₀=1 and rank𝛿₁=2 with cohomologies ℋ⁰=Ker𝛿₀=𝐙,ℋ¹=Ker𝛿₁/Im𝛿₀=0 and ℋ²=𝐙⁴/Im𝛿₁=𝐙².

Vector Representation
There are three edges and three tiles in the 2D representation, and 𝐶¹,𝐶² are free modules of dimension 3 over the ring ℛ₁. The rank of 𝛿₁ over ℛ₁ is 2, and, as abelian groups, we have ℋ⁰=0, ℋ¹=𝐙² and ℋ²=𝐙².

Adding up the contributions of each representations, we get the cohomology of the complex Γ₀: ℋ⁰(Γ₀)=𝐙, ℋ¹(Γ₀)=𝐙², and ℋ²(Γ₀)=𝐙⁴.

The substitution on the vertices is the 2×2 identity matrix. The substitution on 1-chains is 𝜎₁=002𝑟02𝑟²010𝑟², (4.7) and the substitution on 2-chains is𝜎₂=0𝑟²𝑟𝑟0₂01+𝑟²11𝑟+𝑟²010110. (4.8)

In the scalar representation the direct limits of ℋ^𝑘 for 𝑘=0,1 are ℋ^𝑘. A base for ℋ² in this representation is given by the vectors 𝑢₃=(0,0,1,0)^𝑇,𝑢₄=(0,0,0,1)^𝑇, and the classes of 𝑢_𝑘,𝑘=1,2, with 𝑢₁=(0,1,−1,0)^𝑇,𝑢₂=(−1,0,2,−1)^𝑇, are cohomologous to zero. The action of 𝜎^∗₂ applied twice to 𝑢₃ gives 4𝑢₃∼4𝑢₄, and the direct limit yields a contribution of 𝐙[1/2].

In the vector representation 𝑣₃=(1,1−𝑟,−2𝑟)^𝑇 forms a base for ℋ¹. The matrix 𝜎^∗₁ has eigenvalues −𝑟,2𝑟,2𝑟 with eigenvectors (−𝑟²,0,1)^𝑇,(𝑟²,0,2)^𝑇,(0,1,0)^𝑇, and the direct sum of the eigenspaces is not isomorphic to all 𝐙⁶. The action of 𝜎^∗₁ gives a contribution of 𝐙[1/2]² to the first cohomology. A base for ℋ² is given by 𝑢₃=(0,1,0)^𝑇 with 𝑢₁=(1−𝑟,𝑟²,1)^𝑇,𝑢₂=(−1,−𝑟²,0)^𝑇 cohomologous to zero. We have 𝜎₁^∗3𝑢₃∼𝑢₃, and the direct limit of ℋ² is 𝐙². The cohomology groups are thenℋ̌⁰Ω_𝑇=𝐙,ℋ̌¹Ω_𝑇=𝐙12²,ℋ̌²Ω_𝑇=𝐙²1⊕𝐙2. (4.9)

The rotationally invariant part of the cohomology isℋ̌⁰Ω_𝑇=𝐙,ℋ̌¹Ω_𝑇=0,ℋ̌²Ω_𝑇=𝐙12. (4.10)

With the purpose of studying the meaning of the topological invariants in tiling spaces, Sadun shows that there are patches in the equithirds tiling that play a role analogous to return words in one dimension [5]. The control patches are the equilateral triangle and a rhombus, formed by two isosceles triangles, in three different orientations. This case illustrates the more general property that if the rank of the second rational cohomology is 𝑘, then the number of control patches is also 𝑘. The number of appearances of any patch in a sufficiently substituted rhombus in the equithirds tiling is then determined by the number of appearances of the four control patches.

For the tiling 𝑇 we may find four patches that generate the tilings, because they obey certain substitution rules induced by the inflation on the triangular prototiles. The patches 𝑌,𝑍, and their substitution rules are represented in Figure 10. The patch content in the inflation rules is𝑌⟼𝑌𝑍𝑍𝑍^{0𝑥0033𝑓}𝑍⟼𝑍𝑌𝑌𝑌, (4.11) where 𝑍 and ^0𝑥0𝑍^033𝑓 represent rotated versions of 𝑍 by 120 and 240 degrees.

Figure 10 

Inflation rules for the patches 𝑌 and 𝑍 in 𝑇.

Although we have studied in detail only some particular cases, we can see that the general constructions of substitutions proposed in [7, 8] are a rich source of tiling spaces with unusual properties. The tilings with noncrystallographic symmetries studied in this paper, derived with such constructions, have vanishing rotationally invariant first cohomology, but this is not true in general. For instance, the analysis of a tiling 𝑇^∗ obtained with the same basic construction as 𝑇 (which has inflation factor 𝑠₃/𝑠₁) but with scaling factor 𝑠₂/𝑠₁ shows that the associated tiling space has in ℋ̌¹ a nontrivial rotationally invariant piece. The tiling 𝑇^∗ has six prototiles with the same three shapes 𝐴,𝐷,𝐸 as in 𝑇. The mirror images of 𝐴,𝐷, and 𝐸 shape level-1 supertiles are different in 𝑇^∗, while in 𝑇 we have two mirror inflation rules only for 𝐷 (Figure 7). The first cohomology for the tiling space Ω_𝑇∗ in the scalar representation of 𝐙₃ is now 𝐙. An open question is, for general tiling spaces, which property distinguishes them in relation to the existence of a nontrivial rotationally invariant component in the first cohomology group.