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

The computation of topological invariants of tiling spaces

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 [

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 [

A class of tiling spaces with fivefold symmetry has been analyzed in [

An extension of the methods introduced in [

The simplicial arrangements

Line configurations described by (

The line configurations for

Simple and simplicial arrangements for

The simple arrangements

The tiling spaces

In addition to

We study first the cohomology of a ninefold symmetry tiling space

Up to mirror reflection and rotation, the tiling has three triangular prototiles

Inflation rules for the ninefold symmetry tiling.

Level-3 supertiles in the ninefold symmetry tiling.

The rotation group

The three irreducible representations of

In the scalar representation,

There is one vertex in the 2D representation, and

There are no vertices in the 6D representation, and

We have obtained the cohomology of the complex

The substitution matrix on vertices is

The induced matrices on cochains

The cohomology of

Another tiling space with noncrystallographic symmetries and vanishing first cohomology in

Fractal tilings related with Penrose patterns have been studied in [

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

The equithirds tiling was obtained independently by L. Danzer and B. Kalahurka ([

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

We have four tile types

The two irreducible representations of

In the scalar representation,

There are two tiles in the 2D representation, and

The cohomology of the complex

The substitution on the vertex is the identity. The substitution on edges is represented by

In the scalar representation the direct limits of each

In the vector representation the direct limit of

The rotationally invariant part of the cohomology is

Although the paper [

The substitution rules for

Inflation rules for

Vertex configurations in

Level-2 supertiles in

The four tile types are

In this case the edges and the tiles

In this representation,

There are three edges and three tiles in the 2D representation, and

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

The substitution on the vertices is the

In the scalar representation the direct limits of

In the vector representation

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 [

For the tiling

Inflation rules for the patches

Although we have studied in detail only some particular cases, we can see that the general constructions of substitutions proposed in [

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