#### 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 if the underlying rotation group is . When the symmetries are of crystallographic type, the cohomologies are infinitely generated.

#### 1. Introduction

The computation of topological invariants of tiling spaces 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 ฤech cohomology groups. A relevant fact, also from the point of view of applications, is that the ฤech cohomology is related to the gap distribution in the spectrum of the Schroedinger operator with a potential associated to a particular tiling. The ฤech 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 -fold symmetry its minimal dimension is given by the Euler totient function . If the stretching factor of a one-dimensional substitution tiling is a Pisot number of algebraic degree and the rank of the first rational ฤech cohomology is also , 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 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 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 . When the integer first cohomology in 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 -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 orientations: (A) for , (B) for , (C) for , and (D) for . They produce simplicial arrangements , which are the basis for the generation of planar tilings with arbitrarily high symmetry. The systems with odd are included in the systems with even: (B) is included as a subsystem of (A), and (D) is included in (C). The fact that, in general, a system with lines is included in another with 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 , where

The simplicial arrangements and their mirror images can be obtained also by using for certain values of . For fixed we consider the subset formed by the lines with and . When , we get, for , successive rotations of two types of simple arrangements of lines (no more than two lines intersect) and the two types of simplicial arrangements and for even and odd, respectively. In addition to the nontriangular shapes, , have two sets of triangular prototiles: has the prototiles of while and its mirror image have the prototiles of and . In Figure 1(a) we show the case with (from top to bottom) . Two rotated versions of correspond to , the others are simple arrangements of lines obtained when . The case with can be seen in Figure 1(b). Now we have two simplicial arrangements: with 3 prototiles when and with one prototile for . When , the simple arrangement has 6 copies of the prototile appearing in , and when , has seven triangles with 3 different shapes (prototiles in ). In both cases the prototiles have edges with three different lengths as in Section 4.

**(a)**

**(b)**

The line configurations for can be seen in Figure 2(a) when and Figure 2(b) for . We get when , while and are obtained with . The simplicial arrangement corresponds to and we get when . By using the notation in [9], the prototiles appearing in and are (we use capital letters in Section 3 to describe the same tiles), while in and there are seven prototiles . All the prototiles now have edges with four different lengths as in Section 3.

**(a)**

**(b)**

The simple arrangements are on the basis of a construction of algebraic surfaces with many real nodes proposed recently [10]. The surfaces have affine equations , where the plane curve is a product of lines corresponding to a simple arrangement. Real variants of Chmutov surfaces [11] are based on as shown in [12]. On the other hand , and its mirror image , have one more triangle than , 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 , where some singularities disappear. Other deformations of interest, giving smooth surfaces, are obtained by varying .

#### 3. Cohomology Groups of Tiling Spaces with Ninefold Symmetry

The tiling spaces are formed by the closure of the translational orbit of one tiling . A substitution is said to force the border [13] if there is a positive integer such that if and are two tiles of the same type, then the two level- supertiles and 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 , 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 .

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

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

Up to mirror reflection and rotation, the tiling has three triangular prototiles 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 and 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 , where the subindexes denote the vertices appearing on the edge borders. The lengths of are , with .

The rotation group acts freely on edges and tiles. The two vertices satisfy with . We have six tile types , and each appears in 9 orientations. The Anderson-Putnam complex has Euler characteristic .

The three irreducible representations of over the integers are the one-dimensional scalar (), a two-dimensional and a six-dimensional representation. The 2D and the 6D representations have acting by multiplication on the rings and , respectively. In this case the vertex appears only in the scalar representation, the vertex 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 with . The boundary maps for are in all representations and in the scalar and 2D representations, but with appropriate modification having in mind that only the vertex appears in the 2D representation. If the cochain groups are denoted by , then, for , the coboundary maps are the transposes of the matrices given above with replaced with .

*Scalar Representation *

In the scalar representation, . We have and . The cohomologies of in this representation are , and .

*
2D Representation*

There is one vertex in the 2D representation, and are free modules of dimensions over the ring . The rank of over is 1, the matrix has rank 3 over , and, as abelian groups, we have .

*
6D Representation*

There are no vertices in the 6D representation, and are free modules of dimensions 1 and 3 over the ring . The map has rank 3 over and .

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

The substitution matrix on vertices is while the matrix on 1-chains is and the substitution on 2-chains

The induced matrices on cochains are obtained by transposition of the matrices with replaced with . They are isomorphisms, and the direct limit of each is simply . By taking into account all the irreducible representations, we get the following

The cohomology of is the rotationally invariant part of the cohomology of , and we have

Another tiling space with noncrystallographic symmetries and vanishing first cohomology in was analyzed in [6]. The pentagonal tiling space has the same prototiles as the Robinson decomposition of the Penrose tilings: with edges and lengths , with . The inflation rules can be obtained from in Figure 1(a) with arrows as indicated in [6]. The cohomology groups are and the rotationally invariant part is .

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 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 -type level-6 supertile (see Figureโโ12 in [6]). The borders of the new prototile shapes are formed by concatenation of scaled copies of oriented and . The substitution rules for the corresponding pattern can be seen in Figure 5(b).

**(a)**

**(b)**

Tiling spaces with crystallographic symmetries also may have zero first cohomology. For instance the chair tiling space has [14] with . In the next section we study other tiling spaces with crystallographic symmetries, which also have .

#### 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 .

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

We have four tile types , with . The tiles appear in three orientations and in one. The Anderson-Putnam complex has therefore Euler characteristic .

The two irreducible representations of over the integers are the one-dimensional scalar () and a two-dimensional vector representation. The vector representation has acting by multiplication on the ring . The edges and the tiles appear in all representations, while the vertex and the tiles appear only in the scalar representation. The boundary maps are and

*Scalar Representation *

In the scalar representation, . In this case and we have , and .

*Vector Representation*

There are two tiles in the 2D representation, and are free modules of dimensions 2 over the ring . The rank of over is 1 and we have and .

The cohomology of the complex is therefore and . 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 by and the substitution on 2-chains is

In the scalar representation the direct limits of each , under the action of the induced matrices on cochains , are for . We can take a basis for in terms of the canonical basis of formed by with the classes of cohomologous to zero. The eigenvalues of are with eigenvectors . Its action on gives the direct limit .

In the vector representation the direct limit of is . A base for in this representation is given by the vector which is the eigenvector of with eigenvalue 1. The direct limit is now . The cohomology groups for the equithirds tiling space are (see also [5])

The rotationally invariant part of the cohomology is

##### 4.2. The Tiling Spaces

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 that we treat now are based on such system of lines. In the construction discussed in Section 2, it corresponds to the simplicial arrangements ; in Figure 1(b).

The substitution rules for 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 . The substitution forces the border also in two steps: the vertex sequences in the level-2 superedges (Figure 9) are with lengths , , and the vertices in the edge borders are . The rotation group acts freely on the edges and with .

The four tile types are , and each appears in 3 orientations, except which appears in only one. The Euler characteristic of is .

In this case the edges and the tiles appear in all representations of , while the vertices , and the tile appear only in the scalar representation. The boundary maps are

*Scalar Representation*โโ

In this representation, . Here and with cohomologies and .

*Vector Representation*

There are three edges and three tiles in the 2D representation, and are free modules of dimension 3 over the ring . The rank of over is 2, and, as abelian groups, we have , and .

Adding up the contributions of each representations, we get the cohomology of the complex : , , and .

The substitution on the vertices is the identity matrix. The substitution on 1-chains is and the substitution on 2-chains is

In the scalar representation the direct limits of for are . A base for in this representation is given by the vectors , and the classes of , with , are cohomologous to zero. The action of applied twice to gives , and the direct limit yields a contribution of .

In the vector representation forms a base for . The matrix has eigenvalues with eigenvectors , and the direct sum of the eigenspaces is not isomorphic to all . The action of gives a contribution of to the first cohomology. A base for is given by with cohomologous to zero. We have , and the direct limit of is . The cohomology groups are then

The rotationally invariant part of the cohomology is

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 , then the number of control patches is also . 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 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 , and their substitution rules are represented in Figure 10. The patch content in the inflation rules is where and represent rotated versions of by 120 and 240 degrees.

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 obtained with the same basic construction as (which has inflation factor ) but with scaling factor shows that the associated tiling space has in a nontrivial rotationally invariant piece. The tiling has six prototiles with the same three shapes as in . The mirror images of , and shape level-1 supertiles are different in , while in we have two mirror inflation rules only for (Figure 7). The first cohomology for the tiling space in the scalar representation of is now . 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.