\underline{Cover sheet – please remove} – \underline{Some changes to email addresses}

To appear Memoirs of the American Mathematical Society

TOPOLOGICAL INVARIANTS FOR PROJECTION METHOD PATTERNS

Alan Forrest^{1}, John Hunton^{2}, Johannes Kellendonk^{3}

1Department of Mathematics, National University of Ireland,

Cork, Republic of Ireland Current Email: alan forest@base3.com

2The Department of Mathematics and Computer Science,

University of Leicester, University Road, Leicester, LE1 7RH, England Email: J.Hunton@mcs.le.ac.uk

3Fachbereich Mathematik, Sekr. MA 7-2,

Technische Universität Berlin,

10623 Berlin, Germany Email: kellen@math.tu-berlin.de

Current Email: kellendonkj@Cardiff.ac.uk

TABLE OF CONTENTS

General Introduction  1
I Topological Spaces and Dynamical Systems  10
1 Introduction  10
2 The projection method and associated geometric constructions  11
3 Topological spaces for point patterns  16
4 Tilings and point patterns  20
5 Comparing Π_{u} and Π̃_{u}  24
6 Calculating MP̃_{u} and MP_{u}  26
7 Comparing MP_{u} with MP̃_{u}  30
8 Examples and Counter-examples  33
9 The topology of the continuous hull  37
10 A Cantor ℤ^{d} Dynamical System  40
II Groupoids, C^{∗}-algebras, and their Invariants  46
1 Introduction  46
2 Equivalence of Projection Method pattern groupoids  47
3 Continuous similarity of Projection Method pattern groupoids  54
4 Pattern cohomology and K-theory  58
5 Homological conditions for self similarity  61
III Approaches to Calculation I: Cohomology for Codimension One  64
1 Introduction  64
2 Inverse limit acceptance domains  64
3 Cohomology in the case d = N − 1  66
IV Approaches to Calculation II: Infinitely Generated Cohomology  69
1 Introduction  69
2 The canonical projection tiling  69
3 Constructing C-topes  74
4 The indecomposable case  80
5 The decomposable case  86
6 Conditions for infinitely generated cohomology  89
V Approaches to Calculation III: Cohomology for Small Codimension  94
1 Introduction  94
2 Set up and statement of the results  95
3 Complexes defined by the singular spaces  99
4 Group homology  101
5 The spectral sequences  103
6 Example: Ammann-Kramer tilings  112
Bibliography  116

Abstract

This memoir develops, discusses and compares a range of commutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general – any compact set which is the closure of its interior – while in the second half we concentrate on the so-called canonical patterns. The topological invariants used are various forms of K-theory and cohomology applied to a variety of both C^{∗}-algebras and dynamical systems derived from such a pattern.

The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or self-similarity, but one of the main motivations is also to provide an accessible approach to the the K_{0} group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern.

The main results provide complete descriptions of the (unordered) K-theory and cohomology of codimension 1 projection patterns, formulæ for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canonical projection patterns. Computations are made for the Ammann-Kramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One corollary is that, generically, projection patterns cannot be derived via substitution systems.

Mathematics Subject Classification Primary: 52C23

Mathematical Subject Classification Secondary: 19E20, 37Bxx, 46Lxx, 55Txx, 82D25

General Introduction

Of the many examples of aperiodic tilings or aperiodic point sets in Euclidean space found in recent years, two classes stand out as particularly interesting and æsthetically pleasing. These are the substitution tilings, tilings which are self-similar in a rather strong sense described in [GS] [R1] [S1] [AP], and the tilings and patterns obtained by the method of cut and projection from higher dimensional periodic sets described in [dB1] [KrNe] [KD]. In this memoir we consider the second class. However, some of the best studied and most physically useful examples of aperiodic tilings, for example the Penrose tiling [Pe] and the octagonal tiling (see [Soc]), can be approached as examples of either class. Therefore we study specially those tilings which are in the overlap of these two classes, and examine some of their necessary properties.

Tilings and patterns in Euclidean space can be compared by various degrees of equivalence, drawn from considerations of geometry and topology [GS]. Two tilings can be related by simple geometric tranformations (shears or rotations), topological distortions (bending edges), or by more radical adaptation (cutting tiles in half, joining adjacent pairs etc). Moreover, point patterns can be obtained from tilings in locally defined ways (say, by selecting the centroids or the vertices of the tiles) and vice versa (say, by the well-known Voronoi construction). Which definition of equivalence is chosen is determined by the problem in hand.

In this paper, we adopt definitions of equivalence (pointed conjugacy and topological conjugacy, I.4.5) which allow us to look, without loss of generality, at sets of uniformly isolated points (point patterns) in Euclidean space. In fact these patterns will typically have the Meyer property [La1] (see I.4.5). Therefore in this introduction, and often throughout the text, we formulate our ideas and results in terms of point patterns and keep classical tilings in mind as an implicit example.

The current rapid growth of interest in projection method patterns started with the discovery of material quasicrystals in 1984 [SBGC], although these patterns had been studied before then. Quasicrystaline material surprised the physical world by showing sharp Bragg peaks under X-ray scattering, a phenomenon usually associated only with periodic crystals. Projection method patterns share

Received by the editor October 24, 2000. this unusual property and in recent studies they have become the prefered model of material quasicrystals [1] [2]. This model is not without criticism, see e.g. [La2].

Whatever the physical significance of the projection method consruction, it also has great mathematical appeal in itself: it is elementary and geometric and, once the acceptance domain and the dimensions of the spaces used in the construction are chosen, has a finite number of degrees of freedom. The projection method is also a natural generalization of low dimensional examples such as Sturmian sequences [HM] which have strong links with classical diophantine approximation.

General approach of this book In common with many papers on the topology of tilings, we are motivated by the physical applications and so are interested in the properties of an individual quasicrystal or pattern in Euclidean space. The topological invariants of the title refer not to the topological arrangement of the particular configuration as a subset of Euclidean space, but rather to an algebraic object (graded group, vector space etc.) associated to the pattern, and which in some way captures its geometric properties. It is defined in various equivalent ways as a classical topological invariant applied to a space constructed out of the pattern. There are two choices of space to which to apply the invariant, the one C^{∗}-algebraic, the other dynamical, and these reflect the two main approaches to this subject, one starting with the construction of an operator algebra and the other with a topological space with ℝ^{d} action.

The first approach, which has the benefit of being closer to physics and which thus provides a clear motivation for the topology, can be summarized as follows. Suppose that the point set T represents the positions of atoms in a material, like a quasicrystal. It then provides a discrete model for the configuration space of particles moving in the material, like electrons or phonons. Observables for these particle systems, like energy, are, in the absence of external forces like a magnetic field, functions of partial translations. Here a partial translation is an operator on the Hilbert space of square summable functions on T which is a translation operator from one point of T to another combined with a range projection which depends only on the neighbouring configuration of that point. The appearance of that range projection is directly related to the locality of interaction. The norm closure A_{T} of the algebra generated by partial translations can be regarded as the C^{∗}-algebra of observables. The topology we are interested in is the non-commutative topology of the C^{∗}-algebra of observables and the topological invariants of T are the invariants of A_{T}. In particular, we shall be interested in the K-theory of A_{T}. Its K_{0}-group has direct relevance to physics through the gap-labelling. In fact, Jean Bellissard’s K-theoretic formulation of the gap-labelling stands at the beginning of this approach [B1].

The second way of looking at the topological invariants of a discrete point set T begins with the construction of the continuous hull of T . There are various ways of defining a metric on a set of patterns through comparison of their local configurations. Broadly speaking, two patterns are deemed close if they coincide on a large window around the origin 0 ∈ ℝ^{d} up to a small discrepancy. It is the way the allowed discrepancy is quantified which leads to different metric topologies and we choose here one which has the strongest compactness properties, though we have no intrinsic motivation for this. The continuous hull of T is the closure, MT , of the set of translates of T with respect to this metric. We use the notation MT because it is essentially a mapping torus construction for a generalized discrete dynamical system: ℝ^{d} acts on MT by translation and the set Ω_{T} of all elements of MT which (as point sets) contain 0 forms an abstract transversal called the discrete hull. If d were 1 then Ω_{T} would give rise to a Poincaré section, the intersection points of the flow line of the action of ℝ with Ω_{T} defining an orbit of a ℤ action in Ω_{T}, and MT would be the mapping torus of that discrete dynamical system (Ω_{T} , ℤ). For larger d one cannot expect to get a ℤ^{d} action on Ω_{T} in a similar way but finds instead a generalized discrete dynamical system which can be summarized in a groupoid GT whose unit space is the discrete hull Ω_{T}. Topological invariants for T are therefore the topological invariants of MT and of GT and we shall be interested in particular in their cohomologies. We define the cohomology of T to be that of GT . Under a finite type condition, namely that for any given r there are only a finite number of translational congruence classes of subsets which fit inside a window of diameter r, the algebra A_{T} sketched above is isomorphic to the groupoid C^{∗}-algebra of GT . This links the two approaches.

Having outlined the general philosophy we hasten to remark that we will not explain all its aspects in the main text. In particular, we have nothing to say there about the physical aspects of the theory and the description of the algebra of observables, referring here the reader to [BZH] [KePu], or to the more original literature [B1] [B2] [K1]. Instead, our aim in this memoir is to discuss and compare the different commutative and non-commutative invariants, to demonstrate their applicability as providing obstructions to a tiling arising as a substitution, and finally to provide a practical method for computing them; this we illustrate with a number of examples, including that of the Ammann-Kramer (‘3 dimensional Penrose’) tiling. Broadly speaking, one of the perspectives of this memoir is that non-commutative invariants for projection point patterns can be successfully computed by working with suitable commutative analogues.

The subject of this book We work principally with the special class of point sets (possibly with some decoration) obtained by cut and projection from the integer lattice ℤ^{N} which is generated by an orthonormal base of ℝ^{N}. Reserving detail and elaborations for later, we call a projection method pattern T on E = ℝ^{d} a pattern of points (or a finite decoration of it) given by the orthogonal projection onto E of points in a strip (K × E) ∩ ℤ^{N} ⊂ ℝ^{N}, where E is a subspace of ℝ^{N} and K × E is the so-called acceptance strip, a fattening of E in ℝ^{N} defined by some suitably chosen region K in the orthogonal complement E^{⊥} of E in ℝ^{N}. The pattern T thus depends on the dimension N, the positioning of E in ℝ^{N} and the shape of the acceptance domain K. When this construction was first made [dB1] [KD] the domain K was taken to be the projected image onto E^{⊥} of the unit cube in ℝ^{N} and this choice gives rise to the so-called canonical projection method patterns, but for the first three chapters we allow K to be any compact subset of E^{⊥} which is the closure of its interior (so, with possibly even fractal boundary, a case of current physical interest [BKS] [Sm] [Z] [GLJJ]).

It is then very natural to consider not only T but also all point patterns which are obtained in the same way but with ℤ^{N} repositioned by some vector u ∈ ℝ^{N}, i.e., ℤ^{N} replaced by ℤ^{N} + u. Completing certain subsets of positioning vectors u with respect to an appropriate pseudo-metric gives us the continuous hull MT . This analysis shows in particular that MT contains another transversal X_{T} which gives rise to d independant commuting ℤ actions and hence to a genuine discrete dynamical system (X_{T} , ℤ^{d}) whose mapping torus is also MT . This is a key point in relating the K-theory of A_{T} with the cohomology of T ; in the process, the latter is also identified with the Čech cohomology of MT and with the group cohomology of ℤ^{d} with coefficients in the continuous integer valued functions over X_{T}.

The space X_{T} arises in another way. Let V be a connected component of the euclidian closure of π^{⊥}(ℤ^{N}), where π^{⊥} denotes the orthoprojection onto E^{⊥}. We first disconnect V along the boundaries of all π^{⊥}(ℤ^{N})-translates of K (we speak loosely here, but make the idea precise in Chapter I). Then X_{T} can be understood as a compact quotient of the disconnected V with respect to a proper isometric free abelian group action. In the case of canonical projection method patterns, on which we concentrate in the last two chapters, the boundaries which disconnect V are affine subspaces and so define a directed system of locally finite CW decompositions of Euclidean space. With respect to this CW-complex, the integer valued functions over X_{T} appear as continuous chains in the limit. This makes the group cohomology of the dynamical action of ℤ^{d} on X_{T} accessible through the standard machinery (exact sequences and spectral sequences) of algebraic topology.

As mentioned, the interest in physics in the non-commutative topology of tilings and point sets is based on the observation that A_{T} is the C^{∗}-algebra of observables for particles moving in T . In particular, any Hamilton operator which describes this motion has the property that its spectral projections on energy intervals whose boundaries lie in gaps of the spectrum belong to A_{T} as well and thus define elements of K_{0}(A_{T}). Therefore, the ordered K_{0}-group (or its image on a tracial state) may serve to ‘count’ (or label) the possible gaps in the spectrum the Hamilton operator [B2] [BBG] [K1]. One of the main results of this memoir is the determination in Chapter V of closed formulae for the ranks of the K-groups corresponding to canonical projection method patterns with small codimension (as one calls the dimension of V). These formulæ apply to all common tilings including the Penrose tilings, the octagonal tilings and three dimensional icosahedral tilings. Unfortunately our method does not as yet give full information on the order of K_{0} or the image on a tracial state.

Further important results of this memoir concern the structure of a K-group of a canonical projection method pattern. We find that its K_{0}-group is generically infinitely generated. But when the rank of its rationalization is finite then it has to be free abelian. We observe in Chapters III and IV that both properties are obstruction to some kinds of self-similarity. More precisely, infinitely generated rationalized cohomology rules out that the tiling is a substitution tiling. On the other hand, if we know already that the tiling is substitutional then its K-group must be free abelian for it to be a canonical projection method tiling.

No projection method pattern is known to us which has both infinitely generated cohomology and also allows for local matching rules in the sense of [Le]. Furthermore, all projection method patterns which are used to model quasicrystals seem to have a finitely generated K_{0}-group. We cannot offer yet an interpretation of the fact that some patterns produce only finitely many generators for their cohomology whereas others do not, but, if understood, we hope it could lead to a criterion to single out a subset of patterns relevant for quasicrystal physics from the vast set of patterns which may be obtained from the projection method.

We have mentioned above the motivation from physics to study the topological theory of point sets or tilings. The theory is also also of great interest for the theory of topological dynamical systems, since in d = 1 dimensions the dynamical systems mentioned above have attracted a lot of attention. In [GPS] the meaning of the non-commutative invariants for the one dimensional case has been analysed in full detail. Furthermore, substitution tilings give rise to hyperbolic ℤ-actions with expanding attractors (hyperbolic attractors whose topological dimensions are that of their expanding direction)

[AP] a subject of great interest followed up recently by Williams [W] who conjectures that continuous hulls of substitution tilings (called tiling spaces in [W]) are fiber bundles over tori with the Cantor set as fiber. We have not put emphasis on this question but it may be easily concluded from our analysis of Section I.10 that the continuous hulls of projection method tilings are always Cantor set fiber bundles over tori (although these tilings are rarely substitutional and therefore carry in general no obvious hyperbolic ℤ-action). Anderson and Putnam [AP] and one of the authors [K2] have employed the substitution of the tiling to calculate topological invariants of it.

Organization of the book The order of material in this memoir is as follows. In Chapter I we define and describe the various dynamical systems mentioned above, and examine their topological relationships. These are compared with the pattern groupoid and its associated C^{∗} algebra in Chapter II where these latter objects are introduced. Also in Chapter II we set up and prove the equivalence of our various topological invariants; we end the chapter by demonstrating how these invariants provide an obstruction to a pattern being self-similar.

The remaining chapters offer three illustrations of the computability of these invariants. In Chapter III we give a complete calculation for all ‘codimension one’ projection patterns – patterns arising from the projection of slices of ℤ^{d+1} to ℝ^{d} for more or less arbitrary acceptance domains. In Chapter IV we give descriptions of the invariants for generic projection patterns arising from arbitrary projections ℤ^{N} to ℝ^{d} but with canonical acceptance domain. Here, applying the result at the end of Chapter II, we prove the result mentioned above that almost all canonical projection method patterns have infinitely generated cohomology and so fail to be substitution tilings. In Chapter V we examine the case of canonical projection method patterns with finitely generated cohomology, such as would arise from a substitution system. We develop a systematic approach to the calculation of these invariants and use this to produce closed formulæ for the cohomology and K-theory of projection patterns of codimension 1, 2 and 3: in principle the procedure can be iterated to higher codimensions indefinitely, though in practice the formulæ would soon become tiresome. Some parameters of these formulae allow for a simple description in arbitrary codimension, as e.g. the Euler characterisitc (V.2.8). We end with a short description of the results for the Ammann-Kramer tiling.

There is a separate introduction to each chapter where relevant classical work is recalled and where the individual sections are described roughly. We adopt the following system for crossreferences. The definitions, theorems etc. of the same chapter are cited e.g. as Def. 2.1 or simply 2.1. The definitions, theorems etc. of the another chapter are cited e.g. as Def. II.2.1 or simply II.2.1.

A note on the writing of this book Originally this memoir was conceived by the three authors as a series of papers leading to the results now in Chapters IV and V, aiming to found a calculus for projection method tiling cohomology. These papers are currently available as a preprint-series Projection Quasicrystals I-III [FHKI-III] covering most of the results in this memoir. The authors’ collaboration on this project started in 1997 and, given the importance of the subject and time it has taken to bring the material to its current state, it is inevitable that some results written here have appeared elsewhere in the literature during the course of our research. We wish to acknowledge these independent developments here, although we will refer to them again as usual in the body of the text.

The general result of Chapter I, that the tiling mapping torus is also a discrete dynamical mapping torus, and that the relevant dynamics is an almost 1-1 extension of a rotation on a torus, has been known with varying degrees of precision and generality for some time and we mention the historical developments in the introduction to Chapter I. Our approach constructs a large topological space from which the pattern dynamical system is formed by a quotient and so we follow most closely the idea pioneered by Le [Le] for the case of canonical projection tilings. The “Cantorization” of Euclidean space by corners or cuts, as described by Le and others (see [Le] [H] etal.), is produced in our general topological context in sections I.3, I.4 and I.9. In this, we share the ground with Schlottmann [Sch] and Herrmann [He] who have recently established the results of Chapter I in such (and even greater) generality, Schlottmann in order to generalize results of Hof and describe the unique ergodicity of the underlying dynamical systems and Herrmann to draw a connection between codimension 1 projection patterns and Denjoy homeomorphisms of the circle. We mention this relation at the end of chapter III.

Bellissard, Contensou and Legrand [BCL] compare the C^{∗}- algebra of a dynamical groupoid with a C^{∗}-algebra of operators defined on a class of tilings obtained by projection, the general theme of Chapter II. Using a Rosenberg Shochet spectral sequence, they also establish, for 2-dimensional canonical projection tilings, an equation of dynamical cohomology and C^{∗} K-theory in that case. It is the first algebraic topological approach to projection method tiling K-theory found in the literature. We note, however, that the groupoid they consider is not always the same as the tiling groupoid we consider, nor do the dynamical systems always agree; the Penrose tiling is a case in point, where we find that K_{0} of the spaces considered in [BCL] is ℤ^{∞}. The difference may be found in the fact that we consider a given projection method tiling or pattern and its translates, while they consider a larger set of tilings, two elements of which may sometimes be unrelated by approximation and translation parallel to the projection plane.

Acknowledgements. We thank F. Gähler for verifying and assisting our results (V.6) on the computer and for various useful comments and M. Baake for reading parts of the manuscript. The third author would like to thank J. Bellissard for numerous discussions and constant support.

The collaboration of the first two authors was initiated by the William Gordon Seggie Brown Fellowship at The University of Edinburgh, Scotland, and received continuing support from a Collaborative Travel Grant from the British Council and the Research Council of Norway with the generous assistance of The University of Leicester, England, and the EU Network “Non-commutative Geometry” at NTNU Trondheim, Norway. The collaboration of the first and third authors was supported by the Sonderforschungsbereich 288, “Differentialgeometrie und Quantenphysik” at TU Berlin, Germany, and by the EU Network and NTNU Trondheim. The third author is supported by the Sonderforschungsbereich 288 “Differentialgeometrie und Quantenphysik” at TU Berlin. All three authors are most grateful for the financial help received from these various sources.

I Topological Spaces and Dynamical Systems

1 Introduction

In this chapter our broad goal is to study the topology and associated dynamics of projection method patterns, while imposing only few restrictions on the freedom of the construction. From a specific set of projection data we define and examine a number of spaces and dynamical systems and their relationships; from these constructions, in later chapters, we set up our invariants, defined via various topological and dynamical cohomology theories. In Chapter II we shall also compare the commutative spaces of this chapter with the noncommutative spaces considered by other authors in e.g. [B2] [BCL]

[AP] [K2].

Given a subspace, E, acceptance domain, K, and a positioning parameter u, we distinguish two particular ℝ^{d} dynamical systems constructed by the projection method, (MP_{u}, ℝ^{d}) and (MP̃_{u}, ℝ^{d}), the first automatically a factor of the second. This allows us to define a projection method pattern (with data (E, K, u)) as a pattern, T , whose dynamical system, (MT , ℝ^{d}), is intermediate to these two extreme systems. Sections 3 to 9 of this chapter provide a complete description of the spaces and the extension MP̃_{u} ⟶ MP_{u}, showing, under further weak assumptions on the acceptance domain, that it is a finite isometric extension. In section 7 we conclude that this restricts (MT , ℝ^{d}) to one of a finite number of possibilities, and that any projection method pattern is a finite decoration of its corresponding point pattern P_{u} (2.1). The essential definitions are to be found in Sections 2 and 4.

In section 10 we describe yet another dynamical system connected with a projection method pattern, this time a ℤ^{d} action on a Cantor set X, whose mapping torus is the space of the pattern dynamical system. It will be this dynamical system that, in chapters 3, 4 and 5, will allow the easiest computation and discussion of the behaviour of our invariants. For the canonical case with E^{⊥} ∩ ℤ^{N} = 0 this is the same system as that constructed in [BCL].

All the dynamical systems produced in this memoir are almost 1-1 extensions of an action of ℝ^{d} or ℤ^{d} by rotations (Def. 2.15) on a torus (or torus extended by a finite abelian group). In each case the dimension of the torus and the generators of the action can be computed explicitly. This gives a clear picture of the orbits of nonsingular points in the pattern dynamical system. A precursor to our description of the pattern dynamical system can be found in the work of Robinson [R2], who examined the dynamical system of the Penrose tiling and showed that it is an almost 1-1 extension of a minimal ℝ^{2} action by rotation on a 4-torus. Although Robinson used quite special properties of the tiling, Hof [H] has noted that these techniques are generalizable without being specific about the extent of the generalization.

Our approach is quite different from that of Robinson and, by constructing a larger topological space from which the pattern dynamical system is formed by a quotient, we follow most closely the approach pioneered by Le [Le] as noted in the General Introduction. The care taken here in the topological foundations seems necessary for further progress and to allow general acceptance domains. Even in the canonical case, Corollary 7.2 and Proposition 8.4 of this chapter, for example, require this precision despite being direct generalizations of Theorem 3.8 in [Le]. Also, as mentioned in the General Introduction, many of the results of this Chapter are to be found independently in [Sch].

2 The projection method and associated geometric constructions

We use the construction of point patterns and tilings given in Chapters 2 and 5 of Senechal’s monograph [Se] throughout this paper, adding some assumptions on the acceptance domain in the following definitions.

Definitions 2.1 Consider the lattice ℤ^{N} sitting in standard position inside ℝ^{N} (i.e. it is generated by an orthonormal basis of ℝ^{N}). Suppose that E is a d dimensional subspace of ℝ^{N} and E^{⊥} its orthocomplement. For the time being we shall make no assumptions about the position of either of these planes.

Let π be the projection onto E and π^{⊥} the projection onto E^{⊥}. Let Q = \overline{E + ℤ^{N}} (Euclidean closure). This is a closed subgroup

of ℝ^{N}.

Let K be a compact subset of E^{⊥} which is the closure if its interior (which we write IntK) in E^{⊥}. Thus the boundary of K in E^{⊥} is compact and nowhere dense. Let Σ = K + E, a subset of ℝ^{N} sometimes refered to as the strip with acceptance domain K.

A point v ∈ ℝ^{N} is said to be non-singular if the boundary, ∂Σ, of Σ does not intersect ℤ^{N} + v. We write NS for the set of non-singular points in ℝ^{N}. These points are also called regular in the literature.

Let P̃_{v} = Σ ∩ (ℤ^{N} + v), the strip point pattern.

Define P_{v} = π(P̃_{v}), a subset of E called the projection point pattern.

In what follows we assume E and K are fixed and suppress mention of them as a subscript or argument.

Lemma 2.2 With the notation above,

i/ NS is a dense G_{δ} subset of ℝ^{N} invariant under translation by E.

ii/ If u ∈ NS, then NS ∩ (Q + u) is dense in Q + u.

iii/ If u ∈ NS and F is a vector subspace of ℝ^{N} complementary to E, then NS ∩ (Q + u) ∩ F is dense in (Q + u) ∩ F.

Proof i/ Note that ℝ^{N} \ NS is a translate of the set ∪_{v∈ℤ^{N}} (∂K + E + v) (where the boundary is taken in E^{⊥}) and our conditions on K complete the proof.

ii/ NS ∩ (Q + u) ⊃ E + ℤ^{N} + u.

iii/ \overline{(E + ℤ^{N} + u) ∩ F} = (Q + u) ∩ F. □

Remark 2.3 The condition on the acceptance domain K is a topological version of the condition of [H]. We note that our conditions include the examples of acceptance domains with fractal boundaries which have recently interested quasicrystalographers [BKS] [Sm] [Z] [GLJJ].

In the original construction [dB1] [KD] K = π^{⊥}([0, 1]^{N}). We call this the canonical acceptance domain and we reserve the name canonical projection method pattern for the patterns P_{u} produced from this acceptance domain. Sometimes this is shortened to canonical pattern for convenience.

This is closely related to the canonical projection tiling, defined by [OKD] formed by a canonical acceptance domain, u ∈ NS and projecting onto E those d-dimensional faces of the lattice ℤ^{N} + u which are contained entirely in Σ. We write this tiling T_{u}.

The following notation and technical lemma makes easier some calculations in future sections.

Definition 2.4 If X is a subspace of Y, both topological spaces, and A ⊂ X, then we write Int_{X}A to mean the interior of A in the subspace topology of X.

Likewise we write ∂_{X}A for the boundary of A taken in the subspace topology of X.

Lemma 2.5 a/ If u ∈ NS, then (Q+u)∩IntK = Int_{(Q+u)∩E^{⊥}}((Q+ u) ∩ K) and (Q + u) ∩ ∂_{E^{⊥}}K = ∂_{(Q+u)∩E^{⊥}}((Q + u) ∩ K).

b/ If u ∈ NS, then ((Q + u) ∩ E^{⊥}) \ NS = ∂_{(Q+u)∩E^{⊥}}((Q + u) ∩ K) + π^{⊥}(ℤ^{N}).

Proof a/ To show both facts, it is enough to show that (∂_{E^{⊥}}K) ∩ (Q + u) has no interior as a subspace of (Q + u) ∩ E^{⊥}.

Suppose otherwise and that U is an open subset of ∂K ∩ (Q + u) in (Q+u)∩E^{⊥}. By the density of π^{⊥}(ℤ^{N}) in Q∩E^{⊥}, we find v ∈ ℤ^{N} such that u ∈ U + π^{⊥}(v). But this implies that u ∈ ∂K + π^{⊥}(v) and so u ∉ NS - a contradiction.

b/ By defintion the left-hand side of the equation to be proved is equal to (∂_{E^{⊥}}K + π^{⊥}(ℤ^{N})) ∩ (Q + u) which equals (∂_{E^{⊥}}K ∩ (Q + u)) + π^{⊥}(ℤ^{N}) since π^{⊥}(ℤ^{N}) is dense in Q ∩ E^{⊥}. By part a/ therefore we obtain the right-hand side of the equation. □

Condition 2.6 We exclude immediately the case (Q + u) ∩ IntK = ∅ since, when u ∈ NS, this is equivalent to P_{u} = ∅.

Examples 2.7 We note the parameters of two well-studied examples, both with canonical acceptance domain (2.3).

The octagonal tiling [Soc] has N = 4 and d = 2, where E is a vector subspace of ℝ^{4} invariant under the action of the linear map which maps orthonormal basis vectors e_{1} ↦ e_{2}, e_{2} ↦ e_{3}, e_{3} ↦ e_{4}, e_{4} ↦ −e_{1}. Its orthocomplement, E^{⊥}, is the other invariant subspace. Here Q = ℝ^{4} and so many of the distinctions made in subsequent sections are irrelevant to this example.

The Penrose tiling [Pe] [dB1] has N = 5 and d = 2 (although we note that there is an elegant construction using the root lattice of A_{4} in ℝ^{4} [BJKS]). The linear map which maps e_{i} ↦ e_{i+1} (indexed modulo 5) has two 2 dimensional and one 1 dimensional invariant subspaces. Of the first two subspaces, one is chosen as E and the other we name V. Then in fact Q = E ⊕ V ⊕ Δ̃, where Δ̃ = \frac{1}{5}(e_{1} + e_{2} + e_{3} + e_{4} + e_{5})ℤ, and Q is therefore a proper subset of ℝ^{5}, a fact which allows the construction of generalized Penrose tilings using a parameter u ∈ NS \ Q.

Note that we speak of tilings and yet only consider point patterns. In both examples, the projection tiling [OKD] is conjugate to both the corresponding strip point pattern and projection point pattern, a fact proved in greater generality in section 8.

We develop these geometric ideas in the following lemmas. The next is Theorem 2.3 from [Se].

Theorem 2.8 Suppose that ℤ^{N} is in standard position in ℝ^{N} and suppose that ϕ: ℝ^{N} ⟶ ℝ^{n} is a surjective linear map. Then there is a direct sum decomposition ℝ^{n} = V ⊕ W into real vector subspaces such that ϕ(ℤ^{N}) ∩ V is dense in V, ϕ(ℤ^{N}) ∩ W is discrete and ϕ(ℤ^{N}) = (V ∩ ϕ(ℤ^{N})) + (W ∩ ϕ(ℤ^{N})). □

We proceed with the following refinement of Proposition 2.15 of [Se].

Lemma 2.9 Suppose that ℤ^{N} is in standard position in ℝ^{N} and suppose that ϕ: ℝ^{N} ⟶ F is an orthogonal projection onto F a subspace of ℝ^{N}. With the decomposition of F implied by Theorem 2.8, (F ∩ ℤ^{N}) + (V ∩ ϕ(ℤ^{N})) ⊂ ϕ(ℤ^{N}) as a finite index subgroup.

Also, the lattice dimension of F ∩ ℤ^{N} equals dim F − dim V and the real vector subspace generated by F ∩ ℤ^{N} is orthogonal to V.

Proof Suppose that U is the real linear span of Δ = F ∩ℤ^{N}. Since Δ is discrete, the lattice dimension of Δ equals the real space dimension of U.

The argument of the proof of Proposition 2.15 in [Se] shows that each element of F ∩ ℤ^{N} is orthogonal to V. Therefore we have dim_{ℝ}(U) ≤ dim_{ℝ}(F) − dim_{ℝ}(V) immediately.

Consider the rational vector space ℚ^{N}, contained in ℝ^{N} and containing ℤ^{N}, both in canonical position. Let U′ be the rational span of Δ and note that U′ = U ∩ ℚ^{N} and that dim_{ℚ}(U′) = dim_{ℝ}(U). Let U^{′⊥} be the orthocomplement of U′ with respect to the standard inner product in ℚ^{N} so that, by simple rational vector space arguments, ℚ^{N} = U′ ⊕ U^{′⊥}. Thus (U′ ∩ ℤ^{N}) + (U^{′⊥} ∩ ℤ^{N}) forms a discrete lattice of dimension N.

Extending to the real span, we deduce that (U ∩ℤ^{N})+(U^{⊥} ∩ℤ^{N}) is a discrete sublattice of ℤ^{N} of dimension N, hence a subgroup of finite index. Also the lattice dimension of U ∩ ℤ^{N} and U^{⊥} ∩ ℤ^{N} are equal to dim_{ℝ}(U) and dim_{ℝ}(U^{⊥}) respectively.

Let L = (U^{⊥} ∩ ℤ^{N}) be considered as a sublattice of U^{⊥}. It is integral (with respect to the restriction of the inner product on ℝ^{N}) and of full dimension. The projection ϕ restricts to an orthogonal projection U^{⊥} ⟶ U^{⊥} ∩ F and, by construction, U^{⊥} ∩ F ∩ L = 0. Therefore Proposition 2.15 of [Se] applies to show that ϕ(L) is dense in U^{⊥} ∩ F and that ϕ is 1-1 on L.

However ϕ(L) ⊂ ϕ(ℤ^{N}) and so, by the characterisation of Theorem 2.2, we deduce that U^{⊥} ∩ F ⊂ V. However, since U^{⊥} ⊃ V, we have U^{⊥} ∩ F = V.

We have U ∩ ℤ^{N} = F ∩ ℤ^{N} and ϕ(U^{⊥} ∩ ℤ^{N}) = ϕ(ℤ^{N}) ∩ V automatically. Therefore (ϕ(ℤ^{N}) ∩ V) + (F ∩ ℤ^{N}) = ϕ((U^{⊥} ∩ ℤ^{N}) + (U ∩ ℤ^{N})). As proved above, this latter set is the image of a finite index subgroup of the domain, ℤ^{N}, and therefore it is a finite index subgroup of the image ϕ(ℤ^{N}) as required.

The remaining properties follow quickly from the details above. □

Definition 2.10 Let Δ = E^{⊥} ∩ ℤ^{N} and Δ̃ = U ∩ \overline{π^{⊥}(ℤ^{N})} where U is the real vector space generated by Δ.

Note that the discrete group Δ defined here is not the real vector space Δ(E) defined in [Le], but it is a cocompact sublattice and so the dimensions are equal.

Corollary 2.11 With the notation of Theorem 2.8 and taking ϕ = π^{⊥}, \overline{π^{⊥}(ℤ^{N})} = V ⊕ Δ̃ and Q = E ⊕ V ⊕ Δ̃ are orthogonal direct sums. Moreover, Δ is a subgroup of Δ̃ with finite index. □

Example 2.12 For example the octagonal tiling has Δ = 0 and the Penrose tiling has Δ = (e_{1} + e_{2} + e_{3} + e_{4} + e_{5})ℤ, a subgroup of index 5 in Δ̃.

And finally a general result about isometric extensions of dynamical systems.

Definition 2.13 Suppose that ρ: (X, G) ⟶ (Y, G) is a factor map of topological dynamical systems with group, G, action. If every fibre ρ^{−1}(y) has the same finite cardinality, n, then we say that (X, G) is an n-to-1 extension.

The structure of such extensions, a special case of isometric extensions, is well-known [F].

Lemma 2.14 Suppose that ρ: (X, G) ⟶ (Y, G) is an n-to-1 extension and that (X, G) is minimal. Suppose further that there is an abelian group H which acts continuously on X, commutes with the G action,

preserves ρ fibres and acts transitively on each fibre. If (X, G) ⟶^{ρ′}

(Z, G) ⟶^{ρ′′} (Y, G) is an intermediate factor, then (Z, G) is an m-to-1 extension where m divides n, and we can find a subgroup, H′ of H, so that

i/ H/H′ acts continuously on Z, commutes with the G action, preserves ρ′′ fibres and acts transitively on each fibre and

ii/ H′ acts on X as a subaction of H, preserving ρ′ fibres and acting transitively on each fibre.

Proof Given h ∈ H, consider X_{h} = {x | ρ′(x) = ρ′(hx)} which is a closed G-invariant subset of X. Therefore, by minimality, X_{h} = ∅ or X. Let H′ = {h ∈ H | X_{h} = X} which can be checked is a subgroup of H. The properties claimed follow quickly. □

Definitions 2.15 We will call an extension which obeys the conditions of Lemma 2.14 a finite isometric extension.

We say an almost 1-1 extension of topological dynamical systems ρ: (X, G) ⟶ (Y, G) is one in which the set ρ^{−1}(y) is a singleton for a dense G_{δ} of y ∈ Y. In the case of minimal actions, it is sufficient to find just one point y ∈ Y for which ρ^{−1}(y) is a singleton.

We say that an abelian topological group, G, acting on a compact abelian topological group, Z say, acts by rotation if there is a continuous group homomorphism, ψ : G ⟶ Z such that gz = z + ψ(g) for all z ∈ Z and g ∈ G.

3 Topological spaces for point patterns

When v is non-singular, P_{v} forms an almost periodic pattern of points in the sense that each spherical window, whose position is shifted over the infinite pattern, reveals the same configuration at a syndetic (relatively dense) set of positions [Se]. We may formulate this fact precisely in terms of minimal dynamics in a well-known process. Here we note the relevant constructions and lemmas.

Definition 3.1 Let B(r) be the closed ball in E, centre 0 and of radius r with boundary ∂B(r). Given a closed subset, A, of ℝ^{N}, define A[r] = (A ∩ B(r)) ∪ ∂B(r), a closed subset of B(r). Consider the Hausdorff metric d_{r} defined among closed subsets of B(r) and define a metric (after [R1], [S2]) on closed subsets of the plane by

D(A, A′) = inf{1/(r + 1) | d_{r}(A[r], A′[r]) < 1/r}.

We are grateful to Johansen for pointing out that the topology induced by D on subsets of E is precisely the topology induced when E is embedded canonically in its one-point compactification, the sphere of dimension dim E, and the Hausdorff metric is used to compare subsets of this sphere. Such an observation proves quickly the following Proposition which appears first in [Ru] (see also [R1] and [Ra]).

Proposition 3.2 If u ∈ NS, then the sets {P_{v} | v ∈ NS} and {P_{v} | v ∈ u + E} are precompact with respect to D. □

Definition 3.3 Define

MP = \overline{\{P_{v} | v ∈ NS\}}

and

MP_{u} = \overline{\{P_{v} | v ∈ u + E\}},

the completions of the above sets with respect to D. The symbol M is used throughout this paper to indicate a construction such as this: a “Mapping Torus” or continuous hull.

Remark 3.4 The term “continuous hull” (of P_{u} with respect to D) simply refers to the fact that MP_{u} is the D-closure of the orbit of P_{u} under the continuous group E. A similar construction starts not with a point set or a tiling but with an operator on a Hilbert space [B2], this is where the name came from. See [BZH] for details and a comparison.

Note that Δ = 0 if and only if MP = MP_{u} for all u ∈ NS, which happens if and only if MP = MP_{u} for some u ∈ NS.

Also P_{v} forms a Delone set (see [S2]), so we deduce that, for w ∈ E and ||w|| small enough, D(P_{v}, P_{v+w}) = ||w||/(1 + ||w||).

Proposition 3.5 Suppose that w ∈ E, then the map P_{v} ↦ P_{v+w}, defined for v ∈ NS, may be extended to a homeomorphism of MP, and the family of homeomorphisms defined by taking all choices of w ∈ E defines a group action of ℝ^{d} ≅ E on NS.

Also for each u ∈ NS, MP_{u} is invariant under this action of E and E acts minimally on MP_{u}. □

The dynamical system MP_{u} with the action by E ≅ ℝ^{d} is the dynamical system associated with the point pattern P_{u}, analogous to that constructed by Rudolf [Ru] for tilings. We modify this to an action by E on a non-compact cover of MP_{u} as follows.

Definition 3.6 For v, v′ ∈ ℝ^{N}, write \overline{D}(v, v′) = D(P_{v}, P_{v′})+||v − v′||; this is clearly a metric. Let Π be the completion of NS with respect to this metric.

The following lemma starts the basic topological description of these spaces.

Lemma 3.7 a/ The canonical injection NS ⟶ ℝ^{N} extends to a continuous surjection μ: Π ⟶ ℝ^{N}. Moreover, if v ∈ NS, then μ^{−1}(v) is a single point.

b/ The map v ↦ P_{v}, v ∈ NS, extends to a continuous Eequivariant surjection, η: Π ⟶ MP, which is an open map.

c/ The action by translation by elements of E on NS extends to a continuous action of ℝ^{d} ≅ E on Π.

d/ Similarly the translation by elements of ℤ^{N} is \overline{D}-isometric and extends to a continuous action of ℤ^{N} on Π. This action commutes with the action of E found in part c/.

e/ If a ∈ MP and b ∈ ℝ^{N}, then |η^{−1}(a) ∩ μ^{−1}(b)| ≤ 1.

Proof a/ The only non-elementary step of this part is the latter sentence.

We must show that if v ∈ NS then for all ϵ > 0 there is a δ > 0 such that ||w − v|| < δ and w ∈ NS implies that D(P_{w}, P_{v}) < ϵ. However, we know that if B is a ball in ℝ^{N} of radius much bigger than 1/(2ϵ), then (ℤ^{N} + v) ∩ B is of strictly positive distance, say at least 2δ with δ > 0 chosen < ϵ/2, from ∂Σ. Therefore, whenever π(v − w) = 0 and ||v − w|| < δ, we have P_{v} ∩ B = P_{w} ∩ B and hence D(P_{v}, P_{w}) < δ . On the other hand, if π(v − w) ≠ 0 but ||v − w|| < δ then we may replace w by w′ = w + π(v − w), a displacement by less than δ. By the remark (3.4), we deduce that D(P_{w}, P_{w′}) < δ and so we have D(P_{w}, P_{v}) < 2δ < ϵ in general, as required.

b/ The extension to η, and the equivariance and surjectivity, are immediate. The open map condition is quickly confirmed using remark (3.4).

c/ follows from the uniform action of E noted in Remark 3.4. d/ follows similarly where uniform continuity is immediate from the isometry.

Note that e/ is a direct consequence of the definition of the metric \overline{D}. □

Definition 3.8 For u ∈ NS, let Π_{u} be the completion of E + ℤ^{N} + u with respect to the \overline{D} metric.

Lemma 3.9 For u ∈ NS, Π_{u} is a closed E + ℤ^{N}-invariant subspace of Π. If x ∈ Π_{u}, then (E + ℤ^{N})x, the orbit of x under the E and ℤ^{N} actions, is dense in Π_{u}. Consequently

a/ The injection, E + ℤ^{N} + u ⟶ ℝ^{N} extends to a continuous map, equal to the restriction of μ to Π_{u}, μ_{u}: Π_{u} ⟶ ℝ^{N}, whose image is Q + u.

b/ By extending the action by translation by elements of E + ℤ^{N} on E +ℤ^{N} +u, E +ℤ^{N} acts continuously and minimally on Π_{u}. This is the restriction of the action of Lemma 3.7 c/ and d/.

c/ The map v ↦ P_{v}, v ∈ E + ℤ^{N} + u, extends to an open continuous E-equivariant surjection, η_{u}: Π_{u} ⟶ MP_{u}, which is the restriction of η.

d/ If x ∈ Π_{u} and v ∈ E + ℤ^{N} acts on Π_{u} fixing x, then in fact v = 0.

Proof The first sentence is immediate since, by definition, Π_{u} is the closure of an E + ℤ^{N} orbit in Π.

Suppose that x ∈ Π_{u} and that y ∈ E + ℤ^{N} + u which we consider as a subset of Π_{u}. Then there are x_{n} ∈ E + ℤ^{N} + u such that x_{n} → x in the \overline{D} metric. Write β_{n}: Π_{u} ⟶ Π_{u} for the translation action by −x_{n} and write α for the translation action by y. Then we have μ(β_{n}(x)) → 0 and so μ(αβ_{n}(x)) = y + μ(β_{n}(x)) → y.

But, since μ is 1-1 at y ∈ NS by Lemma 3.7 a/, we deduce that \overline{D}(αβ_{n}(x), y) → 0 and so y is in the closure of the E + ℤ^{N} orbit of x. However the orbit of y is dense and so we have the density of the x orbit as well.

The lettered parts follow quickly from this. □

By the results of parts b/ and c/ of Lemma 3.9, we may drop the suffix u from the maps μ_{u} and η_{u} without confusion, and this is what we do unless it is important to note the domain explicitly.

The aim of the next few sections is to fill in the fourth corner of the commuting square

Π_{u}  ⟶^{η} MP_{u}
↓^{μ}  ↓^{?}
(Q + u) ⟶^{?} ?

in a way which illuminates the underlying structure.

4 Tilings and Point Patterns

We now connect the original construction of projection tilings due to Katz and Duneau [KD] with the point patterns that we have been considering until now. We refer to [OKD] and [Se] for precise descriptions of the construction; we extract the points essential for our argument below.

We note two developments of the D metric (3.1) which will be used ahead. The first development is also E.A.Robinson’s original application of D [R1].

Definition 4.1 We suppose that we have a finite set of pointed compact subsets of E which we call the units, and we suppose that we have a uniformly locally finite subset of E, a point pattern. A pattern, T , in E (with these units and underlying point pattern) is an arrangement of translated copies of the units in E, the distinguished point of each copy placed over a point of the point pattern, no point of the point pattern uncovered and no point of the point pattern covered twice. Sometimes, an underlying point pattern is not mentioned explicitly.

For example we could take a tiling of E and let the pattern consist of the boundaries of the tiles with superimposed decorations, i.e. small compact sets, in their interior giving further asymmetries or other distinguishing features. Or we could take a point pattern, perhaps replacing each point with one of a finite number of decorations. See [GS] for a thorough discussion of this process in general.

By taking the union of all the units of the pattern, we obtain a locally compact subset P (T ) of E which can be shifted by elements of E, P (T ) ↦ P (T )+v, and these various subsets of E can be compared using D literally as defined (3.1) (the addition of further decorations can also solve the problem of confusing overlap of adjacent units of the pattern, a complication which we ignore therefore without loss of generality). Under natural conditions (see [Ru] [S2]), which are always satisfied in our examples, the space {P (T ) + v | v ∈ E} is precompact with respect to the D metric and its closure, the continuous hull of T written MT here, supports a natural continuous E action. The pattern dynamical system of T is this dynamical system (MT , E).

Definition 4.2 The second development adapts D to compare subsets of Σ. Recall the notation B(r) for the closed Euclidean r-ball in E (3.1). Let C(r) = π^{−1}(B(r)) ∩ Σ and let dC(r) = π^{−1}(∂B(r)) ∩ Σ.

Given a subset, A, of Σ define A[r] = (A ∩ C(r)) ∪ dC(r). Let d′_{r} be the Hausdorff metric defined among closed subsets of C(r) and define a metric on subsets of Σ by

D′(A, A′) = inf{1/(r + 1) | d′_{r}(A[r], A′[r]) < 1/r}

Let \overline{D}′(v, w) = D′(P̃_{v}, P̃_{w}) + ||v − w||, where we recall that P̃_{v} = Σ ∩ (ℤ^{N} + v).

Let MP̃_{u} be the D′-closure of the space {P̃_{v} | v ∈ E +u}, and let Π̃_{u} be the \overline{D}′ completion of NS ∩ (Q + u). Let MP̃ be the D′ closure of the space {P̃_{v} | v ∈ NS}.

The analogues of Proposition 3.5 and Lemma 3.9 with respect to P̃, MP̃, Π̃_{u}, MP̃_{u} and Q + u, continue to hold and so we define maps μ̃: Π̃_{u} ⟶ Q + u and η̃: Π̃_{u} ⟶ MP̃_{u}.

We use the projection π to compare the strip pattern with the projection pattern. It will turn out that π is a homeomorphism between Π_{u} and Π̃_{u}, but that the definition of Π̃_{u} will be more convenient than that of Π_{u}. Using π we may work with either space.

Theorem 4.3 There are E-equivariant maps π_{∗} induced by the projection π which complete the commuting square

Π̃_{u}  ⟶^{π_{∗}} Π_{u}
↓_{η}  ↓^{η}
MP̃_{u} ⟶^{π_{∗}} MP_{u}

Furthermore we have the following commuting square

Π̃_{u}  ⟶^{π_{∗}} Π_{u}
↓_{μ}  ↓^{μ}
Q + u = Q + u

in which all the labelled maps are 1-1 on NS. □

Consider the example of the canonical projection tiling, T_{u} (2.3). If we know P̃_{u} then we have all the information needed to reconstruct T_{u} by its definition. Conversely, the usual assumption that the projected faces are non-degenerate (see [Le] (3.1)) allows us to distinguish the orientation of the lattice face (in ℤ^{N}) from which a given tile came. Piecing together all the faces defined this way obtains P̃_{u}. So the canonical projection tiling is conjugate (in the sense defined ahead in 4.5) to P̃_{u}.

On the other hand, the well-known Voronoi or Dirichlet tiling [GS] obtained from a point pattern in E is a tiling conjugate to the original point pattern provided we decorate each tile with the point which generates it.

With these two examples of tiling in mind, we consider the pattern P̃_{u} to represent the most elaborate tiling or pattern that can be produced by the projection method, without imposing further decorations not directly connected with the geometry of the construction, and at the other extreme, the point pattern, P_{u}, represents the least decorated tiling or pattern which can be produced by the projection method.

Definition 4.4 For a given E and K as in (2.1), we include in the class of projection method patterns all those patterns, T , of ℝ^{d} such that there is a u ∈ NS and two E-equivariant surjections

MP̃_{u} ⟶ MT ⟶ MP_{u}

whose composition is π_{∗}.

We call (E, K, u) the data of the projection method and by presenting these data we require tacitly that K has the properties of Definition 2.1, that u ∈ NS and that (Q + u) ∩ IntK ≠ ∅ (2.6).

Thus the tilings of [OKD] and the Voronoi tilings discussed above are examples from this class when K = π^{⊥}([0, 1]^{N}). In order to compare these two constructions, or to consider projection method patterns in the general sense of (4.4), we aim to describe π_{∗}: MP̃_{u} ⟶ MP_{u}.

First we adopt the following definitions which possibly duplicate notions already existing in the literature.

Definitions 4.5 Adapting a definition of Le [Le], we say that two patterns, T , T′, in E are topologically conjugate if there is an Eequivariant homeomorphism, MT ↔ MT′.

Say that the two patterns, T , T′, are pointed conjugate if there is an E-equivariant homeomorphism, MT ↔ MT′, which maps T to T′.

A pattern T′ is a finite decoration of a pattern T if there are real numbers r and s so that the following happens: i/ T may be constructed from T′ by a transformation which alters the unit of T at a point v ∈ ℝ^{d} according only to how T appears in the ball v+B(r): and ii/, conversely, if, for any choice of w ∈ ℝ^{d} we know what T′ looks like in the ball w + B(s), then we can construct the remainder of T′ from T by a transformation (depending perhaps on the appearance of T′ in the ball w + B(r)) which alters the unit of T′ at a point v ∈ ℝ^{d} according only to how T′ appears in the ball v + B(r).

Finally, we say that a pattern is a Meyer pattern if the underlying point pattern is a Meyer set, that is a set M for which we can find R and r so that M − M intersects every R-ball in at least one point and intersects every r ball in at most one point.

Remark 4.6 Note that all the patterns we consider in this paper are (pointed conjugate to) Meyer patterns. This is the starting point for Schlottmann’s analysis of the projection method [Sch].

To tie these definitions in to the exisiting literature, we note that topological conjugacy is strictly weaker than local isomorphism (as in [Le] for example) and strictly stronger than equal quasicrystal type [R1]. Pointed conjugacy is strictly stronger than mutual local derivability [BSJ] and topological equivalence [K3], but has no necessary relation with local isomorphism and quasicrystal type. Finite decoration is strictly weaker than local derivability [BSJ].

However, we have the following, an immediate application of the definitions to the fact that an n-to-1 factor map (see 2.13) is an open map [F].

Lemma 4.7 Suppose we have two Meyer patterns, T , T′, in E and a continuous E-equivariant surjection MT′ ⟶ MT which is n-to-1, sending T′ to T . Then T′ is a finite decoration of T . □

5 Comparing Π_{u} and Π̃_{u}

We start by examining π_{∗}: Π̃_{u} ⟶ Π_{u} from (4.3) and seek conditions under which it is a homeomorphism. As the section proceeds we shall find that the conditions can be whittled away to the minimum possible. Recall the space V, one of the orthocomponents of the decomposition of Q in Corollary 2.11. It is the connected component of \overline{π^{⊥}(ℤ^{N})} containing 0.

Lemma 5.1 Suppose that u ∈ NS and that, for all v ∈ Q+u such that v ∈ ∂((V +v)∩IntK) (the boundary taken in V +v), we have (Δ+v)∩ K = {v}; then π_{∗}: Π̃_{u} ⟶ Π_{u} is an E-equivariant homeomorphism.

Proof We ask under what circumstances could we find x ∈ Π_{u} with two preimages under π_{∗} in Π̃_{u}? We would need two sequences v_{n}, w_{n} ∈ (Q + u) ∩ NS both converging to x in the \overline{D} metric such that P̃_{v_{n}} and P̃_{w_{n}} have different \overline{D}′ limits, say A and B respectively. From this we see that AΔB ⊂ ∂Σ (symmetric difference) and yet π(A) = π(B).

Let p ∈ π(AΔB) and consider the set (AΔB)∩π^{−1}(p). As noted above, this set is a subset of the boundary of Σ ∩ π^{−1}(p) ≡ K and each pair of elements is separated by some element of Δ.

Suppose that a ∈ (A \ B) ∩ π^{−1}(p). By construction, there are a_{n} ∈ (Q+u)∩NS ∩P̃_{v_{n}} converging to a implying that a ∈ ∂((Q+u)∩ IntK). But by hypothesis, we deduce B∩π^{−1}(p) = ∅ - a contradiction to the fact that p ∈ π(A) = π(B).

A symmetric argument produces a contradiction from b ∈ (B \ A) ∩ π^{−1}(p). □

Note that if Δ = 0 or, more generally, if K ∩ (K + δ) = ∅ whenever δ ∈ Δ, δ ≠ 0, then the hypothesis of the Lemma is satisfied trivially.

Corollary 5.2 If Δ = 0, then π_{∗}: Π̃_{u} ⟶ Π_{u} is an E-equivariant homeomorphism. □

In special cases the hypothesis is satisfied less trivially. We give a slightly more special condition here.

Lemma 5.3 Suppose that J is the closure of a fundamental domain for Δ in E^{⊥}, and that J = \overline{IntJ} in E^{⊥}. If K is contained in some translate of J, then π_{∗}: Π̃_{u} ⟶ Π_{u} is a homeomorphism. In particular, if K = π^{⊥}([0, 1]^{N}), then π_{∗}: Π̃_{u} ⟶ Π_{u} is a homeomorphism.

Proof For the first part, suppose that a, b ∈ K and 0 ≠ a−b = δ ∈ Δ, then, by construction, a and b sit one in each of two hyperplanes orthogonal to δ between which K lies. Note that then these hyperplanes are therefore both parallel to V and each intersects K only in a subset of ∂K. Therefore, a, b ∈ ∂K and further, since V +a and V +b are contained one in each of the hyperplanes, we have a ∉ ∂((V + a) ∩ IntK) (boundary in V + a) and b ∉ ∂((V + b) ∩ IntK) (similis). Therefore the conditions of Lemma 5.1 are fulfilled vacuously.

In the second part, suppose the domain K = π^{⊥}([0, 1]^{N}) and that δ = (δ_{1}, δ_{2}, ..., δ_{N}) ∈ Δ, δ ≠ 0 (the case Δ = 0 is easy). Consider the set I = {⟨δ, t⟩ | t ∈ K}, where ⟨., .⟩ is the inner product on ℝ^{N}. This is a closed interval. Also, since δ is fixed by the orthonormal projection π^{⊥}, I = {⟨δ, s⟩ | s ∈ [0, 1]^{N}}, from which we deduce that the length of I is ∑_{j} |δ_{j}|. But since |δ_{j}| < 1 implies that δ_{j} = 0, we have ⟨δ, δ⟩ = ∑ |δ_{j}|^{2} ≥ ∑ |δ_{j}| and so K can be fitted between two hyperplanes orthogonal to δ and separated by δ.

Therefore K is contained in a translate of

⋂_{δ∈Δ,δ≠0}{v ∈ E^{⊥} | |⟨v, δ⟩| ≤ (1/2)⟨δ, δ⟩ },

which in turn is contained in the closure of a fundamental domain for Δ. So we have confirmed the conditions of the first part. □

Remark 5.4 Using Lemma 2.5, the condition of Proposition 5.3 is equivalent to the following condition: ((IntK) − (IntK)) ∩ Δ = {0}, where we write A − A = {a − b | a, b ∈ A} for the arithmetic (self- )difference of A, a subset of an abelian group. Compare with 8.2 ahead.

All of these results say that if K is small enough relative to Δ then π_{∗} is a homeomorphism. In fact, we can dispense with all such conditions, and the following construction gives a procedure to reduce the size of a general acceptance domain appropriately.

Theorem 5.5 Suppose that E and K are as in definition 2.1, and that u ∈ NS. Then π_{∗} : Π̃_{u} ⟶ Π_{u} is an E-equivariant homeomorphism.

Proof Suppose that E, K and u are chosen as required and that Δ ≠ 0. The case Δ = 0 is covered by Corollary 5.2.

Suppose that J is the closure of a fundamental domain for Δ in E^{⊥}, such that J = \overline{IntJ} in E^{⊥} and suppose, as we always can by shifting J if necessary, that ∂J ∩ (Q + u) = ∅.

Let K′ = (K + Δ) ∩ J. Then K′ is a subset of E^{⊥} which obeys the conditions required in the original definition of (2.1). Also the placement of J ensures that the points in Q + u, in particular u itself, which are non-singular with respect to K are also non-singular with respect to K′.

Moreover, if we define Σ′ = K′ +E, then, by construction, π(Σ′ ∩ (v + ℤ^{N})) = π(Σ ∩ (v + ℤ^{N})) = P_{u} for all v ∈ ℝ^{N}. Therefore, working with Σ′ instead of Σ, we can retrieve the projection point pattern, P_{u}. So, by Lemma 5.3 and the fact that K′ ⊂ J, we see that π_{∗} is a homeomorphism between the spaces Π_{u} and Π̃_{u}(Σ′), the construction of (4.2) with respect to Σ′.

However, for any v ∈ E + ℤ^{N} + u, we have the equalities: Σ′ ∩ (v+ℤ^{N}) = Σ′ ∩((Σ∩(v+ℤ^{N}))+Δ) and Σ∩(v+ℤ^{N}) = Σ∩((Σ′ ∩(v+ ℤ^{N}))+Δ). Therefore the set Σ′∩(v+ℤ^{N}) can be constructed from Σ∩ (v + ℤ^{N}) = P̃_{v} and vice versa. Moreover, this correspondence defines a \overline{D} metric isometry, between Π̃_{u}(Σ′) and Π̃_{u}, which intertwines π_{∗}. Completing the correspondence gives an isometry between Π̃_{u}(Σ′) and Π̃_{u} which intertwines π_{∗}, as required. □

Remark 5.6 We note a second process of reduction without loss of generality. Until now we have assumed nothing about the rational position of E, but it is convenient to assume and is often required in the literature that E ∩ ℤ^{N} = 0: the irrational case.

If we do not assume this then we can always reduce to the irrational case by quotienting out the rational directions. A simple argument allows us to find in the most general case of projection method pattern, P say, an underlying irrational projection method pattern, P′, with, MP = MP′ × 𝕋^{k} for some value of k; and this torus factor splits naturally with respect to the various constructions and group actions we find later. We leave the details to the reader.

6 Calculating MP̃_{u} and MP_{u}

We now describe MP̃_{u} and MP_{u} as quotients of Π̃_{u} and Π_{u} respectively. Although the last section established an equivalence between Π̃_{u} and Π_{u}, we find it useful to distinguish the two constructions.

First we examine MP̃_{u} and prove a generalisation of (3.8) of [Le].

Proposition 6.1 Suppose that u ∈ NS, then there is an isometric action of ℤ^{N} on Π̃_{u}, which factors by μ̃ to the translation action by ℤ^{N} on Q + u, and MP̃_{u} = Π̃_{u}/ℤ^{N}. Thus we obtain a commutative square of E equivariant maps

Π̃_{u}  \underrightarrow{_{η}}  MP̃_{u}
↓_{μ}  ↓_{μ}
Q + u ⟶ (Q + u)/ℤ^{N}.

The left vertical map is 1-1 precisely at the points in NS ∩ (Q + u). The right vertical map is 1-1 precisely on the same set, modulo the action of ℤ^{N}.

Proof The action of ℤ^{N} on Π̃_{u}, as an extension of the action on Q+u by translation, is easy to define since the maps are \overline{D}′-isometries.

If v, w ∈ NS then it is clear that P̃_{v} = P̃_{w} if and only if v − w ∈ ℤ^{N}. Moreover, there is δ > 0 so that ||v − w|| < δ implies that D′(P̃_{v}, P̃_{w}) ≥ ||v − w||/2.

From this we see that, if P̃_{v} = P̃_{w} and P̃_{v′} = P̃_{w′} and ||v − v′|| < δ/2 and ||w − w′|| < δ/2, then v − w = v′ − w′. The uniformity of δ irrespective of the choice of v, w, v′ and w′ shows that the statement η̃(v) = η̃(w) implies μ̃(v) − μ̃(w) ∈ ℤ^{N}, which is true for v, w ∈ NS ∩ (Q + u), is in fact true for all pairs in Π̃_{u}, the \overline{D}′ closure.

To show the 1-1 properties for the map on the left, suppose that v ∈ Q + u and that p ∈ ∂Σ ∩ (ℤ^{N} + v), i.e. v ∉ NS. Then since K is the closure of its interior and since NS is dense in ℝ^{N} (Lemma 2.2), there are two sequences v_{n}, v′_{n} ∈ NS both converging to v in Euclidean topology and such that p + (v_{n} − v) ∈ Σ and p + (v′_{n} − v) ∉ Σ. This implies that any D′ limit point of P̃_{v_{n}} contains p and any D′ limit point of P̃_{v′_{n}} does not contain p. But both such limit points (which exist by compactness of MP̃) are in μ^{−1}(v) which is a set of at least two elements therefore.

The 1-1 property for the map on the right follows directly from this and the commuting diagram. □

The space (Q+u)/ℤ^{N} and its E action, which is being compared with MP̃_{u}, also has a simple description.

Lemma 6.2 With the data above, (Q+u)/ℤ^{N} is a coset of the closure of E mod ℤ^{N} in ℝ^{N}/ℤ^{N} ≡ 𝕋^{N}. Therefore (Q + u)/ℤ^{N} with its E action is isometrically conjugate to a minimal action of ℝ^{d} by rotation on a torus of dimension N − dim Δ.

Proof The space Q mod ℤ^{N} is equal to the closure of E mod ℤ^{N} and its translate by u mod ℤ^{N} is an isometry which is E equivariant. The action of E on its closure is isometric and transitive, hence minimal, and is by translations. E is a connected subgroup of 𝕋^{N} and so also is the closure of E, which is therefore equal to a torus of possibly smaller dimension. The codimension of this space agrees with the codimension of V + E (the continuous component of Q) in ℝ^{N} which, by Lemma 2.9 and Corollary 2.11, equals dim Δ as required. □

Now we turn to a description of MP_{u} which is similar in form to that of MP̃_{u}, but as to be shown in examples 8.7 and 8.8, need not be equal to MP̃_{u}.

Lemma 6.3 Suppose that u ∈ NS. If v, w ∈ NS ∩ (Q + u) and P_{v} = P_{w} then there are v^{∗} ∈ v + ℤ^{N} and w^{∗} ∈ w + ℤ^{N} such that v^{∗}, w^{∗} ∈ Σ and π(v^{∗}) = π(w^{∗}), and with this choice P̃_{v}+Δ−π^{⊥}(v^{∗}) = P̃_{w} + Δ − π^{⊥}(w^{∗}).

Proof Fix p_{o} ∈ P_{v} = P_{w} and let v^{∗} ∈ P̃_{v} be chosen so that π(v^{∗}) = p_{o} and similarly, let w^{∗} ∈ P̃_{w} be chosen so that π(w^{∗}) = p_{o}. Clearly v^{∗} and w^{∗} obey the conditions required. Also P̃_{w} − π^{⊥}(w^{∗}) and P̃_{v} − π^{⊥}(v^{∗}) are both contained in p_{o} + ℤ^{N} and project under π to the same set P_{v}. Thus the difference of two points, one in P̃_{w} − π^{⊥}(w^{∗}) and the other in P̃_{v} − π^{⊥}(v^{∗}), and each with the same image under π, is an element of Δ as required. □

Proposition 6.4 Suppose that x, y ∈ Π_{u} and that η(x) = η(y), then there is a v ∈ Q and a \overline{D} isometry ϕ: Π_{u} ⟶ Π_{u} so that ϕ(x) = ϕ(y) and the following diagram commutes

Π_{u}  ⟶^{ϕ} Π_{u}
↓^{μ}  ↓^{μ}
Q + u ⟶^{w↦w+v} Q + u

and η_{u}ϕ = η_{u} (here the restriction to Π_{u} is important to note, c.f. 3.9). In this case we deduce v + u ∈ NS.

Conversely, if we have such an isometry in such a diagram and if P_{u+v} = P_{u}, then v + u ∈ NS and η_{u}ϕ = η_{u} automatically.

Proof Suppose w ∈ E + ℤ^{N} and that α_{w}: Π_{u} ⟶ Π_{u} is the map completed from the map z ↦ z + w defined first for z ∈ NS ∩ (Q + u) (see Proposition 3.5). Then, since η(x) = η(y) and η is (E + ℤ^{N})- equivariant, we have η(α_{w}(x)) = η(α_{w}(y)) for all w ∈ E + ℤ^{N}. So, by definition, the map α_{w}(x) ↦ α_{w}(y) defined point-by-point for w ∈ E + ℤ^{N} is a \overline{D} isometry from the (E + ℤ^{N})-orbit of x onto the (E + ℤ^{N})-orbit of y (By Lemma 3.9 d/ the mapping is well-defined). These two orbits being dense (Lemma 3.9) in Π_{u}, this map extends as an isometry onto, ϕ: Π_{u} ⟶ Π_{u}.

Since μ is (E +ℤ^{N})-equivariant, we deduce the intertwining with translation by v = μ(y)−μ(x). Also since ηϕ = η on the E +ℤ^{N} orbit of x, the E-equivariance of η extends this equality over all of Π_{u}.

Conversely suppose we have an isometry which intertwines the translation by v on Q + u. Then for general topological reasons the cardinality of the μ preimage of a point in Q + u is preserved by translation by v and we deduce that NS ∩ (Q + u) is invariant under the translation by v. In particular u + v ∈ NS. The equation follows since it applies, by hypothesis and Lemma 6.3, at u and therefore, by equivariance, at all points in E + ℤ^{N} + u, a dense subset. □

Definition 6.5 For u ∈ NS, let R_{u} = {v ∈ Q | v + u ∈ NS, P_{u+v} = P_{u}}.

Corollary 6.6 Suppose that u ∈ NS and w ∈ NS ∩ (Q + u), then R_{w} = R_{u}. Therefore, if v ∈ R_{u}, then v + w ∈ NS ∩ (Q + u) for all w ∈ NS ∩ (Q + u).

Proof By Lemma 3.9, we know that Π_{w} ⟶^{η_{w}} Q + w equals Π_{u} ⟶^{η_{u}} Q + u and so any isometry of Π_{u} which factors by η through to a translation by v also does the same for Π_{w}. Proposition 6.4 completes the equivalence.

The second sentence follows directly from the definition of R_{w}. □

Remarks 6.7 It would be natural to hope that the condition u + v ∈ NS could be removed from the definition of R_{u}. We have been unable to do this in general. But since NS is a dense G_{δ} set (2.2) and, anticipating Theorem 7.1, R_{u} is countable, we see that for a dense G_{δ} set of u ∈ NS (generically) we can indeed equate R_{u} = {v ∈ Q : P_{u+v} = P_{u}}.

This is bourne out in Corollary 6.6 where we see that R_{u} is defined independently of the choice of u generically, and R_{u} can be thought of as an invariant of Π_{u}. This result also shows that R_{u} is a subset of the translations of ℝ^{N} which leave NS ∩ (Q + u) invariant.

Note that, since μ is 1-1 only on NS, R_{u} could as well have been defined as {v ∈ Q | ημ^{−1}(u + v) = {P_{u}} }.

It is clear that ℤ^{N} ⊂ R_{u}.

Theorem 6.8 If u ∈ NS, then R_{u} is a closed subgroup of Q. Also R_{u} acts by ϕ isometrically on Π_{u} and defines a homeomorphism Π_{u}/R_{u} ≡ MP_{u}. Moreover the R_{u} action commutes with the E-action, so the homeomorphism is E-equivariant.

Proof The main point to observe is that R_{u} consists precisely of those elements v such that there is an isometry ϕ_{v} as in Proposition 6.4 with η_{u}ϕ_{v} = η_{u}. Since the inverse of such an isometry is another such, and the composition of two such isometries produces a third, we deduce the group property for R_{u} immediately. The isometric action is given to us and Proposition 6.4 shows directly that Π_{u}/R_{u} ≡ MP_{u}.

Closure of R_{u} is more involved. Suppose that v_{n} ∈ R_{u} and that v_{n} → v in the Euclidean topology. Then ϕ_{v_{n}} is uniformly Cauchy and so converges uniformly to a bijective isometry, ψ, of Π_{u} which intertwines the translation by v on Q + u. Therefore, if μ^{−1}(u + v) has at least two elements, then so also does ψ^{−1}μ^{−1}(u + v), but this set is contained in μ^{−1}(u), a contradiction since μ^{−1}(u) is a singleton. Therefore u + v ∈ NS and μ^{−1}(u + v) = ψμ^{−1}(u) = lim ϕ_{v_{n}}μ^{−1}(u) = lim μ^{−1}(u + v_{n}) = μ^{−1}(u). Thus v ∈ R_{u} and so R_{u} is closed.

The commutativity with the E action on Π_{u} is immediate from the corresponding commutativity on Q + u. □

7 Comparing MP_{u} with MP̃_{u}

In Section 4 we defined projection method patterns as those whose dynamical system sits intermediate to MP̃_{u} and MP_{u}. We discover in this section how closely these two spaces lie and circumstances under which they are equal.

To compare MP_{u} with MP̃_{u} we start with the fact that Π̃_{u} = Π_{u} (5.5). By Proposition 6.1 and Theorem 6.8, therefore, the problem becomes the comparison of R_{u} with ℤ^{N}. Perhaps surprisingly, under general conditions we find that R_{u} is not much larger than ℤ^{N} and under normal conditions the two groups are equal.

Theorem 7.1 For all u ∈ NS, ℤ^{N} ⊂ R_{u} as a finite index subgroup. In fact, with the notation of (2.10), R_{u} ⊂ ℤ^{N} + Δ̃.

Proof Suppose that v ∈ R_{u}. Then in particular, by (6.7), P_{v+u} = P_{u}. Therefore there is an a ∈ ℤ^{N} such that π(v + u + a) = π(u) and so by translating if necessary, we may assume without loss of generality that v ∈ E^{⊥}; and this defines v uniquely mod Δ.

With this assumption we deduce from Lemma 6.3 that P̃_{u+v} + Δ = P̃_{u} + Δ + v In particular, π^{⊥}(P̃_{u+v}) + Δ = π^{⊥}(P̃_{u}) + Δ + v.

Now each of π^{⊥}(P̃_{u+v}) and π^{⊥}(P̃_{u}) is contained in K a compact set. Suppose that α ∈ Δ^{⊥} and that ⟨v, α⟩ ≠ 0, then there is t ∈ ℤ such that |⟨tv, α⟩| > 2||α|| diamK. However, since tv ∈ R_{u} by Theorem 6.3, we have π^{⊥}(P̃_{tv+u}) + Δ = π^{⊥}(P̃_{u}) + Δ + tv. Applying the function ⟨., α⟩ to both sets produces a contradiction by construction. Thus we have v ∈ U, the space generated by Δ (see 2.9). But if v ∈ R_{u} then v ∈ Q by definition, and so we have v ∈ Δ̃, a group which, by Corollary 2.11, contains Δ with finite index. □

We note, for use in section 10, that therefore R_{u} is free abelian on N generators.

Corollary 7.2 If Δ = 0, then R_{u} = ℤ^{N} and π_{∗}: MP̃_{u} ↔ MP_{u}. □

The following combines Propositions 6.1 and 6.8 and fits the present circumstances to the conditions of Lemma 2.14. Recall the definitions of n-to-1 extension (2.13) and finite isometric extension (2.15).

Proposition 7.3 Suppose that u ∈ NS. The map π_{∗}: MP̃_{u} ⟶ MP_{u} is p-to-1 where p is the index of ℤ^{N} in R_{u}. The group R_{u} acts isometrically on MP̃_{u}, commuting with the E action, preserving π_{∗} fibres and acting transitively on each fibre. This action is mapped almost 1-1 by μ̃ to an action by R_{u} on (Q + u)/ℤ^{N} by rotation, and so we complete a commuting square

\begin{matrix}MP̃_{u} & ⟶^{π_{∗}} & MP_{u} \\ ↓_{μ} & & ↓^{μ} \\ (Q + u)/ℤ^{N} & ⟶^{mod R_{u}/ℤ^{N}} & (Q + u)/R_{u}. \end{matrix}  □

From this and the construction of Lemma 2.14 applied to the case G = E and H = R_{u}, we deduce the main theorem of the section.

Theorem 7.4 Suppose that, E, K and u ∈ NS are data for T , a projection method pattern. Then there is a group S_{T}, intermediate to ℤ^{N} < R_{u}, which fits into a commutative diagram of E equivariant maps

MP̃_{u}  ⟶  MT  ⟶  MP_{u}
↓_{μ}  ↓  ↓^{μ}
(Q + u)/ℤ^{N} ⟶^{mod S_{T} /ℤ^{N}} (Q + u)/S_{T} ⟶^{mod R_{u}/S_{T}} (Q + u)/R_{u}

where the top row maps are finite isometric extensions and the bottom row maps are group quotients.

Conversely, every choice of group S′ intermediate to ℤ^{N} < R_{u} admits a projection method pattern, T , fitting into the diagram above with S′ = S_{T}. □

With the considerations of section 4 (in particular using 4.7) we can count the projection method patterns up to topological conjugacy or pointed conjugacy in the following corollary.

Corollary 7.5 With fixed projection data and the conditions of Theorem 7.4, the set of topological conjugacy classes of projection method patterns is in bijection with the lattice of subgroups of R_{u}/ℤ^{N}. Moreover, each projection method pattern, T , is pointed conjugate to a finite decoration of P_{u}, and P̃_{u} is pointed conjugate to a finite decoration of T . □

We deduce the following result also found by Schlottmann [Sch], which is in turn a generalisation of the result of Robinson [R2] and the topological version of the result of Hof [H].

Corollary 7.6 With the conditions assumed in Theorem 7.4, the pattern dynamical system MT is an almost 1-1 extension (2.15) of a minimal ℝ^{d} action by rotation on a (N − dim Δ)-torus.

Proof It suffices to show that the central vertical arrow in the diagram of Theorem 7.4 is 1-1 at some point. But this is immediate since each of the end arrows is 1-1 at u say. □

8 Examples and Counter-examples

In this section we give sufficient conditions, similar to and stronger than 5.3, under which P_{u} or T is pointed conjugate to P̃_{u}, and show why these conjugacies are not true in general.

Definition 8.1 For data (E, K, u), define B_{u} = \overline{(Q + u) ∩ IntK} (Euclidean closure in E^{⊥}).

Proposition 8.2 Suppose that E, K and u ∈ NS are chosen so that E ∩ ℤ^{N} = 0 and Δ ∩ [(B_{u} − B_{u}) − (B_{u} − B_{u})] = {0}, then R_{u} = ℤ^{N}. In this case, therefore, P_{u} is pointed conjugate to P̃_{u}.

Proof This follows from the fact, deduced directly from the condition given, that if v ∈ NS ∩ (Q + u) and a, b ∈ P_{v}, then we can determine w − w′ whenever w, w′ ∈ P̃_{v} are such that a = π(w) and b = π(w′). Knowing the differences of elements of P̃_{v} forces the position of P̃_{v} in Σ by the density of π^{⊥}(P̃_{v}) in B_{u}. So we can reconstruct P̃_{v} uniquely from P_{v} and we have R_{u} = ℤ^{N}. □

Corollary 8.3 In the case that the acceptance domain is canonical, the condition that the points π(w) | w ∈ {−1, 0, 1}^{N} are all distinct is sufficient to show that R_{u} = ℤ^{N} for all u ∈ NS. In this case, therefore, P_{u} is pointed conjugate with P̃_{u}.

Proof The condition implies that Δ ∩ [(K − K) − (K − K)] = {0} and this gives the condition in the proposition since B_{u} ⊂ K. □

If we are interested merely in the equation between T and P̃_{u}, then the case that the acceptance domain is canonical also allows simple sufficient conditions weaker than 8.3, as we show below.

We observe first that the construction of [OKD] can be extended to admit non-generic parameters, provided that we are comfortable with “tiles” which, although they are convex polytopes, have no interior in E and are unions of faces of the tiles with interior. We retain these degenerate tiles as components of our “tiling”, i.e. as units of a pattern, giving essential information about the pattern dynamics. We call such patterns degenerate canonical projection tilings.

We write e_{j} with 1 ≤ j ≤ N for the canonical unit basis of ℤ^{N}.

Proposition 8.4 In the case of a canonical acceptance domain, the condition that no two points from {π(e_{j}) | 1 ≤ j ≤ N} are collinear is sufficient to show that T_{u}, the canonical (but possibly degenerate) tiling, is pointed conjugate to P̃_{u} for all u ∈ NS.

Proof We show that the conditions given imply that the shape of a tile (even in degenerate cases) determines from which face of the lattice cube it is projected. In fact we shall show that if I ⊂ {1, 2, ..., N} then knowing π(γ^{I}) and the cardinality of I determines I (we write γ^{I} = {∑_{i∈I} λ_{i}e_{i} | 0 ≤ λ_{i} ≤ 1, ∀ i ∈ I}).

Suppose that π(γ^{I}) = π(γ^{J}) and I, J ⊂ {1, 2, .., N} are of the same cardinality. It is possible always to distinguish an edge on the polyhedron π(γ^{I}) which is parallel to a vector π(e_{i}) for some i ∈ I; and i is determined from this edge by hypothesis. The same is true of this same edge with respect to J and so i ∈ J also.

Writing I′ = I \ {i} and J′ = J \ {i} we deduce that π(γ^{I′}) = π(γ^{I}) ∩ (π(γ^{I}) − π(e_{i})) = π(γ^{J}) ∩ (π(γ^{J}) − π(e_{i})) = π(γ^{J′}). Now we can apply induction on the cardinality of I, and deduce that I′ = J′ and so I = J. Induction starts at cardinality 1 by hypothesis.

Now, given this correspondence between shape of tile and its preimage under π, we reconstruct P̃_{u} from T_{u} much as we did in Proposition 8.2 above. To complete the argument we must check that no other element of MP̃_{u} maps onto T_{u} in MT_{u}. But if there were such an element, then the argument above shows that it cannot be of the form P̃_{u′} for u′ ∈ NS. Also, by Theorem 7.4, we deduce that the map MP̃_{u} ⟶ MT_{u} is p-to-1 with p ≥ 2, and so, using Lemma 2.2ii/, we find T_{v} with two preimages of the form P̃_{v} and P̃_{v′}. But this contradicts the principle of the previous sentence. □

The conditions of this proposition include all the non-degenerate cases of the canonical projection tiling usually treated in the literature (including the Penrose tiling), so from the equation MT_{u} = Π̃_{u}/ℤ^{N}, deduced from Proposition 8.4 as a consequence, we retrieve many of the results stated in section 3 of [Le].

Now we turn to conditions under which R_{u} differs from ℤ^{N}. We can extend the argument of 7.1 to give a geometric condition for elements of R_{u}, of considerable use in computing examples.

Lemma 8.5 Suppose that E ∩ ℤ^{N} = 0, u ∈ NS and v ∈ E^{⊥}. Then v ∈ R_{u} ∩ E^{⊥} if and only if v + u ∈ NS and v + (B_{u} + Δ) = B_{u} + Δ.

Proof Suppose that u, v + u ∈ NS. Then \overline{π^{⊥}(P̃_{u})} = \overline{(Q + u) ∩ IntK} and \overline{π^{⊥}(P̃_{u+v})} = \overline{(Q + u + v) ∩ IntK}.

If v ∈ R_{u} ∩ E^{⊥}, then, by Lemma 6.3, we have P̃_{u} + Δ + v = P̃_{u+v} + Δ. Also, since by 7.1, v is in Δ̃, we have Q + u = Q + u + v and π^{⊥}(P̃_{u})+Δ+v = π^{⊥}(P̃_{u+v})+Δ. Putting all these together gives the required equality v + (B_{u} + Δ) = B_{u} + Δ.

Conversely, if v + (B_{u} + Δ) = B_{u} + Δ, then, by the argument of 7.1, v ∈ Δ̃ and so, as above, Q + u = Q + u + v and π^{⊥}(P̃_{u}) + Δ + v = π^{⊥}(P̃_{u+v}) + Δ. So, if a ∈ P̃_{u}, then there is a b ∈ P̃_{u+v} such that π^{⊥}(a)−π^{⊥}(b) ∈ Δ−v. However, since π^{⊥} is 1-1 on ℤ^{N}, we can retrieve the set P̃_{u} as the inverse π^{⊥} image of π^{⊥}(P̃_{u}) ∩ IntK and similarly for P̃_{u+v}. This forces a − b ∈ Δ − v therefore, and so π(a) = π(b). Thus we see that P_{u} = P_{u+v}, as is required to show that v ∈ R_{u} □

Example 8.6 By Corollary 7.2 and the fact that Δ = 0, the octagonal tiling is pointed conjugate to both its projection and strip point patterns.

Also it is from Lemma 8.5 (and not 8.3) that we can deduce that R_{u} = ℤ^{5} for the (generalised) Penrose tiling for all choices of u ∈ NS, and so the generalised Penrose tiling is pointed conjugate to both its projection and strip point patterns.

Lemma 8.5 makes it quite easy to construct counter-examples to the possibility that R_{u} = ℤ^{N} always.

Example 8.7 We start with a choice of E for which Δ ≠ 0 and Δ̃ contains Δ properly. The E used to construct the Penrose tiling is such an example. As in the proof of Lemma 2.6, let U be the real span of Δ and V the orthocomplement of U in E^{⊥}. Choose a closed unit disc, I, in V and let J be the closure of a rectangular fundamental domain for Δ in U. Let K = I + J ⊂ E^{⊥} and note that K has all the propeties required of an acceptance domain in this paper and that, by Lemma 5.1, we have Π_{u} = Π̃_{u}.

With u ∉ V + Δ̃ (equivalently u ∈ NS), the rectilinear construction of K ensures that ((Q + u) ∩ IntK) + Δ is invariant under the translation by any element, v, of Δ̃. Also v+u ∈ NS since the boundary of ((Q + u) ∩ IntK) + Δ in Q + u is invariant under translations by Δ̃. So, with the characterisation of Lemma 8.5, this shows that R_{u} = Δ̃ + ℤ^{N} which is strictly larger than ℤ^{N}.

By varying the shape of J in this example, we can get R_{u}/ℤ^{N} equal to any subgroup of (Δ̃ + ℤ^{N})/ℤ^{N}, and we can make it a nonconstant function of u ∈ NS as well.

Example 8.8 Take N = 3 with unit vectors e_{1}, e_{2} and e_{3}. Let L be the plane orthonormal to e_{1} − e_{2} and let E be a line in L placed so that E ∩ ℤ^{3} = {0}. Then E^{⊥} is a plane which contains e_{1} − e_{2} and we have Δ = {n(e_{1} − e_{2}) | n ∈ ℤ}.

Write e_{1}^{⊥}, e_{2}^{⊥}, and e_{3}^{⊥} for the image under π^{⊥} of e_{1}, e_{2} and e_{3} respectively. Then e_{1}^{⊥} + e_{2}^{⊥} and e_{3}^{⊥} are collinear in E^{⊥} and they are both contained in V (the continuous subspace of V + Δ̃ = \overline{π^{⊥}(ℤ^{3})}). Δ is orthogonal to V and e_{1}^{⊥} − e_{2}^{⊥} = e_{1} − e_{2}. However Δ̃ = {n(e_{1} − e_{2})/2 | n ∈ ℤ} which contains Δ as an index 2 subgroup.

The set K = π^{⊥}([0, 1]^{3}) is a hexagon in E^{⊥} with a centre of symmetry. It is contained in the closed strip defined by two lines, V + a and V + b, where b − a = e_{1} − e_{2}, and it is in fact reflectively symmetric around an intermediate line, V +c where c−a = (e_{1}−e_{2})/2. The boundary of the hexagon on each of V +a and V +b is an interval congruent to e_{3}^{⊥} (i.e. a translate of {te_{3}^{⊥} | 0 ≤ t ≤ 1}). The four other sides are intervals congruent to e_{1}^{⊥} or e_{2}^{⊥}, two of each. The vertices of the hexagon are on V + a, V + b or V + c, two on each.

The point of all this is that there is a choice of non-singular u (in E^{⊥} without loss of generality) such that B_{u} = \overline{(Q + u) ∩ IntK} consists of the two intervals K ∩ (V + a′) and K ∩ (V + b′), where 2a′ = a + c and 2b′ = b + c (we can choose u ∈ (V + a′) ∩ NS for example), and these intervals are a translate by ±(e_{1} − e_{2})/2 of each other. Thus we deduce that B_{u} + Δ = B_{u} + Δ + v for all v ∈ Δ̃.

Upon confirming that v + u ∈ NS for all v ∈ Δ̃ as well, we use 8.5 to show that R_{u} = ℤ^{3} + Δ̃, which contains ℤ^{3} with index 2.

Remark 8.9 We note that Example 8.8 is degenerate and Proposition 8.4 shows why this must be the case. However, under any circumstances, there exist projection method tilings, in the sense of 4.4, pointed conjugate to P_{u} or to P̃_{u}. The point here is that these tilings will not necessarily be constructed by the special method of Katz and Duneau.

Also, leaving the details to the reader, we mention that Example 8.8 and its analogues in higher dimensions are the only counterexamples to the assertion R_{u} = ℤ^{N} when the acceptance domain is canonical and Δ is singly generated (and here we find always that R_{u}/ℤ^{N} is at most a cyclic group of 2 elements). When Δ is higher dimensional we have no concise description of the exceptions allowed.

9 The topology of the continuous hull

Sections 6 and 7 tell us that the continuous hulls MP_{u} and MP̃_{u} are quotients of the same space Π_{u} ≡ Π̃_{u}. One advantage of this equality is that the topology of Π̃_{u} is more easily described than Π_{u} a priori.

Definition 9.1 Let F be a plane complementary, but not necessarily orthonormal, to E and let π′ be the skew projection (idempotent map) onto F parallel to E.

Let K′ = π′(K).

Set F_{u}^{o} = NS ∩(Q+u)∩F and let F_{u} be the \overline{D}′-closure (Def. 4.2) of F_{u}^{o} in Π̃_{u}.

Note that, since R_{u} ⊂ Q, F_{u}^{o} is invariant under translation by π′(r), r ∈ R_{u} and by extension F_{u} supports a continuous R_{u} action. R_{u} acts freely on F_{u} when E ∩ℤ^{N} = 0 (i.e. with x ∈ F_{u}, the equation gx = x implies g = 0).

Similarly, R_{u} acts on E by translation by π(r), r ∈ R_{u}.

Lemma 9.2 With the data above, F_{u} = μ^{−1}(F ∩ (Q + u)) (c.f. (7.3)) and there is a natural equivalence Π̃_{u} ≡ F_{u} × E and a surjection ν: F_{u} ⟶ ((Q + u) ∩ F) which fits into the following commutative square

Π̃_{u} ⟷  F_{u} × E
↓_{μ}  ↓^{ν×id}
Q + u ⟷ ((Q + u) ∩ F) × E.

Moreover these maps are E-equivariant where we require that E acts trivially on F_{u}. The set ν^{−1}(v) is a singleton whenever v ∈ NS ∩ F ∩ (Q + u).

The canonical action of R_{u} on Π̃_{u} is represented in this equivalence as the direct sum (i.e. diagonal) of the action of R_{u} on F_{u} and E described in (9.1).

Proof This follows quickly from the observation that there is a natural, \overline{D}′-uniformly continuous and E equivariant equivalence NS ∩ (Q + u) = F_{u}^{o} + E ≡ F_{u}^{o} × E, which can be completed. □

Definitions 9.3 Let A_{u} be the algebra of subsets (i.e. closed under finite union, finite intersection and symmetric difference) of F_{u}^{o} generated by the sets (NS ∩ (Q + u) ∩ K′) + π′(v) as v runs over ℤ^{N}. It is clear that this algebra is countable and invariant under the action of R_{u}.

Write C^{∗}(A_{u}) for the smallest C^{∗} algebra which contains the indicator functions of the elements of A_{u}.

Let ℤA_{u} be the ring (pointwise addition and multiplication) generated by this same collection of indicator functions.

Let CF_{u} be the group of continuous integer valued functions compactly supported on F_{u}.

These three algebraic objects support a canonical R_{u} action induced by the action of R_{u} on F_{u} described in (8.1) and so we define three ℤ[R_{u}] modules. As ℤ^{N} sits inside R_{u}, this action can be restricted to a canonical subaction by ℤ^{N} thereby defining three ℤ[ℤ^{N}] modules.

Let B_{u} = \{\overline{A} | A ∈ A_{u}\} where the bar refers to \overline{D}′ closure in F_{u}.

And finally, we give the main theorem of this section which is of utmost importance for the remaining chapters.

Theorem 9.4 With data (E, K, u) and Π̃_{u} = Π_{u},

a/ The collection B_{u} is a base of clopen neighbourhoods which generates the topology of F_{u}.

b/ We have the ∗-isomorphisms of C^{∗}-algebras C_{o}(F_{u}) ≅ C^{∗}(A_{u}) and C_{o}(Π̃_{u}) ≅ C^{∗}(A_{u}) ⊗ C_{o}(E) which respect the maps defined in Lemma 8.2.

c/ CF_{u} ≅ ℤA_{u} as a ℤ[R_{u}] module (and by pull-back as a ℤ[ℤ^{N}] module).

d/ F_{u} is locally a Cantor Set.

First we have a lemma also of independent interest in the next section.

Definition 9.5 Write \overline{K} for the \overline{D}′-closure of the set K′∩NS∩(Q+u) (recall K′ from 9.1).

Lemma 9.6 \overline{K} is a compact open subset of F_{u}.

Proof Closure is by definition so compactness follows immediately on observing that K′ ∩ (Q + u) ∩ NS is embedded \overline{D}′-isometrically in the space MP̃_{u} × K′ with metric D′ + ||.|| as the closed subset {(P̃_{v}, v) | v ∈ K′ ∩ (Q + u) ∩ NS}. But MP̃_{u} × K′ is compact.

For openness, we appeal to an argument similar to that of (6.1). Suppose, for a contradiction, that v_{n} is a \overline{D}′-convergent sequence in (F ∩NS)∩K′ and that v′_{n} is a \overline{D}′ convergent sequence in (F ∩NS)\K′ and that both sequences have the same limit x ∈ Π̃_{u}. Therefore v = μ̃(x) is the Euclidean limit of the v_{n} and v′_{n} and so v ∈ ∂K. But by construction P̃_{v_{n}} and P̃_{v′_{n}} have a different D′ limit – a contradiction since the limit in each case must be η̃(x). □

Proof of Theorem 9.4 a/ The sets in B_{u} are clopen by Lemma 9.6 above. Therefore the metric topology on F_{u} we are considering, let us call it τ, is finer than the topology τ′ generated by B_{u}. Both topologies are invariant under the action of ℤ^{N} so that it suffices to show their equivalence on some closed r-ball X of F_{u}. Suppose that τ′ were a Hausdorff topology. Then this equivalence would follow from the continuity of the identity map (X, τ) → (X, τ′), because compactness of (X, τ) implies that of (X, τ′) and so the map would automatically be a homeomorphism. On the other hand, we will proof below that A ↦ \overline{A} yields an isomorphism of Boolean algebras between A_{u} and B_{u}. In particular, B_{u} is closed under symmetric differences so that τ′ being T_{1} already implies that it is Hausdorff.

It remains therefore to check that τ′ is T_{1}, i.e. that the collection B_{u} contains a decreasing set of neighbourhoods around any point in F_{u}.

Certainly, if a ≠ b with a, b ∈ F ∩ (Q + u), then the assumption that Int(K)∩(Q+u) ≠ ∅ (2.6) (hence Int(K′)∩(Q+u) ≠ ∅, interior taken in F) and the facts that K′ is bounded and that π′(ℤ^{N}) is dense in Q ∩ F, imply that there is some v ∈ ℤ^{N} such that a ∈ (Int(K′) ∩ (Q + u)) + π′(v) and b ∉ \overline{(K′ ∩ (Q + u))} + π′(v) (Euclidean closure in F ∩ (Q + u)). i.e. a and b are separated by the topology induced by μ̃(B_{u}).

In particular, if x, y ∈ F_{u} and y ∈ ∩{B ∈ B_{u} | x ∈ B} then μ̃(x) = μ̃(y).

But, if x ≠ y and μ̃(x) = μ̃(y) = v, then, by (3.7)e/, η̃(x) ≠ η̃(y) and we may suppose that there is a point p ∈ η̃(x) \ η̃(y). We use the argument of Lemma 8.6 to show that then there are two sequences v_{n}, v′_{n} ∈ π′(ℤ^{N}) both converging to v in Euclidean topology and such that p + (v_{n} − v) ∈ Σ and p + (v′_{n} − v) ∉ Σ for all n large enough. But then, for such a choice of n, y ∉ \overline{A} (closure of A = (NS ∩ (Q + u) ∩ K′) + π′(p + v_{n}) in \overline{D}′ metric) and x ∈ \overline{A}, a contradiction to the construction of y.

Therefore x = y and so, by the local compactness (Lemma 9.6) of F_{u} we have the required basic property of the collection B_{u}.

b/ This will follow from a/ and the equivalences in Lemma 9.2 if we can show that A_{u} is isomorphic to B_{u} as a Boolean algebra. To show this, it is enough to show that A ↦ \overline{A} (closure in \overline{D}′ metric) is 1-1 on A_{u}; and for this it suffices to prove that if A ∈ A_{u} is non-empty, then its Euclidean closure has interior (in (Q + u) ∩ F).

Note that NS ∩ K′ = NS ∩ Int(K′), so that if A ∈ A_{u} then A is formed of the union and intersection of sets of the form (NS ∩ (Q + u) ∩ Int(K′)) + v (v ∈ π′(ℤ^{N})), and the subtraction of unions and intersections of sets of the form (NS ∩ (Q + u) ∩ K′) + v. With this description and Lemma 2.5, A is equal to NS ∩ Int(\overline{A}) (Euclidean closure, and interior in (Q + u) ∩ F), and from this our conclusion follows.

c/ Elements of CF_{u} are finite sums of integer multiples of indicator functions of compact open sets. Such sets are finite unions of basic clopen sets from the collection in part a/. The isomorphism in part b/ completes the equation.

d/ Given the results of a/ and Lemma 9.6 it is sufficient to show that F_{u} has no isolated points. However, by the argument of part b/ and Lemma 2.5 we see that every clopen subset of F_{u} has μ̃ image with Euclidean interior (in (Q + u) ∩ F) and so cannot be a single point. □

10 A Cantor ZZ^{d} Dynamical System

In this section we describe a ℤ^{d} dynamical system whose mapping torus is equal to MT . First, assuming projection data, (E, K, u) and E ∩ ℤ^{N} = 0, we find a suitable F to which to apply the construction of the previous section.

Definition 10.1 Suppose that G is a group intermediate to ℤ^{N} and R_{u}. The examples in our applications ahead are G = ℤ^{N}, G = R_{u} and G = S_{T}, found in Theorem 7.4.

Fix a free generating set, r_{1}, r_{2}, ..., r_{N}, for G and suppose that the first dim Δ of these generate the subgroup G ∩ E^{⊥} (this can be required by Lemma 2.9).

Let F be the real vector space spanned by r_{1}, r_{2}, ..., r_{n}, where n = N − d.

Note that, since E ∩ G = 0 (by Lemma 2.9 and the assumption E ∩ ℤ^{N} = 0), F is complementary to E and, since n ≥ dim Δ, F contains Δ.

Note that by Theorem 7.1, any two groups intermediate to R_{u} and ℤ^{N}, G and G′ say, differ only by some elements in R_{u} ∩ E^{⊥}, a complemented subgroup of R_{u}. Thus we may fix their generating sets to differ only among those elements which generate G ∩ E^{⊥} or G′ ∩ E^{⊥} respectively. With this convention therefore, the construction of F is independent of the choice of group intermediate to R_{u} and ℤ^{N} and F depends only on the data (E, K, u) chosen at the start.

Also, more directly, the elements r_{dim Δ+1}, ..., r_{N} depend only on the data (E, K, u) chosen at the start.

Definition 10.2 Suppose G_{0} is the subgroup of G generated by r_{1}, r_{2}, ..., r_{n}, and that G_{1} is the complementary subgroup generated by the other d generators. Since n ≥ dim Δ, G_{1} depends only on the data (E, K, u) and not on the choice of G intermediate to ℤ^{N} and R_{u}.

Both groups, G_{1} and G_{0} act on F_{u} and E as subactions of R_{u} (Definition 9.1).

Let X_{G} = F_{u}/G_{0}, a space on which G_{1} acts continuously.

Theorem 10.3 Suppose that we have projection data (E, K, u) such that E∩ℤ^{N} = 0 and G, a group intermediate to ℤ^{N} and R_{u}. Then X_{G} is a Cantor set on which G_{1} acts minimally and there is a commutative square of G_{1} equivariant maps

F_{u}  ⟶^{q}  X_{G}
↓^{ν}  ↓^{ν′}
F ∩ (Q + u) ⟶ (F ∩ (Q + u))/G_{0}.

The set ν^{′−1}(v) is a singleton whenever v ∈ (NS ∩ F ∩ (Q + u))/G_{0}.

The space (F ∩ (Q + u))/G_{0} is homeomorphic to a finite union of tori each of dimension (N − d − dim Δ). Indeed, this space can be considered as a topological group, in which case it is the product of a subgroup of Δ̃/Δ with the (N − d − dim Δ)-torus. The action of G_{1} on this space is by rotation and is minimal.

Proof Assuming we have proved the fact that X_{G} is compact then the commuting square and its properties follow quickly. Therefore we look at X_{G}.

Since G_{0} acts isometrically on F_{u} with uniformly discrete orbits (Theorem 6.8), q is open and locally a homeomorphism and so X_{G} inherits the metrizability of F_{u}, a base of clopen sets and the lack of isolated points (see Theorem 9.4 d/).

Now, let Y_{o} = {∑_{1≤j≤n} λ_{j}r′_{j} | 0 ≤ λ_{j} < 1} ∩ (Q + u), a subset of F ∩ (Q + u). Choose J ⊂ ℤ^{N} finite but large enough that Y_{1} = ∪_{v∈J}((K′ ∩ (Q + u)) + π′(v)) contains \overline{Y}_{o} (Euclidean closure). In particular q(∪_{v∈J}(\overline{K} + π′(v))) = X_{G}, the image of a compact set (Lemma 9.6) under a continuous map. So X_{G} is also compact.

Therefore, we have checked all conditions that show X_{G} is a Cantor set.

Minimality follows from the minimality of the G action on F_{u} which in turn follows from the minimality of the ℤ^{N} + E action on Π̃_{u}, proved analogously to (3.9).

The structure of the rotational factor system follows quickly from the first part of this lemma, the structure of F ∩ (Q + u), and Lemma 9.2. □

Now we describe X_{G} as a fundamental domain of the action of G_{0}.

Definition 10.4 From the details of 10.3 we constuct a clopen fundamental domain for the action of G_{0} on F_{u}. We let G_{0}^{+} = {∑_{1≤j≤n} α_{j}r_{j} | α_{j} ∈ ℕ} and set Y^{+} = ∪_{r∈G_{0}^{+}}(Y_{1} + r) and define Y = Y^{+} \ ∪_{r∈G_{0}^{+},r≠0}(Y^{+} + r).

Define Y_{G} = \overline{ν^{−1}(Y ∩ NS)} (closure in the \overline{D}′ metric), a subset of F_{u}.

The following is immediate from this construction, using Lemma 9.6 and the equivariance of ν and ν′ in Lemma 9.3 with respect to the R_{u} action.

Lemma 10.5 With data (E, K, u), E ∩ ℤ^{N} = 0, and the definitions above, Y is a fundamental domain for the translation action by G_{0} on F ∩ (Q + u). Moreover, Y_{G} is a compact open subset of F_{u}, and a fundamental domain for the action by G_{0} on F_{u}.

There is a natural homeomorphism X_{G} ↔ Y_{G} which is G_{1} equivariant. □

Definition 10.6 Define CX_{G} to be the ring of continuous integer valued functions defined on X_{G}. Also define C(F_{u}; ℤ) to be the ring of continuous integer valued functions defined on F_{u} without restriction on support, uniformity or magnitude (c.f. the definition of CF_{u} from 9.3). As ℤ[G_{0}] modules, the first is trivial and the second is defined as usual using the subaction of the R_{u} action on F_{u}. Both are non-trivial ℤ[G_{1}] modules.

The following combines Lemmas 10.3, 10.5 and Proposition 9.4 and will be of much importance in Chapter II.

Corollary 10.7 With the data of Lemma 10.5,

CF_{u} ≅ CX_{G} ⊗ ℤ[G_{0}]

and

C(F_{u}; ℤ) ≅ Hom_{ℤ}(ℤ[G_{0}], CX_{G})

as ℤ[G_{0}] modules. □

Definition 10.8 Let E′ be the real span of r_{n+1}, ..., r_{N} selected in Definition 10.1. This space contains G_{1} (the integer span r_{n+1}, ..., r_{N}) as a cocompact subgroup.

Recall the definition of dynamical mapping torus [PT] which for the G_{1} action on X_{G} may be equated with

MT (X_{G}, G_{1}) = (X_{G} × E′)/⟨(gx, v) − (x, v − g) | g ∈ G_{1}⟩.

Proposition 10.9 Suppose that we have data (E, K, u) such that E ∩ ℤ^{N} = 0, and G, a group intermediate to ℤ^{N} and R_{u}. With the definitions above, E′ is a d-dimensional subspace of ℝ^{N} complementary to both F and E^{⊥}. Also MT (X_{G}, G_{1}) ≡ Π̃_{u}/G.

Proof The transformation E′ ⟶ E defined by r_{j} ↦ π(r_{j}), n + 1 ≤ j ≤ N is bijective since G_{1} is complementary to G_{0} and hence to the subset G ∩ E^{⊥} of G_{0}. From this we deduce the complementarity immediately.

From Lemma 10.3 we see that G_{0} acts naturally on MT (F_{u}, G_{1}) and that MT (X_{G}, G_{1}) ≡ MT (F_{u}, G_{1})/G_{0}.

To form MT (F_{u}, G_{1}) we take F_{u}×E′ and quotient by the relation (ga, v) ∼ (a, v − g), g ∈ G_{1}, a ∈ F_{u}, v ∈ ℝ^{d}. Applying the inverse of the map of the first paragraph, we can re-express the mapping torus as F_{u} × E quotiented by the relation (ga, w + π(g)) ∼ (a, w), g ∈ G_{1}, a ∈ F_{u}, w ∈ E.

However, the action of G_{1} on the F_{u} is that induced by translation on F_{u}^{o} by elements π′(g) | g ∈ G_{1}. So, working first on the space F_{u}^{o}, we have the equations

MT (F_{u}^{o}, G_{1}) = (F_{u}^{o} × E)/⟨(a + π′(g), w + π(g)) − (a, w) | g ∈ G_{1}⟩

and, since F_{u}^{o} × E = NS, we may write (a, w) = v ∈ NS and so the quotient equals

NS/⟨v + g − v | g ∈ G_{1}⟩ = NS/G_{1}

(recall that NS is G_{1} invariant by Corollary 6.6). Then, by completing, we deduce the equation MT (F_{u}, G_{1}) = Π̃_{u}/G_{1} directly. A further quotient by G_{0} completes the construction. □

Corollary 10.10 Suppose that T is a projection method pattern with data (E, K, u) such that E ∩ ℤ^{N} = 0. Then (X_{T} , G_{1}) is a minimal Cantor ℤ^{d} dynamical system, whose mapping torus MT (X_{T} , G_{1}) is homeomorphic to MT . The pattern dynamical system, (MT , E) is equal to the canonical ℝ^{d} action on the mapping torus (MT (X_{T} , G_{1}), ℝ^{d}) up to a constant time change.

Proof Choose G = S_{T} from Theorem 7.4 which gives MT ≡ Π̃_{u}/G. From this, all but the time change information follows quickly from 10.9 and 10.3, noting that G_{1} ≅ ℤ^{d}.

To compare the two ℝ^{d} actions, we apply the constant time change which takes the canonical ℝ^{d}(≅ E′) action on MT (X_{T} , ℤ^{d}) to the canonical ℝ^{d}(≅ E) action on MT by the isomorphism π|_{E′}: E′ ⟶ E, mapping generators of the G_{1} action r_{j} ↦ π(r_{j}) for n+1 ≤ j ≤ N. □

Examples 10.11 The dynamical system of 10.10 for the octagonal tiling is a ℤ^{2} action on a Cantor set, an almost 1-1 extension of a ℤ^{2} action by rotation on 𝕋^{2} (see [BCL]). For the Penrose tiling, it is also a Cantor almost 1-1 extension of a ℤ^{2} action by rotation on 𝕋^{2} (see [R1]), where we must check carefully that the torus factor has only one component (the only alternative of 5 components is excluded ad hoc).

The correspondence in 10.9 and 10.10 respects the structures found in Theorem 7.4 and so we deduce an analogue.

Definition 10.12 With data (E, K, u) and the selection G = ℤ^{N}, perform the constructions of 10.1 and 10.2, writing the G_{0} and X_{G} obtained there as G_{u} and X_{u} respectively.

Similarly, with G = S_{T}, write G_{0} as G_{T} and X_{G} as X_{T}. Also, with G = R_{u}, write G_{0} as G̃_{u} and X_{G} as X̃_{u}. Note again that F and G_{1} are the same for all three choices of

G.

Corollary 10.13 Suppose that we have data (E, K, u) such that E ∩ ℤ^{N} = 0. Then we can construct two Cantor dynamical systems, (X_{u}, G_{1}) and (X̃_{u}, G_{1}), the latter a finiteisometric extension of the former, together with a compact abelian group, M, which is a finite union of (N − d − dim Δ)-dimensional tori (independent of u) on which G_{1} acts minimally by rotation, and a finite subgroup, Z_{u}, of M.

These have the property that, if T is a projection method pattern with data (E, K, u), then there is a subgroup Z_{T} of Z_{u} and a commuting diagram of G_{1}-equivariant surjections

X̃_{u} ⟶ X_{T} ⟶ X_{u}
↓  ↓  ↓
M ⟶ M/Z_{T} ⟶ M/Z_{u}

In this diagram, the top row consists of finite isometric extensions, the bottom row of group quotients and the vertical maps are almost 1-1.

Taking the G_{1}-mapping torus of this diagram produces the diagram of Theorem 7.4.

Proof With the groups G_{u}, G_{T} and G̃_{u} defined in (10.12). By the remark after 10.1, we know that G_{u} < G_{T} < G̃_{u}. Using the notation of sections 7, 9 and 10, set Z_{u} = G̃_{u}/G_{u} ≅ R_{u}/ℤ^{N} and Z_{T} = G_{T} /G_{u} ≅ S_{T} /ℤ^{N}. Also set M = ((Q + u) ∩ F)/G_{u}, and note that, since ℤ^{N} = G_{1} + G_{u}, a direct sum, G_{u} is in fact independent of the choice of non-singular u. Thus M is independent also of u (the uniform translation by u is irrelevant), so we attach no subscript.

With this notation and the equivalences above, together with the description of the systems, X_{u} and X̃_{u}, using Theorem 10.3, we complete the result. □

II Groupoids, C*-algebras, and their Invariants

1 Introduction

In this chapter we develop the connections between the pattern dynamical systems described in Chapter I and the pattern groupoid. As with the continuous hull, a pattern groupoid, which we write GT^{∗}, can be defined abstractly for any pattern, T , of Euclidean space and we refer to [K1] [K3] for the most general definitions. Here we give a special form for projection method patterns.

The (reduced) C^{∗}-algebra, C^{∗}(GT^{∗}), of this groupoid is a noncommuta-tive version of the mapping torus; this should be regarded as a more precise detector of physical properties of the quasicrystal and the discrete Schröding-er operators for the quasicrystal are naturally members of this algebra.

The initial purpose of this chapter is to compare the noncommutative structure, i.e. , the groupoid, of a pattern with the dynamical systems constructed before. In this regard, we cover similar ground to the work of Bellissard et al. [BCL] but, as noted in the introduction, applied to a groupoid sometimes different. Our Theorem 4.2 is the complete generalization of their connection between C^{∗}- algebraic K-theory and dynamical cohomology, found for projection method tilings in 2 dimensions.

The dynamical system and C^{∗}-algebras we associate to a pattern give rise to a range of ways of attaching an invariant to the pattern T . These include C^{∗} and topological K-theories, continuous groupoid cohomology, Čech cohomology and the dynamical or group (co)homology. The second main result of the chapter is to set up and demonstrate that all these invariants are isomorphic as groups, though the non-commutative invariants contain the richer structure of an ordered group. This structure appears likely to contain information relevant to subsequent investigations, only some of which is recoverable from the other invariants. On the other hand, as we shall see in later chapters, the group (co)homology invariants admit greater ease of computationand is often sufficient for other applications.

GROUPOIDS, C^{∗}-ALGEBRAS, AND THEIR INVARIANTS 47

The final result of the chapter shows that all these common invariants provide an obstruction to the property of self similarity of a pattern. We shall use this obstruction in Chapter IV to show that almost all canonical projection method patterns fail to be substitution systems.

The organisation of this chapter is as follows. In Section 2 we define the various groupoids considered and their C^{∗} algebras, and discuss their equivalences. In Section 3 we consider the notion of continuous similarity of topological groupoids. This is an important equivalence relation for us as continuously similar groupoids have the same groupoid cohomology. In Section 4 we define our invariants and prove them to be additively equivalent and in Section 5 we establish the role our invariants play in discussing self similarity.

2 Equivalence of Projection Method pattern groupoids

First we develop some general results about topological groupoids, appealing to the definitions in [Ren]. These will lead us to the notion of Equivalence of Groupoids which compares most naturally the groupoid C^{∗}-algebras.

Also in this section, we define several groupoids which can be associated to a projection method pattern. We will show that many of these groupoids are related by equivalence.

Definition 2.1 We write the unit space of a groupoid G as G^{o}, and write the range and source maps, r, s: G ⟶ G^{o} respectively. Both these maps are continuous and, due to the existence of a Haar System in all our examples, we note that they are open maps as well.

Recall the reduction of a groupoid. Given a groupoid G with unit space, G^{o}, and a subset, L, of G^{o}, define the reduction of G to L as the subgroupoid _{L}G_{L} = {g ∈ G | r(g), s(g) ∈ L} of G, with unit space, L.

If L is closed then _{L}G_{L} is a closed subgroupoid of G.

We also define G_{L} = {g ∈ G | s(g) ∈ L} and note the maps ρ: G_{L} ⟶ G^{o} and σ: G_{L} ⟶ L defined by r and s respectively.

We say L ⊂ G^{o} is range-open if, for all open U ⊂ G, we have r({x ∈ U : s(x) ∈ L}) open in G^{o}.

Suppose a topological abelian group, H, acts by homeomorphisms on a topological space X, then we define a groupoid called the transformation groupoid, G(X, H), as the topological direct product, X × H, with multiplication (x, g)(y, h) = (x, g + h) whenever y = gx, and undefined otherwise. The unit space is X × {0}.

This last construction is sometimes called the transformation group [Ren] or even the transformation group groupoid, but we prefer the usage to be found in [Pa].

We note that if H is locally compact, then C^{∗}(G(X, H)) can be naturally identified with C_{o}(X)×H, the crossed product [Ren].

Lemma 2.2 Suppose that H is an abelian metric topological group acting homeomorphically on X. Let G = G(X, H) be the transformation groupoid and suppose that L is a closed subset of X ≡ G^{o}.

a/ If H is discrete and countable, then G^{o} is a clopen subset of G, and L is range-open if and only if it is clopen in X.

b/ If there is an ϵ > 0 such that for all neighbourhoods, B ⊂ B(0, ϵ), of 0 in H and all A open in L, we have BA open in X, then L is range-open.

Proof Only part b/ presents complications. Suppose that U is open in X × H. We want to show that r((L × H) ∩ U) is open. Pick x = r(y, h) ∈ r(L×H ∩U) and let C ×(B +h) be a neighbourhood of (y, h) inside U, with B sufficiently small. Then A = s(C) ∩ L is open in L and x ∈ (B + h)A = h(BA) an open subset of X by hypothesis. However, (B + h)A ⊂ r((L × H) ∩ U) by construction, and so we have found an open neighbourhood of x in r((L × H) ∩ U) as required. □

We use the constructions from [MRW] [Rie] without comment. In particular, we do not repeat the definition of (strong Morita) equivalence of groupoids or of C^{∗}-algebras, which is quite complicated. For separable C^{∗}-algebras strong Morita equivalence implies stable equivalence and equates the ordered K-theory (without attention to the scale). All our examples are separable.

Lemma 2.3 Suppose that G is a locally compact groupoid and that L ⊂ G^{o} is a closed, range-open subset which intersects every orbit of G. Then _{L}G_{L} is equivalent to G (in the sense of [MRW]) and the two C^{∗} algebras, C^{∗}(_{L}G_{L}) and C^{∗}(G) are strong Morita equivalent.

Proof It is sufficient to show that G_{L} ⟶^{ρ} G^{o} is a left (G ⟶^{r,s} G^{o})- module whose G ⟶^{r,s} G^{o} action is free and proper, and that G_{L} ⟶^{σ} L is a right (_{L}G_{L} ⟶^{r,s} L)-module whose _{L}G_{L} ⟶^{r,s} L action is free and proper. In short, _{L}G_{L} is an abstract transversal of G and G_{L} a (G,_{L}G_{L})-equivalence bimodule from which we can construct the (C^{∗}(G), C^{∗}(_{L}G_{L}))-bimodule which shows strong Morita equivalence of the two algebras directly, c.f. [MRW] Thm 2.8.

The definition of these actions is canonical and the freedom and properness of the actions is automatic from the fact that L intersects every orbit and from the properness and openness of the maps r, s. Indeed all the conditions follow quickly from these considerations except for the fact that G_{L} ⟶^{ρ} G^{o} is a left (G ⟶^{r,s} G^{o})-module; and the only trouble here is in showing that ρ is an open map. However, this is precisely the problem that range-openness is defined to solve. □

Together with Lemma 2.2 above, this result gives a convenient corollary which unifies the r-discrete and non-r-discrete cases treated separately in [AP].

Corollary 2.4 Suppose that (X, H) and L ⊂ X obey either of the conditions of Lemma 2.2, then, writing G = G(X, H), C^{∗}(_{L}G_{L}) and C^{∗}(G) are strong Morita equivalent. □

Before passing to more special examples, we remark that there is no obstruction to the generalisation of results 2.2, 2.3 and 2.4 to the case of non-abelian locally compact group actions, noting only that, for notational consistency with the definition of transformation groupoid, the group action on a space should then be written on the right.

We define a selection of groupoids associated with projection method patterns, all of them transformation groupoids.

Definition 2.5 Given a projection method pattern, T , with data (E, K, u), fix G = S_{T} as the group obtained from T (Theorem I.7.4) so that MT = Π̃_{u}/G. Recall the definitions of X_{T} = X_{S_{T}}, Y_{T} = Y_{S_{T}} and G_{1} from I.10.2 and I.10.4.

We define in turn: GX_{T} = G(X_{T} , G_{1}), from the G_{1} action on X_{T}, and GF_{T} = G(F_{u}, G), using the action of G on F_{u}, both defined in (I.8.1).

Also define GΠ̃_{T} = G(Π̃_{u}, E + G).

All but the last of these groupoids are r-discrete (see [Ren]).

Lemma 2.6 Suppose T is a projection method pattern with data (E, K, u) such that E ∩ ℤ^{N} = 0. The groupoids GF_{T} and GX_{T} are each a reduction of GΠ̃_{T} to the closed range-open sets, F_{u} and Y_{T}.

Proof It is clear that GF_{T} is a reduction of GΠ̃_{T} to F_{u}. To prove that this a range-open set using Lemma 2.2, we take an open subset of F_{u} and examine the action of small elements of E + G on it. Only the E action enters our consideration and then it is clear from (I.9.2) that if B is an open subset of E and A is an open subset of F_{u}, then as topological spaces, BA ≡ A × B, which is clearly open in Π̃_{u}.

Recall the notation of (I.10.12), and the homeomorphism X_{T} ↔ Y_{T}, found by Lemma I.10.5, putting G = S_{T}. In effect, this homeomorphism equates X_{T} with a fundamental domain of the G_{T} action on F_{u}. This homeomorphism is G_{1}-equivariant if we equat e the G_{1} action on Y_{T} with the induced action of G/G_{0} on F_{u}/G_{0} = X_{T} ≡ Y_{T}. But this is precisely the correspondence needed to equate G(X_{T} , G_{1}) with the reduction of GF_{T} to Y_{T} considered as a subset of the unit space of GF_{T}. Thus GX_{T} is the reduction of GΠ̃_{T} to Y_{T}, and since Y_{T} is clopen in F_{u} the same argument as above shows that Y_{T} is closed and range-open in Π̃_{u}. □

Now we define a groupoid connected more directly with the pattern, T .

Definition 2.7 From Definition I.4.4, recall the two maps MP̃_{u} ⟶ MT ⟶^{∗} MP_{u} whose composition is π_{∗}. Without confusion we name the second (starred) map π_{∗} as well.

We also define a map η_{T} which is the composite Π̃_{u} \underrightarrow{_{η}} MP̃_{u} ⟶ MT .

Note that η(x) = π_{∗}(η_{T} (x)) for all x ∈ Π̃_{u} and that, being a composition of open maps (I.3.9), η_{T} is an open map.

Define the discrete hull of T as Ω_{T} = {S ∈ MT | 0 ∈ π_{∗}(S)}. The pattern groupoid, GT , is the space {(S, v) ∈ Ω_{T} × E | v ∈

π_{∗}(S)} inheriting the subspace topology of Ω_{T} × E. The restricted multiplication operation is (S′, v′)(S, v) = (S′, v + v′), if S = v′S′, undefined otherwise. The unit space is GT^{o} = {(S, 0) | S ∈ Ω_{T} }, homeomorphic to Ω_{T}.

Also define E_{u}^{⊥} = μ^{−1}(E^{⊥}), a space which is naturally homeomorphic to F_{u}; a correspondence made by extending the application of π^{⊥}, inverted by the extension of π′.

Lemma 2.8 Suppose T is a projection method pattern with data (E, K, u) such that E ∩ ℤ^{N} = 0. The groupoid GT is isomorphic to a reduction of GΠ̃_{T} to a closed range-open set.

Proof Let L be a compact open subset of E_{u}^{⊥} so that η_{T} (L) = Ω_{T} and η_{T} is 1-1 on L. This can be constructed as follows. Define L_{o} = \overline{NS ∩ K ∩ (Q + u)} where the closure is taken with respect to the \overline{D}′ metric - a clopen subset of E_{u}^{⊥} by (I.9.6). Let L = L_{o} \∪\{gL_{o} | g ∈ G ∩ E^{⊥}, g ≠ 0} (using the G action on Π̃_{u}).

We claim that the reduction of GΠ̃_{T} to L is isomorphic to the pattern groupoid defined above.

Suppose that (x; g, v) ∈ GΠ̃_{T} and x ∈ L and (g + v)x ∈ L, then 0 ∈ η(x) = π_{∗}(η_{T} (x)) and 0 ∈ η((g + v)x). But note that the action by v ∈ E on x ∈ NS is vx = x − v and so η((g + v)x) = η(gx) − v = π_{∗}(η_{T} (gx)) − v = π_{∗}(η_{T} (x)) − v. Thus 0, v ∈ π_{∗}(η_{T} (x)) and the map ψ: (x; g, v) ↦ (η_{T} (x), v) is well defined _{L}GΠ̃_{T} _{L} ⟶ GT . The E and G equivariance of the maps used to define ψ show that the groupoid structure is preserved.

Conversely, if 0, −v ∈ π_{∗}(η_{T} (x)), then there are, by construction of L, g, g′ ∈ G such that gx, (g′ + v)x ∈ L. Thus (gx; g′ − g, v) ∈ LGΠ̃_{T} _{L} showing that ψ is onto. Also, the g, g′ are unique by the construction of L above, and so ψ is 1-1. The continuity of ψ and its inverse is immediate, so we have a topological groupoid isomorphism, as required.

Thus we have shown that GT is isomorphic to a reduction of GΠ̃_{T} to the set L which is clearly closed.

Also, L is a subset of E_{u}^{⊥}, transverse to E, so that the same argument as 2.6 shows that L is range open.

It remains to show that L hits every orbit of GΠ̃_{T} and for this it is sufficient to show that for any x ∈ Π̃_{u}, Gx ∩ (L × E) ≠ 0 (where we exploit the equivalence: Π̃_{u} ≡ E_{u}^{⊥} × E (I.9.2). But this is immediate from the fact that L × E is a clopen subset of Π̃_{u} (I.9.2), and by minimality of the G + E action on Π̃_{u} (as in I.3.9). □

Combining the Lemmas above, we obtain the following.

Theorem 2.9 Suppose that T is a projection method pattern with data (E, K, u) such that E ∩ℤ^{N} = 0. The C^{∗}-algebras C^{∗}(GT ), C_{o}(F_{u})×G and C(X_{T})×G_{1} are strong Morita equivalent and thus their ordered K-theory (without attention to scale) is identical. □

Remark 2.10 We can compare the construction above with the “rope” dynamical system constructed by the third author [K2] exploiting the generalised grid method introduced by de Bruijn [dB1]. The rope construction actually shows that, in a wide class of tilings including the canonical projection tilings, there is a Cantor minimal system (X, ℤ^{d}) such that G(X, ℤ^{d}) is a reduction of GT . By comparing the details of the proof above with [K2] it is possible to show directly that, in the case of non-degenerate canonical projection tilings, the rope dynamical system is conjugate to (X_{T} , G_{1}).

We note that the construction of Lemma 2.8 depends only on the data (E, K, u) and on G and from this we deduce the following.

Corollary 2.11 Suppose we specify projection data (E, K, u), such that E ∩ ℤ^{N} = 0. Then, among projection method patterns, T , with these data, the dynamical system (MT , E) determines GT up to groupoid isomorphism.

Thus among projection method patterns with fixed data, the dynamical invariants are at least as strong as the non-commutative invariants. □

Finally we reconnect the work of this section with the original construction of the tiling groupoid [K1].

Definition 2.12 Recall the notation A[r] = (A ∩ B(r)) ∪ ∂B(r) etc. defined in I.3.1 and I.4.2, for r ≥ 0 and A ⊂ ℝ^{N} or E. Given two closed sets, A, A′ define the distance D_{o}(A, A′) = inf{1/(r + 1) | r > 0, A[r] = A′[r]}.

As a metric this can be used to compare point patterns in E or ℝ^{N} (as in I.3.1 and I.4.2), or decorated tilings in E as described in I.4.1.

Given a tiling, T , of E, bounded subsets which are the closure of their interior, the construction of the discrete hull in [K1] starts by placing a single puncture generically in the interior of each tile according to local information (usually just the shape, decoration and orientation of the tile itself–the position of the puncture would then depend only on the translational congruence class of the tile). So we form the collection of punctures, τ(T ) of T , a discrete subset of points in E.

We consider the set Ω_{T}^{o} = {T + x | 0 ∈ τ(T + x) = τ(T ) + x}, and define a modified hull, which we write Ω_{T}^{∗} in this section, as the D_{o} completion of this selected set of shifts of T .

As in [K1] we consider only tilings T which are of finite type (or of finite pattern type as in [AP] or of finite local complexity) which means that for each r the set {T′[r]|T′ ∈ Ω_{T}^{o} } is finite. It is well known that canonical projection tilings are finite type.

From this hull, we define the groupoid, GT^{∗} exactly as for GT : GT^{∗} = {(S, v) ∈ Ω_{T}^{∗} × E | v ∈ τ(S)} with the analogous rule for partial multiplication.

The assumption of local information dictates more precisely that up to isomorphism GT^{∗} is independent of the punctures chosen and the map τ is continuous, E-equivariant, and 1-1 from Ω_{T}^{∗} with D_{o} metric to the space of Delone subsets of E also with D_{o} metric.

Remark 2.13 Although phrased in terms of tilings, this definition can in fact be applied to patterns as well, where the idea of puncture becomes now the association of a point with each unit of the pattern (I.4.1). In this case the finite type condition is equated with the condition that τ(T ) is Meyer (see [La1]), and this is sufficient to prove the analogues of all the Lemmas below. However, we continue to use the language of tilings and, since every projection method pattern is pointed conjugate to a decorated finite type tiling (decorating the Voronoi tiles for P_{u} for example (I.7.5)), we lose no generality in doing so.

We note that when a projection method pattern T is in fact a tiling, the two definitions of hull (2.7 and above) given here seldom coincide nor do we obtain the same groupoids (but we note the important exception of the canonical projection tiling in 2.16 ahead). The remainder of this section shows that, never-the-less, the two groupoids, GT and GT^{∗}, are equivalent. We start by comparing D and D_{o}.

Lemma 2.14 Suppose that T is a finite type tiling as above, then Ω_{T}^{o} is precompact with respect to D_{o}. Further D and D_{o} generate the same topology on Ω_{T}^{o}.

Proof The precompactness of Ω_{T}^{o} is proved in [K1].

For any two tilings, we have D(T , T′) ≤ D_{o}(T , T′) by definition, and so the topology of D_{o} is always finer than that of D.

Conversely, as a consequence of the finite type condition of T there is a number δ_{o} < 1 such that if 0 < ϵ < δ_{o}, then T + x, T + x′ ∈ Ω_{T}^{o} and D(T + x, T + x′) < ϵ together imply that T + x and T + x′ actually agree up to a large radius (1/ϵ − 1 will do) and we conclude D_{o}(T + x, T + x′) < 2ϵ as required. □

Consequently, Ω_{T}^{∗} is canonically a subspace of MT and we can consider its properties as such.

Lemma 2.15 With respect to the E action on MT , both Ω_{T} and Ω_{T}^{∗} are range-open.

Proof With the notation of the proof of Lemma 2.8, Ω_{T} = η_{T} (L), where L is a compact open subset of E_{u}^{⊥}. As in 2.8, L is range-open in Π̃_{u} and, since η_{T} is an open E-equivariant map, we deduce the same of η_{T} (L).

For Ω_{T}^{∗}, we note that (as in Section 1, 3.4 before) by the finite type condition of T , there is a number δ_{o} so that, if x, x′ ∈ E, T′ ∈ MT and 0 < ||x − x′|| < δ_{o}, then D(T′ − x, T′ − x′) ≥ ||x − x′||/2 and D_{o}(T′ − x, T′ − x′) = 1. In particular, since Ω_{T}^{∗} is D_{o}-compact and hence a finite union of radius 1/2 D_{o}-balls, we deduce that the map Ω_{T}^{∗} ×B(δ_{o}/2) ⟶ MT , defined as (T′, x) ↦ T′+x, is locally injective and hence open. From here the range-openness of Ω_{T}^{∗} is immediate. □

Theorem 2.16 If T is at once a tiling and a projection method pattern with data (E, K, u), such that E ∩ ℤ^{N} = 0, then the tiling groupoid GT^{∗} as defined in Def. 2.12 is equivalent to GT (Def. 2.7). Thus the respective C^{∗}-algebras are strong Morita equivalent also.

If T is a non-degenerate canonical projection tiling there is a puncturing procedure inducing an isomorphism between the two groupoids.

Proof Recall the definition of transformation groupoid and the action of E on MT and consider G(MT , E). Using Lemmas 2.3 and 2.15, it suffices to show that the groupoids GT and GT^{∗} are each a reduction of G(MT , E) to the sets Ω_{T} and Ω_{T}^{∗} respectively. But this is immediate from their definition.

To treat the case of canonical projection tilings, we note that its tiles are parallelepipeds, and that the point pattern consists of their vertices. Hence if we fix a small generic vector then adding this vector to each vertex gives exactly one puncture for each tile. Translating an element of Ω_{T} by that vector therefore produces an element of Ω_{T}^{∗} if we use these punctures to define the latter. The map so defined is clearly a continuous bijection which is E-equivariant (with respect to the restricted E-action). Therefore it induces an isomorphism between GT and GT^{∗}. □

3 Continuous similarity of Projection Method pattern groupoids

The aim of this section is to compare our groupoids in a second way: by continuous similarity. This gives most naturally an equivalence between groupoid cohomology.

We will show that many of the groupoids we associate with a projection pattern are related in this way also. Further background facts about groupoids and their cohomology and the idea of similarity may be found in [Ren].

Definition 3.1 Two homomorphisms, ϕ, ψ: G ⟶ H between topological groupoids are continuously similar if there is a function, Θ: G^{o} ⟶ H such that

Θ(r(x))ϕ(x) = ψ(x)Θ(s(x)).

Two topological groupoids are continuously similar if there exist homomorphisms ϕ: G ⟶ H, ψ: H ⟶ G such that Φ_{G} = ψϕ is continuously similar to id_{G} and Φ_{H} = ϕψ is continuously similar to id_{H}.

In all our examples, produced by Lemma 3.3 ahead, the function Θ is continuous but note that this is not required by Definition 3.1. Our interest in this relation lies in the following fact which we exploit in Section 4; see [Ren] for the definition of continuous cohomology H^{∗}(G; ℤ) of a topological groupoid G.

Proposition 3.2 ([Ren], with necessary alterations for the continuous category) If G and H are continuously similar then H^{∗}(G; ℤ) = H^{∗}(H; ℤ). □

It turns out that the construction of continuous similarities follows closely the reduction arguments in the examples that interest us.

Lemma 3.3 Suppose that (X, H) is a free topological dynamical system (i.e., hx = x implies that h is the identity), with transformation groupoid G = G(X, H), and that L, L′ are two closed subsets of G^{o}. Suppose there are continuous functions, γ: L ⟶ H, δ: L′ ⟶ H which define continuous maps α: L ⟶ L′ and β: L′ ⟶ L by αx = γ(x)x and βx = δ(x)x. Then _{L}G_{L} and _{L′}G_{L′} are continuously similar.

Proof Construct the two homomorphisms, ϕ:_{L}G_{L} → _{L′}G_{L′} and ψ:_{L′}G_{L′} → _{L}G_{L} by putting ϕ(x, g) = (αx, γ(gx) + g − γ(x)) and ψ(y, h) = (βy, δ(hy) + g − δ(y)).

A quick check confirms that these are homomorphisms, and they are both clearly continuous. Moreover, ϕψ(y, h) = (αβy, γ((δ(hy) + h−δ(y))β(y)) +δ(hy)+h−δ(y)−γ(βy), a rather complicated expression which can be simplified if we note that γ((δ(hy)+h−δ(y))β(y)) = γ((δ(hy) + h)y) = γ(βhy), and define σ(y) to be the element of H such that σ(y)y = αβy. Then σ(y) = δ(y) + γ(βy), by definition, and so σ(hy) = γ(hβ(y)) + δ(hy) = γ((δ(hy) + h − δ(y))β(y)) + δ(hy). This gives ϕψ(y, h) = (αβy, σ(hy) + h − σ(y)).

It is now easy to see that ϕψ is continuously similar to the identity on_{L′}G_{L′} using the transfer function, Θ : L′ ⟶ _{L′}G_{L′} given by Θ(y) = (αβy, −σ(y)). This Θ happens also to be continuous.

Reciprocal expressions give the similarity between ψϕ and the identity on _{L}G_{L}. □

Remark 3.4 Note that if L′ = G^{o}, then Lemma 3.3 can be reexpressed in the following form. If L is a closed subset of G^{o} for which there is a continuous map γ: G^{o} ⟶ H such that γ(x)x ∈ L for all x ∈ G^{o}, then LG_{L} is continuously similar to G. (The condition on L implies that L intersects every H-orbit of (G^{o}, H), but the converse is not true.)

We apply this lemma and remark in two ways as we examine continuous similarities between the various groupoids of section 2.

Lemma 3.5 Suppose that T is a projection method pattern and write G for GΠ̃_{T}. If L is a clopen subset of F_{u} then _{L}G_{L} is continuously similar to G.

Proof It suffices to find the function γ in the remark.

Pick an order ≻ on G = S_{T} in which every non-empty set has a minimal element. The set EL = {vy : v ∈ E, y ∈ L} is naturally homeomorphic to E × L by Lem. I.9.2, and hence is clopen in Π̃_{u}. By the minimality of the E+G action on Π̃_{u}, (which is proved analogouly to Lem. I.3.9), we have ∪_{h∈G}hEL = Π̃_{u}, so that for each x ∈ Π̃_{u}, there is a ≻-minimal h ∈ G such that hx ∈ EL. Let γ_{0}(x) be this g, and note that by the freedom and isometric action of G and the clopenness of EL, this function x ↦ γ_{0}(x) is continuous and maps Π̃_{u} to EL.

Now, given γ_{0}(x)x ∈ EL, then there is a unique γ_{1}(x) ∈ E such that γ_{1}(x)γ_{0}(x)x ∈ L, and it is clear that x ↦ γ_{1}(x) is continuous as a map Π̃_{u} ⟶ E. The desired map γ can now be taken as this composite. □

Lemma 3.6 Suppose that T is a finite type tiling for which we have chosen a puncturing and which is also a projection method pattern. Then GT and GT^{∗} are continuously similar.

Proof We construct the maps γ and δ as follows.

Recall the metric D_{o}(A, B) = inf{1/(r+1) : A[r] = B[r]} and the argument of Lemma 2.14 which shows that D and D_{o} are equivalent on each of the sets Ω_{T} and Ω_{T}^{∗}. (Actually the argument refers only to the second space, but the fact that π_{∗}T is a Meyer pattern (see [La1] and Remark 2.13) allows it to be applied directly to the first space as well.)

We may assume without loss of generality that, for each S ∈ MT , each point of π_{∗}(S) is in the interior of a tile of S (if not we shift all the tiles in T by a uniform short generic displacement and start again equivalently).

Suppose that S ∈ Ω_{T}. We know that 0 ∈ π_{∗}(S) and that by assumption there is a unique tile in S which contains the origin in its interior. This tile has a puncture at a point v say, and so S − v ∈ Ω_{T}^{∗}. So we have defined a map from Ω_{T} to E, γ: S ↦ −v which is clearly continuous with respect to the D_{o} metric. Moreover the map, α: S ↦ S − v has range Ω_{T}^{∗}.

Conversely, let r be chosen so that every ball in E of radius r contains at least one point of π_{∗}(T ) = P_{u}. Consider the sets π_{∗}(S) ∩ B(r), as S runs over Ω_{T}^{∗} and note that there are only finitely many possibilities, i.e. the set J = {π_{∗}(S) ∩ B(r) : S ∈ Ω_{T}^{∗} } is a finite collection of non-empty finite subsets of B(r). Furthermore, by the continuity of π_{∗} on Ω_{T}^{∗} with respect to the D_{o} metric, for each C ∈ J, the set {S ∈ Ω_{T}^{∗} : π_{∗}(S) ∩ B(r) = C} is clopen.

For each C ∈ J choose an element v = v(C) ∈ C, and define δ(S) = −v(π_{∗}(S) ∩ B(r)); this is continuous by construction. Then S + δ(S) ∈ Ω_{T}. □

Now, appealing to 2.6 and 2.8, we can gather the results of this section into the following corollary.

Corollary 3.7 Suppose that T is a projection method pattern. Then GT , GΠ̃_{T}, GF_{T} and GX_{T} are all continuously similar. If T is also a finite type tiling, then these are all continuously similar to the tiling groupoid, GT^{∗}, of [K2].

Proof Lemmas (3.5) and (2.6) and (2.8) show that GT , GF_{T} and GX_{T} are all continuously similar to GΠ̃_{T}. The second part is a restatement of (3.6). □

4 Pattern cohomology and K-theory

We are now in a position to define our topological invariants for projection method patterns and prove their isomorphism as groups.

Definition 4.1 For a projection method pattern, T , in ℝ^{d}, we define for each m ∈ ℤ the following groups.

(a) H^{m}(GT , ℤ), the continuous groupoid cohomology of the pattern group-oid GT ;

(b) H^{m}(MT ), the Čech cohomology of the space MT ;

(c) H_{d−m}(G_{1}, CX_{T}) and H^{m}(G_{1}; CX_{T}), the group homology and cohomology of G_{1} with coefficients CX_{T} (I.10.6);

(d) H_{d−m}(S_{T}; CF_{u}) and H^{m}(S_{T}; C(F_{u}; ℤ)), the group homology and cohomology of S_{T} with coefficients CF_{u} (I.9.3) or C(F_{u}; ℤ) (I.10.6);

(e) K_{d−m}(C^{∗}(GT )), the C^{∗} K-theory of C^{∗}(GT );

(f) K_{d−m}(C(X_{T}) × G_{1}), the C^{∗} K-theory of the crossed product C(X_{T}) × G_{1};

and, for finite type tilings (2.12),

(g) the continuous groupoid cohomology H^{m}(GT^{∗}, ℤ).

Theorem 4.2 For a projection method pattern T and for each value of m, the invariants defined in (4.1)(a) to (d) are all isomorphic as groups. If T is also a finite type tiling, then these are also isomorphic to that defined in (4.1)(g).

The invariants defined in (4.1)(e) and (f) are each isomorphic as ordered groups. Finally, all these invariants are related via isomorphisms of groups such as

K_{m}(C(X_{T}) × G_{1}) ≅ ⨁_{j=−∞}^{∞} H^{m+d+2j}(G_{1}; CX_{T}).

These invariants are, in all cases, torsion free, and those in parts (a) to (d) and (g) are non-zero only for integers m in the range 0 ≤ m ≤ d.

Proof It is immediate from the definition [Ren] that if W is a locally compact space on which a discrete abelian group G acts freely by homeomorphisms then the continuous groupoid cohomology H^{∗}(G(W, G); ℤ) is naturally isomorphic to the group cohomology H^{∗}(G, C(W; ℤ)) with coefficients the continuous compactly supported integer-valued functions on W, with ℤ[G]-module structure dictated naturally by the G action on W. This proves the equality of (a) (and (g) where appropriate) with the cohomology versions of (c) and (d) from (3.7) and the fact that GF_{T} and GX_{T} are transformation groupoids.

By (I.10.10) the space MT is homeomorphic to the mapping torus MT (X_{T} , G_{1}) and as noted in [FH] the Čech cohomology H^{∗}(MT (X_{T} , G_{1})) is isomorphic to the group cohomology H^{∗}(G_{1}, CX_{T}) (this is standard and follows, for example, by induction on the rank of G_{1} with the induction step passing from ℤ^{r} to ℤ^{r+1} coming from the comparison of the Mayer-Vietoris decomposition of MT (X_{T} , G_{1}) along one coordinate with the long exact sequence in group cohomology coming from the extension ℤ^{r} → ℤ^{r+1} → ℤ). This proves the isomorphism of (4.1)(b) with the cohomological invariant of (4.1)(c).

The isomorphism of H^{m}(G_{1}, CX_{T}) with H_{d−m}(G_{1}; CX_{T}) is simply Poincaré duality for the group G_{1} ≅ ℤ^{d}.

By Cor. I.10.7, a decomposition of CF_{u} as a ℤ[G_{T}] module is given by CX_{T} ⊗ ℤ[G_{T}] where CX_{T} is a trivial ℤ[G_{T}] module. Standard homological algebra now tells us that

\begin{matrix}H_{p}(S_{T}; CF_{u}) ≅ & H_{p}(G_{1} ⊕ G_{T}; CX_{T} ⊗ ℤ[G_{T}]) \\ ≅ & H_{p}(G_{1}; CX_{T}) \end{matrix}

establishing the isomorphism of (4.1)(c) and (d) in homology. A similar argument also works in cohomology for C(F_{u}; ℤ).

The isomorphism of (4.1)(e) and (f) follows from the Morita equivalence of the underlying C^{∗}-algebras in (2.7) and the isomorphism

K_{m}(C(X_{T}) × G_{1}) ≅ ⨁_{j=−∞}^{∞} H^{m+d+2j}(G_{1}; CX_{T})

is one of the main results of [FH].

The torsion-freedom of these invariants also follows from the results of [FH], while the vanishing of the (co)homological invariants outside the range of dimensions stated is immediate from their identification with the (co)homology of the group G_{1} ≅ ℤ^{d}.

We make one further reduction of the complexity of the computation of these invariants. Recall first the construction of I.2.9 and I.10.1, in particular the equation F ∩ Q = V ⊕ Δ̃ splitting F into continuous and discrete directions, and in which π′(ℤ^{N}), the projection of ℤ^{N} onto F parallel to E, is dense. Recall also the map μ̃: Π_{u} ⟶ Q + u defined in (I.4.3) for each u ∈ NS.

Definition 4.3 The restriction of μ̃ to F_{u} is written ν: F_{u} ⟶ F ∩(Q+ u) = (F ∩Q)+π′(u); this map is π′(ℤ^{N})-equivariant and |ν^{−1}(v)| = 1 precisely when v ∈ NS ∩ F ∩ (Q + u) (see Lemma I.9.2).

Let Γ_{T} = {v ∈ S_{T} : π′(v) ∈ V } and CV_{u} = {f ∈ CF_{u} : ν(supp(f)) ⊂ V + π′(u)}, where supp refers to the support of the function. This is consistent with setting V_{u} = {x ∈ F_{u} : ν(x) ∈ V + π′(u)} and taking CV_{u} as the continuous integer valued functions on V_{u} with compact support. There is a natural decomposition CF_{u} = CV_{u} ⊗_{ℤ} ℤ[Δ̃].

Lemma 4.4 As a subgroup of S_{T} = G_{T} + G_{1} (the decomposition of I.10.1), Γ_{T} satisfies Γ_{T} = (Γ_{T} ∩ G_{T}) ⊕ G_{1}. Moreover, Γ_{T} is complemented in S_{T} by a group Γ_{Δ}, naturally isomorphic to Δ̃.

With this splitting, the action of ℤ[S_{T}] = ℤ[Γ_{T}] ⊗ ℤ[Γ_{Δ}] on CF_{u} = CV_{u} ⊗_{ℤ} ℤ[Δ̃] is the obvious one, and hence there is an isomorphism of homology groups H_{∗}(S_{T}; CF_{u}) ≅ H_{∗}(Γ_{T}; CV_{u}).

Proof The decompositions and restrictions on G_{T} and G_{1} follow from the definition and the original construction of (I.10.1). The conclusion in homology is the same homological argument as used in the previous proof. □

We note that since S_{T} ⊃ ℤ^{N} and π^{⊥}(ℤ^{N}) is dense in Q∩F, the group Γ_{T} acts minimally on V and hence on V_{u}.

Corollary 4.5 With the data above,

K_{n}(C^{∗}(GT )) = ⨁_{j=−∞}^{∞} H_{n+2j}(Γ_{T}; CV_{u}).  □

This is, in fact, the most computationally efficient route to these invariants and, with the exception of Chapter III, the one we shall use in the remainder of this memoir.

5 Homological conditions for self similarity

To motivate the direction we now move in, we give an immediate application of these invariants. In this section we show that the (co)homological invariants defined in Section 4 provide an obstruction to a pattern arising as a substitution system. This result will be used in Chapter IV to show that almost all canonical projection tilings fail to be self similar.

We adopt the construction of substitution tilings in [AP] considering only finite type tilings whose tiles are compact subsets of ℝ^{d} homeomorphic to the closed ball. A substitution procedure as in [AP] is based on a map which assigns to each tile of a tiling T a patch of tiles (a tiling of a compact subset of ℝ^{d}) which covers the same set as the tile. Moreover, the map is ℝ^{d}-equivariant in the sense that translationally congruent tiles are mapped to translationally congruent patches. This is useful if there is a real constant λ > 1 such that the tiling which is obtained from T , by first replacing all tiles with their corresponding patches and then stretching the resulting tiling by λ (keeping the origin fixed), belongs to MT . This procedure of replacing and stretching can then be applied to all tilings of MT thus defining the substitution map σ : MT → MT which is assumed to be injective. As an aside, Anderson and Putnam show that under this condition σ is a hyperbolic homeomorphism. They establish the following property for substitution tilings, i.e. tilings which allow for a substitution map.

Theorem 5.1 [AP] Suppose that T is a finite type substitution tiling of ℝ^{d}. Then MT is the inverse limit of a stationary sequence

Y ⟵^{γ} Y ⟵^{γ} ⋯

of a compact Hausdorff space Y with a continuous map γ. □

Let us describe Y. Call a collared tile a tile of the tiling T decorated with the information of what are its adjacent tiles and a collared prototile the translational congruence class of a collared tile. As a topological space we identify the collared prototile with a tile it represents as a compact subset of ℝ^{d}, regardless of its decoration. By the finite type condition there are only finitely many collared prototiles. Let Ỹ be the disjoint union of all collared prototiles. Y is the quotient of Ỹ obtained upon identifying two boundary points x_{i} of collared prototiles t_{i}, i = 1, 2, if there are two collared tiles t̂_{i} in the tiling, t̂_{i} of class t_{i}, such that x̂_{1} = x̂_{2} where x̂_{i} is the point on the boundary of t̂_{i} whose position corresponds to that of x_{i} in t_{i}. If the tiles are polytopes which match face to face (i.e. two adjacent tiles touch at complete faces) then Y is a finite CW complex whose highest dimensional cells are the (interiors of) the tiles represented by the collared prototiles and whose lower dimensional cells are given by quotients of the set of their faces of appropriate dimension. The map γ above is induced by the substitution. Going through the details of its construction in [AP] one finds that in the case where tiles are polytopes which match face to face it maps faces of dimension l to faces of dimension l thus providing a cellular map. We are then in the situation assumed for the second part of [AP].

Corollary 5.2 Suppose that T is a finite type tiling whose tiles are polytopes which match face to face. Then for each m, the rationalized Čech cohomology H^{m}(MT ) ⊗ ℚ has finite ℚ-dimension.

Proof Like [AP] we obtain from (5.1) and the conclusion that Y is a finite CW complex and γ a cellular map that H^{m}(MT ) = lim_{→} H^{m}(Y). So H^{∗}(MT ) ⊗ ℚ = lim_{→} (H^{∗}(Y) ⊗ ℚ). Thus the ℚ dimension of H^{∗}(MT ) ⊗ ℚ is bounded by that of H^{∗}(Y) ⊗ ℚ and this is finite since Y is a finite CW complex. □

The conclusion of (5.2) applies to much more general pattern constructions. Note that the only principle used is that the space MT of the tiling dynamical system is the inverse limit of a sequence of maps between uniformly finite CW complexes. We sketch a generalization whose details can be reconstructed by combining the ideas to be found in [Pr] and [Fo].

Finite type substitution tilings are analysed combinatorially by Priebe in her PhD Thesis [Pr] where the useful notion of derivative tiling, generalized from the 1-dimensional symbolic dynamical concept [Du], is developed. We do not pursue the details here except to note that the derivative of an almost periodic finite type tiling is almost periodic and finite type, and that the process of deriving can be iterated.

Suppose that T is an almost periodic finite type tiling. By means of repeated derivatives, and adapting the analysis of [Fo] for periodic lattices, we may build a Bratteli diagram, B, for T . Its set of vertices at level t is formally the set of translation classes of the tiles in the t^{th} derivative tiling, and the edges relating two consecutive levels, t and t + 1 say, are determined by the way in which the tiles of the (t + 1)^{th} derivative tiling are built out of the tiles of the t^{th} derivative tiling. The diagram B defines a canonical dimension group, K_{0}(B). Adapting the argument of [Fo] we can define a surjection K_{0}(B) ⟶ H^{d}(MT ) and hence a surjection K_{0}(B) ⊗ ℚ ⟶ H^{d}(MT ) ⊗ ℚ.

In [Pr] it is shown that the repeated derivatives of a finite type aperiodic substitution tiling have a uniformly bounded number of translation classes of tiles and are themselves finite type. In particular, the number of vertices at each level of its Bratteli diagram B is bounded uniformly. Thus K_{0}(B) ⊗ ℚ is finite dimensional over ℚ, being a direct limit of uniformly finite dimensional ℚ vector spaces, and so we reprove (5.2) for the case m = d .

It is worth extracting the full power of this argument since it applies to a wider class than the substitution tilings.

Theorem 5.3 Suppose that T is a finite type tiling of ℝ^{d} whose repeated derivatives have a uniformly bounded number of translation classes of tiles, then H^{d}(GT ) ⊗ ℚ is finite dimensional over ℚ. □

Therefore in Chapter IV, when we take a pattern, T , compute its rationalized homology H_{0}(Γ_{T}; CV_{u}) ⊗ ℚ and find it is infinite dimensional, we know we are treating a pattern or tiling which is outwith the class specified in Theorem 5.3, and a fortiori outside the class of finite type substitution tilings.

III Approaches to Calculation I: Cohomology for Codimension One

1 Introduction

Our goal in this chapter is to demonstrate the computability of the invariants introduced in Chapter II and we do so by looking at the case where N = d + 1. In this case the lattice Δ is always trivial whenever E ∩ ℤ^{N} = 0.

Recall that when Δ = 0 the projection pattern is determined by a small number of parameters – the dimensions d and N of the space E and the lattice ℤ^{N} ⊂ ℝ^{N}, the slope of E in ℝ^{N} and the shape of the acceptance domain K. We shall restrict ourselves to specific acceptance domains in later chapters, but the main result of this chapter (3.1) characterises the invariants of patterns on ℝ^{d} arising as projection from ℝ^{d+1} for more or less arbitrary acceptance domains K.

This chapter thus gives a complete answer to the so-called codimension 1 patterns (i.e., N −d = 1). After restricting the shape of the acceptance domain we shall give in Chapter V an alternative analysis of this case together with formulæ for the ranks of the invariants in the codimension 2 and 3 cases, when these are finite.

When N − d = 1 one of the most important features of K is its number of path components. To facilitate our computations we examine in Section 2 a general technique which sometimes simplifies the computations of projection pattern cohomology when the acceptance domain is disconnected. We prove our main results in Section 3.

We note that the case d = 1 gives the classical Denjoy counterexample systems whose ordered cohomology is discussed in [PSS], an observation which has also been made by [He]. The result in this chapter, for d = 1 and H_{∗} finitely generated, can be deduced from that paper [PSS] a task which has been carried out in [He].

2 Inverse limit acceptance domains

Suppose that K and K_{i}, i = 1, 2, …, are compact subsets of E^{⊥} each of which is the closure of its interior, and suppose that IntK = ∪_{i}IntK_{i} is a disjoint union and that ∂K = ∪∂K_{i}. Let K_{i}^{∗} = ∪_{j≤i} K_{j},

so that IntK_{i}^{∗} = ∪_{j≤i} IntK_{j} is a disjoint union and ∂K_{i}^{∗} = ∪_{j≤i} ∂K_{j}.

We define NS^{i} = ℝ^{N} \ (E + ℤ^{N} + ∂K_{i}^{∗}) and Σ^{i} = K_{i}^{∗} + E. So for each u ∈ NS^{i} we have P̃_{u}^{i} = ℤ^{N} ∩ Σ^{i} and P_{u}^{i} = π(P̃_{u}^{i}). From these we construct Π̃_{u}^{i}, Π_{u}^{i}, MP_{u}^{i} and so on, as usual. In fact, in the following, we shall be interested only in the strip pattern P̃_{u}^{i}.

Provided u is non-singular for K, and hence is non-singular for all K_{i}^{∗}, we can take a space F complimentary to E in ℝ^{N} and a corresponding group G_{u} (Def. I.10.2) which will play their usual roles for all sets of projection data (E, ℝ^{N}, K_{i}^{∗}) and (E, ℝ^{N}, K). For each domain, K_{i}^{∗}, we construct the corresponding F_{u}^{i} etc. The following lemma follows easily from the definitions.

Lemma 2.1 Suppose j < i is fixed throughout the statement of this lemma. Then NS^{i} is a dense subset of NS^{j} and NS = ∩_{k}NS^{k}. Moreover, for u ∈ NS^{i}, we have a natural continuous E + ℤ^{N} equivariant surjection Π̃_{u}^{i} ⟶ Π̃_{u}^{j}, and a natural continuous E equivariant surjection MP̃_{u}^{i} ⟶ MP̃_{u}^{j}; this latter is described equivalently by the formula S ↦ S ∩ Σ^{j}.

We also have an ℤ^{N}-equivariant map F_{u}^{i} ⟶ F_{u}^{j}, and a G_{u}- equivariant map X_{u}^{i} ⟶ X_{u}^{j}. All these maps respect the commutative diagrams of Chapter I and they map many-to-one only when the image is in (the appropriate embedding of) NS^{j} \ NS^{i}. □

Theorem 2.2 With the notation and assumptions above, we have the following equivariant homeomorphisms.

(a) Π̃_{u} ≅ lim_{←} Π̃_{u}^{i}, E + ℤ^{N} equivariantly;

(b) MP̃_{u} ≅ lim_{←} MP̃_{u}^{i}, E-equivariantly;

(c) F_{u} ≅ lim_{←} F_{u}^{i}, ℤ^{N}-equivariantly;

(d) X_{u} ≅ lim_{←} X_{u}^{i}, G_{u}-equivariantly.

Proof Once again the results are straightforward from the definitions. The map MP̃_{u} ⟶ MP̃_{u}^{i} is again equivalently written S ↦ S ∩ Σ^{i}. □

The following is now a direct consequence of (4.2), (2.2)(b) and the continuity of Čech cohomology on inverse limits.

Corollary 2.3 There is a natural equivalence

H^{∗}(GP̃_{u}) = lim_{→} H^{∗}(GP̃_{u}^{i}).  □

3 Cohomology in the case d=N–1

In this section we determine the cohomology for projection method patterns when d = N − 1. It is the only case for which we have such a complete answer.

Here E is a codimension 1 subspace of ℝ^{N} and so E ∩ ℤ^{N} = 0 implies that Δ = 0. Therefore, MP_{v} = MP_{u} = MP̃_{u} for all u, v ∈ NS and so, given E and K, there is only one projection pattern torus MP to consider, no need to parametrise by u, and an equation S_{P} = ℤ^{N} (I.8.2). With this in mind, we shall avoid further explicit mention of any particular non-singular point u.

Write e_{1}, ..., e_{N} for the usual unit vector basis of ℝ^{N}, which are also the generators of ℤ^{N}. Choose the space F as that spanned by e_{N}, and so G_{T} = ⟨e_{N}⟩ and G_{u} = ⟨e_{1}, e_{2}, ..., e_{N−1}⟩. Recall that we write K′ = π′(K) ⊂ F, where π′ is the skew projection onto F parallel to E and that π′ maps K homeomorphically to K′, preserving the boundary, ∂K′ = π′(∂K) ≅ ∂K.

Now any compact subset of E^{⊥} ≡ ℝ which is the closure of its interior is a countable union of closed disjoint intervals; and K is such a set. Thus ∂K and hence ∂K′ is countable. Pick A = {p_{1}, p_{2}, ...}, p_{j} ∈ ∂K′, to be a set of representatives of π′(ℤ^{N}) orbits of ∂K′, a countable and possibly finite set. Write k ∈ ℤ_{+}∪∞ for the cardinality of A.

Theorem 3.1 If T is a projection method pattern with d = N − 1 and E ∩ ℤ^{N} = 0, then

H^{m}(GT ) = H^{m}(𝕋^{N} \ k points ) = \{ \begin{matrix}ℤ^{\binom{N}{m}} & for m ≤ N − 2, \\ ℤ^{N+k−1} & for m = N − 1, \\ 0 & otherwise. \end{matrix}

An infinite superscript denotes the countably infinite direct sum of copies of ℤ.

Proof We know that IntK′ is the union of a countable number of open intervals, whose closures, K_{i}, are disjoint. We use the notation and results of Section 2, setting K_{i}^{∗} = ∪_{j≤i} K_{j} as the finite union of disjoint closed intervals, ∪_{j≤i}[s_{j}e_{N}, r_{j}e_{N}] say. As MT = MP̃, by (2.3)

it is enough to compute the direct limit lim_{→} H^{∗}(MP̃^{i}).

We consider the process of completion giving rise to the space MP̃^{i} which we consider as Π̃^{i} (the completion of the non-singular points NS^{i}) modulo the action of ℤ^{N}. The limit points introduced in Π̃^{i} arise as the limit of patterns P_{x_{n}}^{i} as x_{n} approaches a singular point, either from a positive e_{N} direction, or from a negative one. To be more precise, suppose that x_{n} = x+t_{n}e_{N} ∈ NS is a sequence converging to x ∈ ℝ^{N} in the Euclidean topology. If (t_{n}) is an increasing sequence, then lim_{n→∞}P̃_{x_{n}}^{i} exists in the D metric and is the point pattern (x +

ℤ^{N}) ∩ ( ∪_{j≤i}(s_{j}e_{N}, r_{j}e_{N}] + E).

Likewise, if (t_{n}) is a decreasing sequence then lim_{n→∞}P̃_{x_{n}}^{i} is the

point pattern (x + ℤ^{N}) ∩ ( ∪_{j≤i}[s_{j}e_{N}, r_{j}e_{N}) + E). These two patterns

are the same if and only if x ∈ NS^{i}. If x ∉ NS^{i} then these two patterns define the two D limit points in Π̃^{i} over x ∈ ℝ^{N}. Thus the quotient MP̃^{i} ⟶ 𝕋^{N} is 1-1 precisely when mapping to the set NS^{i}/ℤ^{N}, and otherwise it is 2-1; we can picture the map intuitively as a process of “closing the gaps” made by cutting 𝕋^{N} along the finite set of hyperplanes (∂K_{i}^{∗} + E)/ℤ^{N}, c.f. [Le].

We examine the space MP̃^{i} in more detail. Given r > 0, consider the space M_{r}^{i} = {(S ∩ B(r)) ∪ ∂B(r) : S ∈ MP̃_{u}^{i}} endowed with the Hausdorff metric d_{r} among the set of all closed subsets of B(r), the closed ball in ℝ^{N} with centre 0 and radius r. By construction M_{r}^{i} is compact and, for s ≥ r and i ≥ j, there are natural restriction maps M_{s}^{i} ⟶ M_{r}^{j}, whose inverse limit for fixed i = j is MP̃^{i}, and whose inverse limit over all i and r by (2.2) is MP̃. The map M_{r}^{i} ⟶ 𝕋^{N} given by P̃_{v} ↦ v mod ℤ^{N}, v ∈ NS^{i}, factors the canonical quotient MP̃^{i} ⟶ 𝕋^{N}.

Define C_{r}^{i} as the set {v ∈ 𝕋^{N} : (v + ℤ^{N}) ∩ (∂Σ^{i} ∩ IntB(r)) ≠ ∅}. As before, M_{r}^{i} ⟶ 𝕋^{N} is 2-1 precisely on those points mapped to C_{r}^{i} and otherwise is 1-1.

The intersection ∂Σ^{i} ∩ B(r) is, for all r large enough compared with the diameter of K_{i}^{∗}, equal to a finite union of codimension 1 discs, parallel to E, and of radius at least r−1, and at most r. Each of these discs has centre π^{⊥}(a) for some a ∈ ∂K_{i}^{∗}. Consider this collection of discs modulo ℤ^{N} and select two, say with centres π^{⊥}(a) and π^{⊥}(b), where a, b ∈ ∂K_{i}^{∗}. Then, for r very large and a − b ∈ π′(ℤ^{N}), these discs will overlap modulo ℤ^{N}. Since there are a finite number of such pairs in ∂K_{i}^{∗} to consider, we have a universal r such that if a, b ∈ ∂K_{i}^{∗} and a − b ∈ π′(ℤ^{N}), then the disc with centre π^{⊥}(a) overlaps, modulo ℤ^{N}, the disc with centre π^{⊥}(b). If a − b ∉ π′(ℤ^{N}), then these discs will not overlap, modulo ℤ^{N}. Hence for r sufficiently large, ∂Σ^{i} ∩ B(r) mod ℤ^{N} has precisely |A_{i}| components.

For the same r, C_{r}^{i} mod ℤ^{N} is also a finite union of discs of radius at least r − 1 and at most r; likewise C_{r}^{i} has exactly |A_{i}| components, in direct correspondence with the elements of A_{i}.

The description above of the limiting points in Π̃^{i} as we approach C_{r}^{i} in a direction transverse to E, shows that M_{r}^{i} is homeomorphic to 𝕋^{N} with a small open neighbourhood of C_{r}^{i} removed. There is a natural homotopy equivalence with the space 𝕋^{N} \ C_{r}^{i}.

We can now examine what happens as we let first r and then i tend to infinity in this construction. For the above sufficiently large r, the map M_{r+1}^{i} ⟶ M_{r}^{i} is, up to homotopy, the injection from 𝕋^{N} \C_{r+1}^{i} to 𝕋^{N} \C_{r}^{i}, and this is simply, up to homotopy, the identity from 𝕋^{N} \|A_{i}| points to itself. Hence H^{∗}(M_{r}^{i}) = H^{∗}(𝕋^{N} \|A_{i}| points) and H^{∗}(M_{r}^{i}) ⟶ H^{∗}(M_{r+1}) is the identity showing that H^{∗}(MP̃^{i}) is the cohomology of the torus with |A_{i}| punctures.

Finally, for each i and for r sufficiently large (depending on i) the map M_{r}^{i+1} ⟶ M_{r}^{i} is that induced by the inclusion of A_{i} in A_{i+1}, and this corresponds in the above description to the adding of a new puncture for each element of A_{i+1} \ A_{i}. In cohomology, the map H^{p}(M_{r}^{i−1}) ⟶ H^{p}(M_{r}^{i}) is thus the identity for p ≠ N − 1, and in dimension d gives rise to the direct system of groups and injections ⋯ ⟶ ℤ^{N−1+|A_{i}|} ⟶ ℤ^{N−1+|A_{i+1}|} ⟶ ⋯ which gives the required formula. □

We give an alternative proof of this theorem from a different perspective in Chapter V.

We note that the pattern dynamical system (X, G_{1}) is in fact a Denjoy example [PSS] generalized to a ℤ^{N−1} action and dislocation along k separate orbits.

Corollary 3.2 Suppose that Γ is a dense countable subgroup of ℝ finitely generated by r free generators. Suppose we Cantorize ℝ by cutting and splitting along k Γ-orbits (as described e.g. in [PSS]) to form the locally Cantor space R′ on which Γ acts continuously, freely and minimally. Consider the Γ-module C of compactly supported integer valued functions defined on R′. Then H^{∗}(Γ, C) = H^{∗}(𝕋^{r} \ k points; ℤ). □

IV Approaches to Calculation II: Infinitely Generated Cohomology

1 Introduction

In this chapter we restrict ourselves to the classical projection tilings with canonical acceptance domains K (so K is the projection of the unit cube in ℝ^{N}). We examine the natural question of when such tilings arise also as substitution systems and show that the invariants of chapter 2 are effective and computable discriminators of such tiling properties.

Much of this chapter is devoted to giving a qualitative description of the cohomology of canonical projection method patterns. The main result is formulated in Theorem 2.9. It gives a purely geometric criterion for infinite generation of pattern cohomology and for infinite rank of its rationalisation. As a corollary of this, by the obstruction proved in II.5, we deduce that almost all canonical projection method patterns fail to be substitution systems; in fact for vast swathes of initial data all such patterns fail to be self similar.

The restriction to the canonical acceptance domain allows for a second, geometric description of the coefficient groups CF_{u} (I.9.3) and CV_{u} (II.4.3) in terms of indicator functions on particular polytopes, we call them C_{u}-topes. In Section 2 we introduce this viewpoint, setting up the notation and definitions sufficient to state the main theorem. From here until the end of Section 5, our aim is to prove Theorem 2.9 by establishing criteria for the existence of infinite families of linearly independent C_{u}-topes. In Sections 3 and 4 we construct such families in the indecomposable case, and complete the analysis for the general case in Section 5. The final Section 6 gives some general classes of patterns where the conditions of Theorem 2.9 are satisfied, so proving the generic failure of self-similarity for canonical projection method patterns.

2 The canonical projection tiling

For the first time in our studies, we narrow our attention to the classical projection method tilings of [OKD] [dB1]. This section outlines the simplifications to be found in this case, and describes the main result of the remainder of this paper; a sufficient condition for infinitely generated H^{d}(GT ). In Chapter V we will see that this condition is also neccessary.

Therefore we have data (K, E, u) where K = π^{⊥}([0, 1]^{N}) and u ∈ NS. From Section I.8 we see that, but for a few exceptional cases, we have S_{T} = ℤ^{N}, and if we elect either to exclude these exceptions (as most authors do) or to include them only in their most decorated form (MT = MP̃_{u}), we make S_{T} = ℤ^{N} a standing assumption.

With Theorem I.9.4 we have a description of the topology of CF_{u}: it is generated by intersection and differences of shifts of a single compact open set \overline{K}, formed by completing π′K ∩(Q+u)∩NS (Lem. I.9.6). Topologically CF_{u} ≡ CV_{u} × Δ̃ and CV_{u} inherits the subspace topology and a stabilizing subaction Γ_{T} of S_{T} (see II.4.3).

With the choice of K as a canonical acceptance domain above, we may follow more closely the work of Le [Le] and give other more geometrical descriptions of the elements of CF_{u} and CV_{u}.

Definition 2.1 For each J ⊂ {1, 2, ..., N}, we construct a subspace e^{J} = ⟨e_{j} : j ∈ J⟩ (the span) of ℝ^{N}, where {e_{j}} is the standard unit basis of ℝ^{N} or ℤ^{N}.

Write dim F = n.

Let I = {J ⊂ {1, 2, ..., N} : dim π′(e^{J}) = n − 1} and define I^{∗} to be the set of elements of I minimal with respect to containment.

Define ℤ_{n−1}^{N} = ∪{e^{J} + v : |J| = n − 1, v ∈ ℤ^{N}}, i.e. the n − 1dimensional skeleton of the regular cubic CW decomposition of ℝ^{N}.

The following Lemma gives some combinatorial information about I^{∗} and describes the singular points in F - they are formed by unions of affine subspaces of F of codimension 1.

Lemma 2.2 With the construction above,

i/ I^{∗} is a sub-collection of the n−1-element subsets of {1, 2..., N}. Also every subspace of F of the form π′(e^{J}), with |J| = n − 1, is contained in π′(e^{J′}) for some J′ ∈ I^{∗}.

ii/ ℝ^{N} \ NS = π^{′−1}π′(ℤ_{n−1}^{N}) and π′(ℤ_{n−1}^{N}) = F \ NS.

Proof i/ Straightforward.

ii/ With the data above, π′(K) is a convex polytope in F, with interior, each of whose extreme points is of the form π′(v) where v ∈ {0, 1}^{N}. By i/, each of the faces of π′(K) is contained in some π′(e^{J} + v) where v ∈ {0, 1}^{N} and J ∈ I^{∗}. Also by i/, we have π′(ℤ_{n−1}^{N}) = π′(∪{e^{J} + v : v ∈ ℤ^{N}, J ∈ I^{∗}}).

However, by definition, F \ NS is the union of the faces of those polytopes of the form π′(K+v), v ∈ ℤ^{N}. Thus we deduce immediately that π′(ℤ_{n−1}^{N}) = π′(∪{e^{J} + v : v ∈ ℤ^{N}, J ∈ I^{∗}}) ⊃ F \ NS.

Conversely, a simple geometric argument shows that for each J ∈ I^{∗}, there is a face of π′(K) which is contained in π′(e^{J} + v) for some v ∈ ℤ^{N}.

Assuming the claim, we have, for each J ∈ I^{∗} and each v ∈ ℤ^{N}, some suitable shift of π′(K), π′(K + w) say, w ∈ ℤ^{N}, one of whose faces, Φ say, contains the point π′(v) as an extreme point, and Φ ⊂ π′(e^{J} + v). With this same Φ, therefore, we see that ∪{Φ + π′(w) : w ∈ ℤ^{N}} ⊃ ∪{π′(e^{J} + w) : w ∈ ℤ^{N}}. Thus π′(ℤ_{n−1}^{N}) = ∪{π′(e^{J} + v) : v ∈ ℤ^{N}, J ∈ I^{∗}} ⊃ F \ NS. So we are done. □

Definition 2.3 Recall the subspace V from (I.2) and write dim V = m.

Now consider the set I^{∗}(V) = {J ∈ I^{∗} : dim(π′(e^{J}) ∩ V) = m − 1}.

The following Lemma shows that the sets I^{∗}(V) can be found canonically from the sets I^{∗} and describes the singular points in V + π′(u) preparatory to a description of V_{u} (See II.4.3). Once again, the singular points in V +π′(u) form a union of affine hyperplanes in V +π′(u).

Lemma 2.4 i/ I^{∗}(V) = {J ∈ I^{∗} : π′(e^{J}) ∩ V ≠ V }.

ii/ If u ∈ NS, then (V + π′(u)) \ NS = (V + π′(u)) ∩ π′(ℤ_{n−1}^{N}) = (V + π′(u)) ∩ (∪{π′(e^{J} + v) : v ∈ ℤ^{N}, J ∈ I^{∗}(V)}).

Proof Follows directly from definition of V and Lemma 2.2 □

With these notations in mind, we are well equipped to describe the topology of V_{u}. Although V_{u} is best described as a subspace of F_{u} formed from placing V as the affine subspace V +π′(u) in F, we prefer to shift the whole construction back to V by applying a uniform shift by −π′(u). There are some advantages later in having an origin and a vector space structure.

Definition 2.5 Given u ∈ NS, we use 2.4 i/ to define a set, C_{u}, of m − 1-dimensional affine subspaces of V whose elements are of the form (π′(e^{J} + v) ∩ (V + π′(u))) − π′(u) where J ∈ I^{∗}(V) and v ∈ ℤ^{N}. Such a space may also be written, π′(e^{J} + v − u) ∩ V.

We say that a subset of V is a C_{u}-tope, if

i/ it is compact and is the closure of its interior, and

ii/ it is a polytope, each of whose faces is a subset of some element of C_{u}.

We shall also say that a subset, B, of V_{u} is a C_{u}-tope if B is clopen and ν(B) − π′(u) is a C_{u}-tope subset of V in the sense above (recall ν from (I.9.2) above).

We shall show that the C_{u}-topes generate the topology on V_{u}. To help this we describe a third possible topology as it is found in [Le].

For each element, α, of C_{u}, a hyperplane in V, consider the halfspaces H_{α}^{±} defined by it. The sets H_{α}^{±} ∩ NS can be completed in the \overline{D} metric to a closed and open subset of V_{u} which we write \overline{H}_{α}^{±}. We also call these subsets of V_{u} half-spaces.

Proposition 2.6 The following three collections of subsets of V_{u} are the same:

i/ The collection of C_{u}-topes.

ii/ The collection of compact open subsets of V_{u}.

iii/ The collection of those finite unions and intersections of halfspaces in V_{u} which are compact.

Proof Every C_{u}-tope in V_{u} is clearly an element of collection iii/ since every C_{u}-tope in V is a finite intersection and union of half spaces in V.

Conversely, every open half-space in V is a countable, locally finite, union of the interiors of C_{u}-topes (in V). Therefore any precompact intersection of half spaces can be formed equivalently from some union and intersection of a finite collection of C_{u}-topes, i.e. is a C_{u}-tope, and each C_{u}-tope in V_{u} is an intersection of clopen halfspaces.

Now compare collection ii/ with iii/ and i/.

Suppose that \overline{H} is a half-space in V_{u} defined by the hyperplane W ∈ C_{u}. As noted in 2.5, W has the form π′(e^{J} + v − u) ∩ V for some J ∈ I^{∗}(V) and v ∈ ℤ^{N}, and our choice of half-space gives a corresponding choice of half-space in V on one side or other of W. Since J ∈ I^{∗}(V) ⊂ I^{∗} (2.4 i/ ), this in turn defines a choice of halfspace in F one side or other of π′(e^{J} + v − u). Write H for the closed half-space of F chosen this way.

Consider the collection of N-dimensional cubes in the ℤ^{N} lattice whose image under the projection π′ is contained entirely in H + π′(u). By Lemma 2.4 ii/, the boundary of H + π′(u) is an (n − 1)-dimensional hyperplane in F, the image under π′ of an (n − 1)- dimensional subspace of ℝ^{N} contained in ℤ_{n−1}^{N}. Therefore the cubes collected above line up against this boundary exactly and, projected under π′, they cover H + π′(u) exactly. In particular, H + π′(u) is covered by Γ_{T} -translates of π′([0, 1]^{N}). Restricting this construction to the space (V + π′(u)) ∩ NS, we find that \overline{H} is a locally finite countable union of shifts of \overline{K}. In short, collection iii/ is contained in collection ii/.

Conversely, we see that π′([0, 1]^{N})∩NS = π′((0, 1)^{N})∩NS so we are sure that the intersection (V + π′(u)) ∩ ((π′([0, 1]^{N})) + π′(v)) (for any choice of v ∈ ℤ^{N}) is a polytope subset of V + π′(u) with interior. The faces of this polytope are clearly subsets of NS, i.e. contained in elements of C_{u} shifted along with V by π′(u). Thus every Γ_{T} translate of \overline{K} intersects V_{u} either non-emptily or as a C_{u}-tope. Collection ii/ is contained in collection i/ therefore. □

Since V_{u} is locally compact, we can describe the topology on V_{u}, defined in II.4.3, equivalently as that generated by C_{u}-topes, or as that generated by half-spaces. This gives an alternative description of CV_{u}.

Corollary 2.7 Let V′ = (V + π′(u)) ∩ NS) − π′(u). The group CV_{u} is naturally isomorphic to the group of integer-valued functions, V′ ⟶ ℤ, generated by indicator functions of sets of the form C ∩ V′ where C is a C_{u}-tope. This group isormophism is a ring and ℤ[Γ_{T}]- module isomorphism as well. □

Now we describe an important set of points.

Definition 2.8 Write P for the set of points in V which can found as the 0-dimensional intersection of m elements of C_{u}. Note that, under the assumptions on C_{u}, P is a non-empty countable set, invariant under shifts by Γ_{T}.

Say that P is finitely generated if P is the disjoint union of a finite number of Γ_{T} orbits, infinitely generated otherwise.

We may now express the main theorem of this Chapter.

Theorem 2.9 Given a canonical projection method pattern, T and the constructions above. If P is infinitely generated, then H^{d}(GT ) is infinitely generated and H^{d}(GT ) ⊗ ℚ is infinite dimensional.

In Chapter V (Theorem V.2.4) we prove the converse i.e. that H^{d}(GT ) infinitely generated is equivalent to P infinitely generated.

We complete the proof of Theorem 2.9 in Section 5, but the final step and basic idea, of the proof can be presented already.

Proposition 2.10 Suppose that G is a torsion-free abelian group and that H is an abelian group, with H/2 := H/2H its reduction mod 2. Suppose that there is a group homomorphism, ϕ : G ⟶ H/2, such that Imϕ is infinitely generated as a subgroup of H/2 (equivalently infinite dimensional as a ℤ/2 vector space); then G is infinitely generated as an abelian group, and G ⊗ ℚ is infinitely generated as a ℚ vector space.

Proof It is sufficient to prove the statement concerning G ⊗ ℚ.

Suppose that ϕ(s_{n}) is a sequence of independent generators for Imϕ and suppose that there is some relation

∑_{n=1}^{m} q_{n}s_{n} = 0

for q_{n} ∈ ℚ. Since G is torsion-free, we can assume the q_{n} are integers and have no common factor; in particular, they are not all even. Applying the map ϕ then gives a non-trivial relation among the ϕ(s_{n}), a direct contradiction, as required. □

Therefore, to prove 2.9, we shall find a homomorphism from H^{d}(GT ), equivalently H_{0}(Γ_{T} , CV_{u}), to an infinite sum of ℤ/2 whose image is infinitely generated. This is completed in full generality in Theorem 5.4. In order to construct independent elements of H_{0}(Γ_{T} , CV_{u}) and its image, we must consider the geometry of the C_{u}-topes defined in 2.5.

3 Constructing C-topes

To prove that a group or ℚ-vector space is infinitely generated, we must produce independent generators. In the case of H_{0}(Γ_{T} , CV_{u}), we must find elements of CV_{u} which remain independent modulo Γboundaries. In any case, we must at least produce some elements of CV_{u}, and in this section we start with constructions of the simplest objects in this space: indicator functions of C_{u}-topes.

Rather than refer constantly to the original tiling notation, we abstract the construction conveniently, basing our development on a general collection of affine hyperplanes, C, of a vector space, V, with group action, Γ. Always, the example in mind is C = C_{u} (2.5), V (I.2) and Γ = Γ_{T} (II.4.3), but the construction is potentially more general. However, the first few definitions and constructions are the slightest generalization of those of section 2.

Definition 3.1 Suppose that V is a vector space of dimension m and that Γ is a finitely generated free abelian group acting minimally by translation on V. Thus we write w ↦ w + γ for the group action by γ ∈ Γ, and we may think of Γ as a dense subgroup of V without confusion.

Suppose that C is a countable collection of affine subspaces of V such that each W ∈ C has dimension m − 1, and such that, if W ∈ C and γ ∈ Γ, then W + γ ∈ C.

We suppose that the number of Γ orbits in C is finite.

If W ∈ C, then we define a unit normal vector, λ(W) (with respect to some inner-product). The set N (C) = {λ(W) : W ∈ C} is finite and we suppose that we have chosen the λ(W) consistently so that −λ(W) ∉ N (C).

We suppose that N (C) generates V as a vector space.

We can consider the intersections of elements of C.

Definition 3.2 Given 0 ≤ k ≤ m−1, define C^{(k)} to be the collection of k-dimensional affine subspaces of V formed by intersection of elements of C. Thus C can be written C^{(m−1)}, and, to be consistent with the notation of section 2, C^{(0)} can be written P.

Define a singular flag, F, to be a sequence of affine subspaces, (θ_{0}, θ_{1}, ..., θ_{m−1}) of V such that θ_{j} ∈ C^{(j)} for all 0 ≤ j ≤ m − 1, and θ_{j} ⊂ θ_{j+1} for all 0 ≤ j ≤ m−1. The set of singular flags is written J_{o}.

It is clear that each C^{(k)} and J_{o} supports a canonical Γ action. We write J = J_{o}/Γ, the set of Γ orbits in J_{o}.

Definition 3.3 We say that a subset, C, of V is a C-tope, if i/ C is compact and is the closure of its interior, and ii/ C is a polytope, each of whose (m − 1)-dimensional faces is a

subset of some element of C.

Definition 3.3 Convex C-topes are finite objects whose geometry and combinatorics are immediately and intuitively related. Therefore we think of the definition of face, edge and vertex, and more generally k-dimensional face, in this case, as the intuitive one. Likewise the idea of incidence of edge on vertex, face on edge, etc is intuitive.

Suppose that C ⊂ V is a convex C-tope and that F = (θ_{j}) is a singular flag. We say that F is incident on C if each θ_{k} contains a k-dimensional face of C.

We write [F] for the Γ-orbit class of F. We say that F is uniquely incident on C if F is incident on C, but no other F′ ∈ [F] is incident on C.

The main aim of this section is to build convex C topes on which certain singular flags are uniquely incident. This will not happen in every possible circumstance however, and we develop an idea of decomposability which will break up the space V into a direct sum of spaces on which such constructions can be made. In section 5 we shall show how to recombine these pieces.

Construction 3.5 Suppose that A is a finite set of non-zero vectors in V which spans V, and no pair of which is parallel. The example we have in mind is N (C), the set of normals.

A decomposition of A is a partition A = A_{1} ∪ A_{2} such that V_{1} ∩ V_{2} = 0 where each V_{j} is the space spanned by A_{j}, j = 1, 2.

A is indecomposable if no such decomposition is possible. It is not hard to show that every set A has a unique partition into indecomposable subsets.

Suppose that B ⊂ A is a basis for V. Then, by stipulating that B is an orthonormal basis, we define an inner product which we write as square brackets: [., .]_{B}.

Then we define a finite graph G(B; A) with vertices B and an edge from x to y whenever there is a z ∈ A \ B such that [x, z]_{B} ≠ 0 and [y, z]_{B} ≠ 0. We do not allow loops.

The following is elementary.

Lemma 3.6 Suppose that A spans V, then the following are equivalent:

i/ A is indecomposable

ii/ for all bases B ⊂ A, G(B; A) is connected

iii/ for some basis B ⊂ A, G(B; A) is connected. □

Remark 3.7 Note that if ϕ : V ⟶ V is a linear bijection, then A is indecomposable if and only if ϕ(A) is indecomposable. Therefore, the condition N (C) indecomposable can be stipulated without reference to a particular inner product, although an inner product must be used to define the normals. We use this freedom in Theorem 3.10 ahead.

Construction 3.8 Suppose that A and B are as above, giving an inner product [., .]_{B} to V and defining a graph G(B; A). Choose b ∈ B and let W_{b} be the hyperplane in V orthogonal to b and let π_{b} be the orthogonal projection of V onto W_{b}.

Consider the sets π_{b}(A) and π_{b}(B). Apart from 0 = π_{b}(b) the latter equals B \ b precisely, an othonormal basis for W_{b}. The former contains B \ b and other vectors which may be of various lengths and pairs of which may be parallel. From this set, we form A_{b} a new set of vectors in W_{b} by taking, for each class of parallel elements of π_{b}(A) a single unit length representative (ignoring 0). If the class in question is one which contains some b′ ∈ B \ b, then we let b′ be the representative chosen. Write B_{b} = B \ b, so that B_{b} ⊂ A_{b} is a basis.

We consider the sets A_{b}, B_{b} in W_{b} and form the graph G(B_{b}; A_{b}) with respect to the inner product [., .]_{B_{b}}, the restriction of the inner product [., .]_{B} to W_{b}.

If C is a collection of affine hyperplanes in V, then we define C_{b} to be collection of affine hyperplanes in W_{b} of the form W_{b} ∩W where W ∈ C is chosen so that W is not parallel to W_{b}.

The following useful lemma comes straight from the definition.

Lemma 3.9 i/ The graph G(B_{b}; A_{b}) is formed from G(B; A) by removing the vertex b and all its incident edges from G(B; A).

ii/ If A = N (C), then A_{b} = N (C_{b}).

Theorem 3.10 Suppose that m = dim V > 1 and that 0 ∉ A ⊂ V spans V, and A has no parallel elements. Suppose that B ⊂ A is a basis for V and that G(B; A) is connected, then there is a closed convex polytope, C, of V, with interior, such that

i/ The normal of each face of C is an element of A.

ii/ v is a vertex of C which is at the intersection of exactly m faces of C and each of these faces is normal to some element of B.

iii/ The vertex v is uniquely defined by property ii/.

(All normals are taken with respect to the inner product [., .]_{B}.)

Proof We suppose, without loss of generality, that v = 0 and proceed by induction on m = |B| ≥ 2.

Suppose that |B| = 2. By graph connectedness, we find a ∈ A\B with non-zero components in each B coordinate direction. Thus we can construct easily a triangle, C, in V with the required properties.

For larger values of |B|, we proceed as follows.

As in Construction 3.8, define, for each b ∈ B, W_{b} to be the hyperplane normal to b and let π_{b} : V ⟶ W_{b} be the orthogonal projection.

A simple argument shows that we can find b_{o} ∈ B so that G(B; A) \ b_{o} (i.e. removing the vertex b_{o} and all incident edges) is connected. Thus, appealing to the description of Lemma 3.9 and by induction on |B|, we can find a convex compact polytope subset, C_{1}, of W_{b_{o}} which obeys conditions i/ to iii/ above with respect to the basis B \b and hyperplanes with normals parallel to some π_{b_{o}}(a) : a ∈ A\b.

Suppose that W′ is a hyperplane in W_{b_{o}} which contains a face of C_{1} and has normal π_{b_{o}}(a) for some a ∈ A. Then W′ is in fact of the form W ∩ W_{b_{o}}, where W is the unique hyperplane in V, normal to a, containing the space W′. Indexing the faces of C_{1} with j say, each face is contained in W′_{j} with normal π_{b_{o}}(a_{j}). So construct a hyperplane W_{j} in V as above, with normal a_{j}. In the case of the faces of C_{1}, incident on 0 and normal to b ∈ B \ b_{o} say, we make sure that we choose a_{j} = b, so that in this case W_{j} = W_{b} as defined above.

Now we build C_{2}, a convex closed subset of V, as follows. For each face j of C_{1}, let H_{j} be the closed half-space defined by W_{j}, containing the set C_{1}. Let C_{2} = ∩_{j}H_{j}, where the intersection is indexed over all faces of C_{1}, so that C_{1} = C_{2} ∩ W_{b_{o}}.

C_{2} is almost certainly unbounded, but it is important to note that none of its faces is normal to b_{o}. Furthermore, by construction of C_{1}, there is only one edge in C_{2} which sits at the intersection of m − 1 hyperplanes normal to some element of B \ b_{o}, and this edge contains the point 0 in its interior and is parallel to b_{o}.

To produce a bounded set, we intersect C_{2} with a compact convex polytope C_{4} whose faces are mostly normal to elements of B. We construct C_{4} as follows.

For each b ∈ B \ b_{o}, let H_{b} be the half space of V defined by the hyperplane W_{b} and containing C_{1}. Now let W_{b}^{+} an affine translation of W_{b} beyond the other side of C_{1}. Precisely, we argue as follows: since W_{b} contains a face of the convex set C_{1}, either [b, c]_{B} ≥ 0 for all c ∈ C_{1}, or [b, c]_{B} ≤ 0 for all c ∈ C_{1}; we construct W_{b}^{+} in the first case leaving the second case to symmetry. Since C_{1} is compact, there is an upper bound r > [b, c]_{B} for all c ∈ C_{1}. Let W_{b}^{+} = rb + W_{b}. Now let H_{b}^{+} be the half space defined by W_{b}^{+} and containing C_{1}.

Let H_{b_{o}} be a half space of V defined by W_{b_{o}}; it doesn’t matter which one. The intersection,

C_{3} = ⋂_{b∈B} H_{b} ∩ ⋂_{b∈B\b_{o}} H_{b}^{+}

is therefore a semi-infinite rectangular prism whose semi-infinite axis runs parallel to b_{o} and whose base contains C_{1}.

We now form an oblique face to truncate this prism. Let a_{o} ∈ A \ b_{o} be chosen so that [a_{o}, b_{o}]_{B} ≠ 0, as can indeed be done by the assumed connectedness of G(B; A). Let W_{a_{o}} be an affine hyperplane of V with normal a_{o}; we shall detail its placement soon. Note that, by constuction, W_{a_{o}} intersects every line parallel to b_{o}, in particular we may place W_{a_{o}} so that it intersects (the interior of) all the edges of C_{3} parallel to b_{o}. Let H_{a_{o}} be the half-space defined by W_{a_{o}} and containing C_{1}. Let C_{4} = H_{a_{o}} ∩ C_{3}.

The properties of C_{4} are summarized: C_{4} is compact and convex: all its faces, except exactly one, are normal to some element of B; exactly one of its faces is normal to b_{o} and that face contains C_{1}: 0 is a vertex of C_{4} and it is at the intersection of exactly m faces each normal to some element of B: the other vertices of C_{4} for which this can be said are in W_{b_{o}} but are all outside C_{1}.

Let C = C_{2} ∩ C_{4}. This is clearly a compact convex polytope in V and C_{1} is a face of C. By construction, the point 0 is a vertex of C and obeys property ii/. Any other vertex of C with property ii/ must be found in C_{1} since there are no other faces of C_{2} or C_{4} normal to b_{o}. However, no other vertex of C_{1} can have property ii/ by its original definition. □

Although the theorem above makes no reference to group actions, we can adapt it for use in section 4.

Theorem 3.11 Suppose C is a collection of hyperplanes in V, dim V > 1, with Γ action as in Def. 3.1, and suppose that N (C) is indecomposable. Suppose that F is a singular flag (3.2). Then there is a convex C-tope on which F is uniquely incident.

Proof A singular flag is a descending sequence of singular spaces and so we can also express it as the sequence of spaces,

W_{1}, W_{1} ∩ W_{2}, ∩_{1≤i≤3}W_{i}, …, ∩_{1≤i≤k}W_{i}, …, ∩_{1≤i≤m}W_{i},

where W_{i} : 1 ≤ i ≤ m are transverse elements of C. Let {v} = ∩_{1≤i≤m}W_{i}.

Fix some inner product [., .] in V so that the W_{i} are orthogonally transverse and let B = {λ(W_{i}) : 1 ≤ i ≤ m} , where the normal λ is taken with respect to this inner product. B is an orthonormal basis for V. We define N (C) with respect to this inner product (recall Remark 3.7 above) and, to fit it into past notation, let A = N (C).

By hypothesis, A is indecomposable. Thus the graph G(B; A) is connected, by 3.6, and so we may form by 3.10 a convex polytope, C_{o}, in V with the properties outlined in 3.10.

i/ The normal of each face of C_{o} is an element of A.

ii/ v is a vertex of C_{o} which is at the intersection of exactly m faces of C_{o} and each of these faces is normal to some element of B.

iii/ The vertex v is uniquely defined by property ii/.

However, we know that the orbit of an element, W, of C is dense in V in the sense that for every affine hyperplane, W′, of V, parallel to W, and every ϵ > 0, there is an W′′, in the Γ orbit class of W, such that W′ and W′′ are separated by a vector of length at most ϵ. Therefore we may adjust C_{o} slightly without disturbing the combinatorial properties of its faces to form a C-tope, C, with the same properties. The vertex v need not be disturbed atall.

However, then it is clear that F is uniquely incident on C. □

4 The indecomposable case

We continue to consider the abstracted situation of section 3 and reintroduce the analysis of 2.6 and 2.7 as a definition.

Definition 4.1 In the space, V′ = V \ ∪{W : W ∈ C}, define A_{C} to be the collection of subsets of V′ of the form C ∩ V′, where C is a C-tope, and with the empty set thrown in as well.

We write CV_{C} for the ring of integer-valued functions generated by indicator functions of elements of A_{C}.

Compare this definition with the construction of the topology of V_{u} in section 2.

Recall the set of singular flags, J_{o}, for V and C as above, the Γ action on J_{o}, and the set J of Γ orbits in J_{o}. This transfers by a coordinatewise action to a Γ action on groups such as ⊕_{J_{o}}ℤ/2, the J_{o}-indexed direct sum of ℤ/2. In this case (⊕_{J_{o}}ℤ/2)/Γ = ⊕_{J} ℤ/2 canonically.

The main results of this section (4.2) and (4.6) are two similar technical results. Here is the first.

Proposition 4.2 With the constructions and notation of section 3 we suppose that m = dim V > 1 and N (C) is indecomposable. Then, there is a Γ-equivariant homomorphism

ξ_{o} : CV_{C} → ⊕_{J_{o}}ℤ/2

with the following property: for each singular flag F there is an element e ∈ CV_{C} so that ξ_{o}(e) has value 1 at coordinate F and value 0 at all other coordinates F′ ∈ [F].

The proof follows directly from section 3. Our aim is to build a Γequivariant homomorphsism from CV_{C} to ⊕_{J_{o}}ℤ/2 with certain further properties. We use the following lemma to make the construction.

Lemma 4.3 i/ From 4.1 above, A_{C} is an algebra generated by sets of the form, C ∩ V′, where C is a convex C-tope.

ii/ There is a one-to-one correspondance between a group homomorphisms ξ : CV_{C} ⟶ G to an abelian group G and maps ξ′ from convex C-topes to G with the property that if C_{1} and C_{2} are interior disjoint convex C-topes and if C_{1} ∪ C_{2} is convex, then ξ′(C_{1} ∪ C_{2}) = ξ′(C_{1}) + ξ′(C_{2}).

iii/ If Γ acts homomorphically on G then ξ is Γ-equivariant whenever the corresponding ξ′ is Γ-equivariant.

Proof i/ Every C-tope can be decomposed into a finite interiordisjoint union of convex C-topes.

ii/ Using the decomposition of part i/ we can define a homomorphism ξ : A_{C} ⟶ G as follows: Take a C-tope, C, break it into interiordisjoint cover by convex C-topes, C_{j}. Let ξ(C ∩ V′) = ∑ ξ′(C_{j}).

To see that this is well-defined, consider a second such decomposition C = ∪_{i}C′_{i}. Let D_{i,j} be the closure in V of V′ ∩ C′_{i} ∩ C_{j}; this is either empty or a convex C-tope. With the convention ξ′(∅) = 0, the condition on ξ′ extends inductively to show that ξ′(C_{j}) = ∑_{i} ξ′(D_{i,j}) and ξ′(C′_{i}) = ∑_{j} ξ′(D_{i,j}). Therefore ∑_{j} ξ′(C_{j}) = ∑_{i} ξ′(C′_{i}) since both are equal to ∑_{i,j} ξ′(D_{i,j}).

It is clear that ξ is additive for interior-disjoint unions in A_{C}, and so extends to CV_{C}. It is straightforward to construct a map ξ′ from ξ.

iii/ Follows quickly from the construction of ii/. □

Proof of 4.2 It is easy to define a map ξ′ on convex C-topes which reflects the geometric idea of incidence of a singular flag. Suppose C is a convex C-tope, then define ξ′(C) to be an element of ⊕_{J_{o}}ℤ/2 whose entry at the F coordinate (F ∈ J_{o} of course) is 1 if and only if F is incident on C.

It is clear that ξ′ is Γ-equivariant.

To show that that ξ′ is additive, in the sense of Lemma 4.3 ii/, we consider two interior disjoint convex C-topes C_{1} and C_{2}, whose union is a convex C-tope, and fix a particular singular flag, F. Three cases should be checked, the third breaking down into two subcases.

Note first the general principle that a singular subspace (of any dimension, t say) containing a face (of dimension t) of C_{1} ∪ C_{2} must therefore contain a face (of dimension t) of C_{1} or a face (of dimension t) of C_{2}. The converse, of course, is not true as faces between C_{1} and C_{2} meet and fall into the interior of a higher dimensional faces of C_{1} ∪ C_{2}; and this is generally the only way that faces can be removed from consideration. So we have a second general principle that a singular subspace contains a face of C_{1} ∪ C_{2} when it contains a face of C_{1} and no face of C_{2} (matching dimensions always).

Case a/: F is incident neither on C_{1} nor on C_{2}. By our first general principle, it is immediate that F is not incident on C_{1} ∪ C_{2}.

Case b/: F is incident on precisely one of C_{1} or C_{2}. The second general principle above shows then that F is incident on C_{1} ∪ C_{2}.

Case c/: F = (θ_{j})_{0≤j<N} is incident on both C_{1} and C_{2}. Suppose that θ_{t} is the singular subspace (of dimension t) containing a face F_{1} (of dimension t) of C_{1} and a face F_{2} (of dimension t) of C_{2}. We analyse two possibilities: i/ F_{1} = F_{2}: ii/ F_{1} and F_{2} are interior disjoint (as subsets of θ_{t}). Note that we have used the convexity of C_{1} ∪ C_{2} and other assumed properties to make this dichotomy.

Case ci/: Here F_{1} is no longer a face of C_{1} ∪ C_{2} and θ_{t} does not contain a face of C_{1} ∪ C_{2}. Thus F is not incident on C_{1} ∪ C_{2}.

Case cii/: Here θ_{t} contains the face F_{1} ∪ F_{2} of C_{1} ∪ C_{2}. However, consider θ_{t−1} (note that t ≥ 1 automatically in case ii/) which contains a face of both C_{1} and C_{2} of dimension t−1. In fact, by convexity, this face must be F_{1} ∩ F_{2}. However, this set is not a t − 1 dimensional face of C_{1} ∪C_{2}, having been absorbed into the interior of the t-dimensional face F_{1} ∪ F_{2}.

Either way, in case c/ F is not incident on C_{1} ∪ C_{2}.

Combining all these three cases gives the additivity mod 2 required of ξ′. Therefore, we define a Γ-equivariant homomorphism ξ_{o} : CV_{C} ⟶ ⊕_{J_{o}}ℤ/2 as required.

The element e is provided by the construction of 3.11. Consider a convex C-tope, C produced by Theorem 3.11 from the singular flag F. The indicator function of C is an element of CV_{C} which we will write e. The properties claimed of e in the theorem follow automatically from unique incidence of F on C. □

Γ-equivariance of the homomorphism from Proposition 4.2 allows us to build the commuting diagram

CV_{C}  ⟶^{ξ_{o}} ⊕_{J_{o}}ℤ/2
↓^{q}  ↓^{q}
H_{0}(Γ; CV_{C}) ⟶^{ξ_{∗}} ⊕_{J}ℤ/2

to define ξ_{∗}, where the maps q quotient by the action of Γ. In the indecomposable case therefore, Theorem 5.4 below is a direct consequence of this observation and the reader who is only interested in the indecomposable case can jump directly to that theorem replacing the first words ”Given the data above” by the first sentence of Proposition 4.2. In the decomposable case, however, we need a lot more work to establish such a diagram.

We prove an analogous technical result for the case dim V = 1. It seems necessary and slightly surprising that its proof is not as directly geometric.

For the remainder of the section, therefore, we assume that dim V = 1, but, for technical reasons, we also relax the assumption that Γ is a subgroup of V. Rather Γ is a free abelian group acting minimally by translation, but some of these translations may be 0. In this case we decompose Γ = Γ_{0} ⊕ Γ_{1} where Γ_{1} acts minimally and freely and Γ_{0} fixes every point in V.

We note that in the case dim V = 1, the flags are simply single points from P (sections 2 and 3). Thus there is a canonical correspondence between J_{o} and P, which is clearly Γ-equivariant.

Construction 4.4 Consider the direct sum group L = ⊕_{J} ℤ whose elements can be considered as collections of integers indexed by elements of J, or equivalently by Γ-orbit classes in P. Let h_{j} : j = 1, 2, .., k (with the obvious adaption for k = ∞) be the canonical free generating set for L.

Let P_{1}, ...P_{k} be the Γ-orbit classes of P and choose x_{0} ∈ P_{1} and, for each 1 ≤ j ≤ k, also choose x_{j} ∈ P_{j} such that x_{j} > x_{0} (recall that P ⊂ V = ℝ in the case dim V = 1).

According to definition 4.1 above, V′ = V \ P, and the sets I_{j} = [x_{0}, x_{j}] ∩ V′ are elements of A_{C}. Thus the functions f_{j} which indicate respective I_{j}, are elements of CV_{C}.

This allows us to define a homomorphism β : L ⟶ CV_{C}, defined β(h_{j}) = f_{j} for each 1 ≤ j ≤ k. Consider ⊕_{Γ_{1}}L, the Γ_{1} indexed direct sum of copies of L, whose elements we shall consider as Γ_{1}-indexed elements of L, (g_{γ})_{γ∈Γ_{1}}, all but a finite number of which are 0. Note that, by correponding the coordinate indices Γ-equivariantly, ⊕_{Γ_{1}}L is Γ-equivariantly isomorphic to ⊕_{J_{o}}ℤ. (The Γ_{0} component of the action acts trivially).

Now consider the homomorphism β^{∗} : ⊕_{Γ_{1}}L ⟶ CV_{C}, defined as

β^{∗}((g_{γ})_{γ∈Γ_{1}}) = ∑_{γ∈Γ_{1}} γβ(g_{γ})

Lemma 4.5 With the construction above, β^{∗} is Γ-equivariant. i/ β^{∗} : ⊕_{Γ_{1}}L ⟶ CV_{C} is injective. ii/ The image of β^{∗} is complemented in CV_{C}.

Proof The proof relies on a construction whose proper generalization is made in Chapter 5. In the case dim V = 1, however, it is easy enough to describe directly.

Consider those subsets, I_{a,b}, of A_{C} formed from a C-tope interval [a, b] ∩ V′ (much as the sets I_{j} were formed above). In this case a, b ∈ P necessarily. Therefore to this set, we associate an element of ⊕_{P}ℤ namely one which is 0 at every coordinate except the a coordinate, where it is 1, and at the b coordinate, where it is −1. We write this element δ(I_{a,b}) = 1_{a} − 1_{b} in an obvious notation. This function is additive in the sense of Lemma 4.3 and so extends to a homomorphism δ : CV_{C} ⟶ ⊕_{P}ℤ. Moreover, δ is clearly Γ-equivariant on such intervals, and hence it is equivariant when extended to a group homomorphism.

Recall the elements, f_{j}, of CV_{C} defined before. Consider an equation of the form ∑_{1≤j≤k,γ∈Γ_{1}} t_{j,γ}γf_{j} = 0 (where t_{j,γ} ∈ ℤ equal 0 for all but a finite set of indices). If we apply δ to this, we find ∑_{1≤j≤k,γ∈Γ_{1}} t_{j,γ}γδ(f_{j}) = 0, a sum in ⊕_{P}ℤ, and we can now use the equation δ(f_{j}) = 1_{x_{0}} − 1_{x_{j}}.

Decompose ⊕_{P}ℤ = ⊕_{j} ⊕_{P_{j}} ℤ and examine what happens on each ⊕_{P_{j}}ℤ.

Consider first an index j ≥ 2. In the component ⊕_{P_{j}}ℤ for such a j, the sum reduces to ∑_{γ∈Γ_{1}} t_{j,γ}γ1_{x_{j}} = 0 whence t_{j,γ} = 0 for all γ ∈ Γ_{1} since Γ_{1} acts freely on P_{j}.

The case remaining is j = 1 for which the sum reduces to

( ∑_{1≤j≤k,γ∈Γ_{1}} t_{j,γ}γ1_{x_{0}}) − ( ∑_{γ∈Γ_{1}} t_{1,γ}γ1_{x_{1}}) = 0

however, by the last paragraph, this reduces to ∑_{γ∈Γ_{1}} t_{1,γ}γ(1_{x_{0}} − 1_{x_{1}}) = 0. But x_{1} = γ_{1}(x_{0}) for some γ_{1} ∈ Γ_{1} and so we find γ_{1}f = f where f = ∑_{γ∈Γ_{1}} t_{1,γ}γ1_{x_{0}}. This implies that f = 0 (Γ_{1} acts freely on ⊕_{P}ℤ) and so we have t_{1,γ} = 0 for all γ ∈ Γ_{1} as well.

In conclusion, the equation ∑_{1≤j≤k,γ∈Γ_{1}} t_{j,γ}γf_{j} = 0 implies t_{j,γ} = 0 for all j and γ, and so i/ follows.

To show ii/ we note first that CV_{C} is isomorphic to a countable direct sum of C(X; ℤ) ≡ ⊕_{∞}ℤ (X Cantor) and so is itself free abelian. Therefore to prove complimentarity of the image of β^{∗} it’s enough to show that CV_{C}/Imβ^{∗} is torsion free.

Therefore, we argue to the contrary and suppose that we have an element g = (g_{γ})_{γ∈Γ_{1}} ∈ ⊕_{Γ_{1}}L for which β^{∗}(g) = tf for some t ∈ ℤ, t ≥ 2, and f ∈ CV_{C}. This equation may be written ∑_{1≤j≤k,γ∈Γ_{1}} t_{j,γ}γf_{j} = tf, where β(g_{γ}) = ∑_{1≤j≤k} t_{j,γ}f_{j}.

The analysis we have just completed in part i/ can be performed equally well modulo t. Therefore each t_{j,γ} = 0 mod t and we deduce that each β(g_{γ}) = tf_{γ} for some f_{γ} ∈ β(L). In particular f = ∑_{γ∈Γ_{1}} γf_{γ} ∈ β^{∗}(⊕_{Γ_{1}}L). Thus every element of β^{∗}(⊕_{Γ_{1}}L) which can be divided in CV_{C} can also be divided in β^{∗}(⊕_{Γ_{1}}L), i.e. the quotient is torsion-free as required. □

This gives the key to the second and final technical result of this section.

Proposition 4.6 Assume the constructions and notation of 4.4 and section 3: in particular we suppose that dim V = 1. Then there is a surjective Γ-equivariant homomorphism ξ_{o} : CV_{C} → ⊕_{J_{o}}ℤ/2.

Proof Lemma 4.5 builds an injective Γ-equivariant homomorphism β^{∗} : ⊕_{J_{o}}ℤ ⟶ CV_{C} whose image is complemented in CV_{C}. Therefore we have automatically a surjective Γ-equivariant homomorphism ξ : CV_{C} ⟶ ⊕_{J_{o}}ℤ reversing this (i.e. ξβ^{∗} = identity). Extending ξ by the reduction mod 2, ⊕_{J_{o}}ℤ ⟶ ⊕_{J_{o}}ℤ/2, gives us the homomorphism ξ_{o} : CV_{C} ⟶ ⊕_{J_{o}}ℤ/2. □

5 The decomposable case

Proposition 4.2 is very close to a proof of Theorem 2.9 in the case that N (C) is indecomposable. Rather than finish the argument for that case, we clear up the decomposable case and so complete the most general proof. Once again we exploit the generalities of section 3.

Construction 5.1 Suppose that N (C) is decomposable and that N_{1}, …, N_{k} are its components. Let V_{j} be the vector space spanned by N_{j}, defined for each 1 ≤ j ≤ k. Without loss of generality (see Remark 3.7) we can impose an inner product on V which will make the V_{j} mutually orthogonal and we do this from the start.

Therefore V = ∑_{j} V_{j} = ⊕_{j}V_{j} is an orthogonal direct sum. Let π_{j} be the orthogonal projection onto V_{j} with kernel ∑{V_{i} : i ≠ j}.

Let C_{j} be the set of those elements of C whose normal is contained in V_{j}. Thus, for each j, N_{j} = N (C_{j}), an indecomposable subset of V_{j}. The Γ action on C leaves each of the sets C_{j} invariant.

The natural Γ action on V_{j} is more complicated than mere restriction. We consider π_{j}(Γ) as a subgroup of V_{j} and let Γ act by translation: γ(v) = v + π_{j}(γ). We call this the projected action.

On each V_{j} we construct the sets of singular flags, J_{oj} and J_{j} = J_{oj}/Γ, and the spaces CV_{j C_{j}^{∗}} according to section 3, using the projected action of Γ on V_{j} and the singular hyperplanes C_{j}^{∗} = {W ∩V_{j} : W ∈ C_{j}}. Note again that, for each j, N (C_{j}^{∗}) = N (C_{j}) is indecomposable as a spanning subset of V_{j}. Moreover, P_{j}, the intersection point set defined using C_{j}^{∗} in V_{j}, is equal to π_{j}(P) which in turn is equal to P ∩ V_{j}.

We abbreviate the ℤ[Γ] module CV_{j C_{j}^{∗}} as CV_{j} in what follows.

Lemma 5.2 Given the constructions above, we have the following relations:

i/ There is a canonical bijection C ↔ ∪_{j}C_{j}^{∗}.

ii/ There is a canonical Γ-equivariant bijection Π_{j}P_{j} ↔ P (direct product of sets), where Γ acts on the direct product diagonally by its projected actions.

iii/ V ≡ ⊕_{j}V_{j} is a canonical orthogonal direct sum on which the usual Γ action on V is retrieved as the diagonal action of the projected actions of Γ.

iv/ CV ≡ ⊗_{j}CV_{j} canonically as a ℤ[Γ]-module, where the module action is given by the diagonal action of the projected actions.

v/ There is a canonical Γ-equivariant surjection σ : J_{o} ⟶ Π_{j}J_{oj} (direct product of sets) and a natural Γ-equivariant injection τ : Π_{j}J_{oj} ⟶ J_{o} whose composition στ is the identity.

Proof The first four parts all follow from the fact that, for each j, the singular spaces in V break into two classes. Those spaces which are parallel to V_{j} do not impinge on its singular geometry at all. And those singular spaces, W, which are not parallel, intersect V_{j} in exactly the same way as they project to V_{j}, i.e. π_{j}(W) = W ∩ V_{j}.

Part v/ is more involved. Consider a singular flag, F, in V with the singular plane set C. The bijection of part i/ can be detailed: each element of C is of the form V_{1} +V_{2} +...+V_{j−1} +W_{j} +V_{j+1} +..+V_{k}, an orthogonal sum, where W_{j} ∈ C_{j}^{∗}. Thus any intersection of elements of C can be written as an orthogonal sum W_{1} + W_{2} + ... + W_{k}, where, for each j, W_{j} is an intersection of elements from C_{j}^{∗}.

Now consider the sequence of singular spaces listed in F and their orthogonal decomposition as above. As we read from point to hyperplane, their dimension rises by exactly one and so the orthogonal summands rise in dimension, but only in one of the summands and only by one dimension. Suppose in the orthogonal decomposition of this increasing sequence, we ignore all directions but the jth say. Then we find a nested sequence of singular spaces in V_{j} whose dimension rises by at most 1 at each step, either going all the way up to V_{j} or (for precisely one value of j) stopping one dimension short. Extract the sequence as a strictly increasing subsequences, neglecting the last term if it happens to be V_{j}, and call the result F_{j}. For each j, F_{j} will be a singular flag in V_{j} with respect to the singular planes C_{j}^{∗}. The map F ↦ (F_{1}, F_{2}, ..., F_{k}) is σ.

To show it’s onto, we construct τ. Given F_{j} = (W_{j,0}, ..., W_{j,m_{j}−1}) (where dimension of V_{j} is m_{j}) consider the sequence: W_{1,0} + W_{2,0} + ...+W_{k,0}, W_{1,1}+W_{2,0}+...+W_{k,0}, W_{1,2}+W_{2,0}+...+W_{k,0},..., W_{1,m_{1}−1}+ W_{2,0} + ... + W_{k,0}, V_{1} + W_{2,0} + ... + W_{k,0}, V_{1} + W_{2,1} + ... + W_{k,0}, ..., V_{1} + W_{2,m_{2}−1} + ... + W_{k,0}, V_{1} + V_{2} + W_{3,0} + ... + W_{k,0},..., V_{1} + V_{2} + ...V_{k−1} + W_{k,m_{k}−2}, V_{1} + V_{2} + ...V_{k−1} + W_{k,m_{k}−1}, where all sums are orthogonal sums in V. This is a singular flag, in V with respect to C, which we shall call τ(F_{1}, F_{2}, ..., F_{k}).

It is immediate from definition that σ and τ are Γ-equivariant and that στ is the identity. □

Note that although P may have an infinite number of Γ-orbits, each of the P_{j} may have only finitely many Γ orbits under the projected action. This subtle problem cuts out the possibility of an easy argument by induction on the number of indecomposable components of N (C).

Applying τ coordinate-wise, we find

Corollary 5.3 There is a Γ-equivariant homomorphism τ^{+} : ⊗_{j} ⊕_{J_{oj}} ℤ/2 ⟶ ⊕_{J_{o}}ℤ/2. □

Now we are ready for the main result of this section.

Theorem 5.4 With the data above, there is a homomorphism ξ_{∗}: H_{0}(Γ; CV_{C}) ⟶ ⊕_{J} ℤ/2 such that for all v ∈ P, there is a singular flag F = (θ_{j})_{1≤j<m} ∈ J_{o} so that θ_{0} = {v}, and an element e_{v} ∈ H_{0}(Γ, CV_{C}) such that ξ_{∗}(e_{v}) has value 1 at coordinate [F] (i.e. the Γ-orbit class of F, an element of J (3.3)).

Proof Suppose that N (C) = ∪_{j}N (C_{j}) is an indecomposable partition, forming the spaces V_{j}, of dimension m_{j} = dim V_{j}, etc as above.

Note that on each V_{j}, Γ acts by translation by elements of π_{j}(Γ) and that action may or may not be free. Proposition 4.6 applies in this case by hypothesis. Also the proof of Proposition 4.2 does not depend on the freedom of the Γ action. No complication arises therefore if we stick with the projected Γ action on each V_{j} even though the action may not be free.

For each j, π_{j}(Γ) is the projection of a dense subset of V and so itself is dense in V_{j}. Therefore the Γ action on each V_{j} is minimal and rank(π_{j}(Γ)) > 1.

Given v ∈ P, we find π_{j}(v) = v_{j} ∈ P_{j}. By Propositions 4.2 and 4.6 we find for each j a homomorphism

ξ_{oj} : CV_{j} ⟶ ⊕_{J_{oj}}ℤ/2,

a singular flag F_{j} in V_{j} such that F_{j0} = {v_{j}}, and an element e_{v,j} ∈ CV_{j} so that ξ_{o}(e_{v,j}) has value 1 in the coordinate F_{j} and 0 in the coordinates of all other flags from the Γ-orbit [F_{j}] of F (if dim V_{j} = 1 flags are just points and the latter properties follow from surjectivity). The homomorphisms ξ_{oj} are independent of the choice of v.

The equation CV_{C} = ⊗_{j}CV_{j} allows us to build the homomorphism

⊗ξ_{oj} : CV_{C} ⟶ ⊗_{j} ⊕_{J_{oj}} ℤ/2

and using τ^{+} of Cor 5.3, we can continue this homomorphism to ⊕_{J_{o}}ℤ/2. This homomorphism is clearly Γ-equivariant and so we deduce a quotiented homomorphism: ξ_{∗} : H_{0}(Γ; CV_{C}) ⟶ ⊕_{J} ℤ/2 which completes a square as required

\begin{matrix}CV_{C} & ⟶^{τ^{+}(⊗_{j}ξ_{oj})} & ⊕_{J_{o}}ℤ/2 \\ ↓^{q} & & ↓^{q} \\ H_{0}(Γ; CV_{C}) & ⟶^{ξ_{∗}} & ⊕_{J} ℤ/2 \end{matrix} .

Given v, the element e_{v} = ⊗_{j}e_{v,j} ∈ CV_{C} is mapped by τ^{+}(⊗_{j}ξ_{oj}) to an element with value 1 at the coordinate F = τ(F_{1}, F_{2}, ..., F_{k}) and with value 0 at coordinates F′ = τ(F′_{1}, F′_{2}, ..., F′_{k}) where F′_{j} ∈ [F_{j}] but F ≠ F′. In particular, τ^{+}(⊗_{j}ξ_{oj})(e_{v}) has value 0 at all coordinates of the Γ-orbit [F] of F except at F itself. Hence qτ^{+}(⊗_{j}ξ_{oj}) : CV_{C} ⟶ ⊕_{J} ℤ/2 maps e_{v} to an element with value 1 at the coordinate [F]. The zero dimensional element of F is {v} by construction. □

Proof of Theorem 2.9 Suppose that P is infinitely generated and that v_{1}, v_{2}, ... are representatives of distinct Γ-orbit classes. By Theorem 5.4, we find a homomorphism ξ_{∗} : H_{0}(Γ; CV_{C}) ⟶ ⊕_{J} ℤ/2, singular flags F_{j} ∈ J_{o} so that the zero dimensional element of F_{j} is {v_{j}}, and elements e_{j} ∈ H_{0}(Γ, CV_{C}) such that ξ_{∗}(e_{j}) has value 1 at coordinate [F_{j}].

By taking a subsequence if necessary we may assume that for i < j, all the values ξ_{∗}(e_{i}) have value 0 at the [F_{j}] coordinate. In particular, we have ensured that the set {ξ_{∗}(e_{j}) : j ≥ 1} is ℤ/2 independent in ⊕_{J} ℤ/2. By Proposition 2.10 we have H_{0}(Γ, CV_{C}) ⊗ ℚ infinitely generated.

Now to prove Theorem 2.9, we apply this analysis to the construction of section 2, using the same V, and setting C = C_{u} and Γ = Γ_{T}. □

6 Conditions for infinitely generated cohomology

To apply Theorem 2.9 we must be able to count the orbits in P. This is a geometric exercise, and each case will have its own peculiarities. We present in this section elementary general conditions which are sufficient to give infinite orbits in P.

Recall the general set-up from IV.3, and the construction of P as points which are the proper intersection of m = dim V hyperplanes picked from C.

Definition 6.1 Suppose that W_{1}, .., W_{m} is a set of hyperplanes chosen from C, intersecting in a single point p. For each subset A of {1, ⋯, m} the intersection

W_{A} := ⋂_{i∈A} W_{i}

has dimension m − |A|. We write A^{c} = {1, ⋯, m} \ A and define Γ_{A} = {x ∈ V |∃γ ∈ Γ : {x + p} = W_{A} ∩ (W_{A^{c}} + γ)}. Finally let Γ^{A} ⊂ Γ be the stabilizer of W_{A}.

We think of Γ_{A} as the projection of Γ onto W_{A} along W_{A^{c}}. The following is straightforward from the definitions.

Lemma 6.2 With the notation above:

i/ Γ_{A} is a group with Γ^{A} as a subgroup.

ii/ Γ_{A} + p = P ∩ W_{A}.

iii/ If q ∈ P ∩ W_{A}, then (Γ + q) ∩ W_{A} = Γ^{A} + q.

iv/ If A_{1}, A_{2} ⊂ {1, ⋯, m} are disjoint, then Γ^{A_{1}∪A_{2}} = Γ^{A_{1}} ∩Γ^{A_{2}}.

This gives immediately an easy way to determine whether we have an infinite number of orbits in P.

Proposition 6.3 If, for some choice of W_{1}, ..., W_{m} and A ⊂ {1, ⋯, m}, the stabilizer Γ^{A} has infinite index in Γ_{A} (equivalently, if rk Γ^{A} < rk Γ_{A}), then P is infinitely generated.

Proof By Lemma 6.2 ii/ and iii/ the orbits in P which intersect W_{A} are enumerated precisely by the cosets of Γ^{A} in Γ_{A}. This is infinite by assumption. □

We can pursue the construction above a little further to get even a sharper condition for infinitely generated P. Given A_{1}, A_{2} ⊂ {1, ⋯, m}, note that A_{1} ∪A_{2} = {1, 2, .., m} implies W_{A_{1}} ∩W_{A_{2}} = {p} and therefore Γ_{A_{1}} ∩ Γ_{A_{2}} = {0}. This together with Lemma 6.2 gives the following result.

Lemma 6.4 For every choice of W_{1}, ..., W_{m} as above and for every pair of sets, A_{1}, A_{2} ⊂ {1, ⋯, m} we have,

i/ if A_{1} ∩ A_{2} = ∅, then

rk Γ^{A_{1}} +rk Γ^{A_{2}} −rk Γ^{A_{1}∪A_{2}} ≤ rk Γ ≤ rk Γ_{A_{1}} +rk Γ_{A_{2}} −rk (Γ_{A_{1}} ∩Γ_{A_{2}}),

ii/ if A_{1} ∪ A_{2} = {1, 2, .., m}, then
Γ^{A_{1}} + Γ^{A_{2}} ⊂ Γ^{A_{1}∩A_{2}} ⊂ Γ_{A_{1}∩A_{2}} ⊂ Γ_{A_{1}} + Γ_{A_{2}}

both sums being direct. □

Corollary 6.5 If P is finitely generated, then for every choice of W_{1}, ..., W_{m} and for every pair of sets, A_{1}, A_{2} ⊂ {1, ⋯, m} we have,

i/ if A_{1} ∩ A_{2} = ∅, then

rk Γ^{A_{1}} + rk Γ^{A_{2}} − rk Γ^{A_{1}∪A_{2}} = rk Γ,
ii/ if A_{1} ∪ A_{2} = {1, 2, .., m}, then
rk Γ^{A_{1}} + rk Γ^{A_{2}} = rk Γ^{A_{1}∩A_{2}}.

Proof Suppose A_{1} ∩ A_{2} = ∅. Then Γ^{A_{1}∪A_{2}} = Γ^{A_{1}} ∩ Γ^{A_{2}} ⊂ Γ_{A_{1}} ∩ Γ_{A_{2}} which, by Proposition 6.3, implies rk Γ_{A_{1}∪A_{2}} ≤ rk (Γ_{A_{1}} ∩ Γ_{A_{2}}). Hence, by Lemma 6.4 rk Γ ≤ rk Γ_{A_{1}} + rk Γ_{A_{2}} − rk Γ_{A_{1}∪A_{2}} and now Proposition 6.3 and Lemma 6.4 i/ allow to conclude i/. ii/ follows directly from Proposition 6.3 and Lemma 6.4 ii/. □

Definition 6.6 Let us now look at a situation in which we pick more hyperplanes, W = {W_{1}, ⋯, W_{f}}, f > m = dim V, from C demanding that the set N (W) of normals of the planes (defined as in Definition 3.1 but for the subset W ⊂ C) is indecomposable in the sense of 3.5. This requires that N (W) spans V and by (3.6) is equivalent to the fact that all graphs G(B; N (W)) with B ⊂ N (W) a basis for V are connected. Let us denote by I_{l} the collection of subsets A ⊂ {1, ⋯, f} of m − l elements such that W_{A} has dimension l (compare with 2.1). Note that any A ∈ I_{0} defines a basis B_{A} by the normals to all W_{i}, i ∈ A.

Theorem 6.7 With W as above suppose that P is finitely generated. Then, for all A ∈ I_{l}

rk Γ^{A} = l \frac{rk Γ}{dim V} .

In particular dim V divides rk Γ.

Proof Fix i and choose an A ∈ I_{0} which contains i (this is clearly possible). Applying Corollary 6.5 i/ iteratively we obtain

∑_{i∈A} rk Γ^{{i}} = (m − 1)rk Γ.  (6.1)

By hypothesis there exists a j ∈ A such that the vertices of G(B_{A}, N (W)) corresponding to the normals of W_{i} and W_{j} are linked. Hence there is a k ∉ A such that the normal of W_{k} has nonvanishing scalar product [·, ·] with the normals of W_{i} and W_{j}. This implies that, both, (A\{i}) ∪ {k} and (A\{j}) ∪ {k} belong to I_{0}. Hence we can apply (6.1) to both sets to conclude ν_{i} = ν_{j}. By assumption G(B_{A}; N (W)) is connected so that a repetition of the argument shows that ν_{i} does not depend on the choice of i ∈ A. This proves the theorem for l = m − 1 (l = m is clear). The statements for l < m − 1 follow now inductively from Corollary 6.5 i/. □

Theorem 6.7 contains as a special case a condition for canonical projection method patterns as to whether they have infinitely generated cohomology which can quickly be checked.

Corollary 6.8 Suppose that T is a tiling in ℝ^{d}, topologically conjugate (I.4.5) to a canonical projection method pattern with data (E, u), and suppose that E ∩ ℤ^{N} = 0. If N − rk Δ is not divisible by N − rk Δ − d then H_{0}(GT_{u}) ⊗ ℚ is infinite dimensional, and so T_{u} is not a substitution tiling.

Proof In the tiling case rk Γ_{T_{u}} = N −rk Δ and dim V = N −d−rk Δ. □

Examples 6.9 (Cf. Examples I.2.7) The Octagonal tiling, a canonical projection tiling with N = 4, Δ = 0 and d = 2, and the Penrose tiling, a canonical projection tiling with N = 5, Δ ≅ ℤ and d = 2, are both substitutional and hence have finitely generated cohomology. Theorem 6.7 therefore tells us that the stabilizers of the hyperplanes (here lines) have rank two.

It is clear that in the generic placement of planes the ranks of the stabilizers of the intersections of hyperplanes will have smaller rank than compatible with the last theorem. Thus we deduce:

Theorem 6.10 Suppose that T is a tiling in ℝ^{d}, topologically conjugate to a canonical projection method pattern with data (E, u) and N > d + 1, and suppose that E is in generic position. Then H_{0}(GT )⊗ℚ is infinite dimensional, and T is not a substitution tiling. □

V Approaches to Calculation III: Cohomology for Small Codimension

1 Introduction

The last chapter was devoted to the case in which the cohomology groups of a canonical projection tiling were infinitely generated. Now we turn to the opposite case. In particular we shall assume that P has only finitely many Γ orbits, i.e. is finitely generated (IV.2.8). In that case we find that the cohomology groups of canonical projection tilings are finitely generated free abelian groups and we can provide explicit formulae for their ranks if the codimension is smaller or equal to 3. We obtain a formula for their Euler characteristic even in any codimension. Although we saw that finitely generated P is non-generic this seems to be the case of interest for quasicrystal physics. In particular we will present calculations for the Ammann-Kramer tiling as an example. It is a three dimensional analog of the Penrose tilings and often used to model icosahedral quasicrystals.

The material presented here extends [FHK] to the codimension 3 case. This is important, because all known tilings which are used to model icosahedral quasicrystals are obtained from projection out of a 6-dimensional periodic structure with codimension 3. Unlike in [FHK] we use here a spectral sequence derived from a double complex to prove our formulae.

Perhaps the first use of spectral sequences in calculation of tiling cohomology or K-theory is found in [BCL]. However, we note that the spectral sequence we use here differs significantly from the spectral sequence found in [BCL]. While the sequence of [BCL] produces an isomorphism of the K-theory of the groupoid C^{∗}-algebra of the tiling with its cohomology in 2 dimensions (as is generalized in Chapter II), our sequence represents a geometric decomposition of the tiling cohomology itself which is a powerful calculating tool.

In Section 2 we recall the set-up and state the main results (Thms. 2.4, 2.5, 2.8 for arbitrary codimension and Thm. 2.7 for codimension 3). In Section 3 we explain one part of the double complex, namely the one which is related to the structure of the set of singular points, and in Section 4 we recall the other half which is simply group homology. We put both together in Section 5 where we employ the machinery of spectral sequences to proof our results. In principle there is no obstruction to pushing the calculation further to higher codimension, it is rather the complexity which becomes overwhelming to do this in practice. In the final section we sketch how our formulae apply to the Ammann-Kramer tiling.

2 Set up and statement of the results

In the second chapter we defined the cohomology of a tiling as the cohomology of one of its groupoids (it turned out that it does not matter which one we take) and interpreted it in various forms, in particular as a standard dynamical invariant of one of the systems. This dynamical system is of the following type: Consider a dense lattice Γ of rank N in a euclidian space V as it arises generically if one takes N > dim V vectors of V and considers the lattice they generate. Let K be a compact set which is the closure of its interior and consider the orbit S = ∂K + Γ ⊂ V of ∂K under Γ. We showed in Chapters I and II how such a situation arises for projection method patterns and how in this case the rather simple dynamical system (V, Γ) (Γ acting by translation) extends to a dynamical system (\overline{V} , Γ) which coincides with the old one on the dense G_{δ}-set V \S. \overline{V} is locally a Cantor set and obtained from V upon disconnecting it along the points of S. We are interested in calculating the homology groups H_{p}(Γ, C\overline{V}) of the group Γ with coefficients in the compactly supported integer valued continuous functions over \overline{V}. Already for the results of Chapter IV we specialized to the situation in which S is a union of a collection C of hyperplanes, the collection consisting of finitely many Γ_{T} -orbits. We denoted there \overline{V} by V_{C}. We will restrict our attention to this case here too. In the context of projection method patterns (Def. I.4.4), where Γ = Γ_{T} and V = V + π′(u) (II.4.3), this means that we consider only polytopal acceptance domain K and such that the orbit of a face under the action of Γ contains the hyperplane it spans (this is hypothesis H3 in [FHK]). The Γ_{T} -orbits of these hyperplanes constitute the set C. In particular, we rule out fractal acceptance domain although this case might be important for quasicrystal physics.

Set-up 2.1 All our results below depend only on the context described by the data (V, Γ, W), a dense lattice Γ of finite rank in a Euclidean space V with a finite family W = {W_{i}}_{i=1,···,f} of (affine) hyperplanes whose normals span V. Thus they are not specific for the tilings considered here but can be applied e.g. also to the situation of [FHK]. Our aim is to analyse the homology groups H_{p}(Γ, CV_{C}) where C := {W + γ : W ∈ W, γ ∈ Γ}.

We extend the notation of IV.6.6.

Definitions 2.2 Given a finite family W = {W_{i}}_{i=1,···,f} of hyperplanes, recall that J_{l} is the collection of subsets A ⊂ {1, ⋯, f} of dim V − l elements such that W_{A} has dimension l. On W_{A} we have an action of Γ^{|A|}:

x⃗ · W_{A} := ⋂_{i∈A}(W_{i} − x_{i}).

We call x⃗ · W_{A} a singular space or singular l-space if we want to specify its dimension l. Let P_{l} be the quotient Γ^{|A|} × J_{l}/ ∼ with (x⃗, A) ∼ (x′, A′) if x⃗·W_{A} = x′·W_{A′}. It is in one to one correspondence to the set of singular l-spaces. We denote equivalence classes by [x⃗, A] and the space x⃗ · W_{A} also by W_{[x⃗,A]}. On P_{l} we have an action of Γ: y · [x⃗, A] = [y · x⃗, A] where (y · x⃗)_{i} = y + x_{i}. We denote the orbit space P_{l}/Γ by I_{l}. Since this action coincides with the geometric action of Γ (by translation) on the singular l-spaces we can use the elements of I_{l} to label the Γ-orbits of singular l-spaces. Note that the map {1, ⋯, f} → I_{dim V −1} which assigns to i the orbit of [0, {i}] is surjective but not necessarily injective. The stabilizer of a singular l-space depends only on its orbit class, if the label of its orbit is Θ ∈ I_{l} we denote the stabilizer by Γ^{Θ}.

Fix Θ̂ ∈ P_{l+k}, l + k < dim V and let PΘ̂ ; l := {Ψ̂ ∈ P_{l}|WΨ̂ ⊂ WΘ̂}. Then Γ^{Θ} (Θ the orbit class of Θ̂) acts on PΘ̂ ; l (diagonally, by the same formula as above) and we let IΘ̂ ; l = PΘ̂ ; l/Γ^{Θ}, the orbit space. It labels the Γ^{Θ}-orbits of singular l-spaces in the l +k-dimensional space WΘ̂. We can naturally identify IΘ̂ ; l with IΘ̂′ ; l if Θ̂ and Θ̂′ belong to the same Γ-orbit and so we define I_{l}^{Θ}, for the class Θ ∈ I_{l+k}. I_{l}^{Θ} is the subset of I_{l} labelling those orbits of singular l-spaces which have a representative that lies in singular space whose label is Θ. Finally we denote

L_{l} = |I_{l}|, L_{l}^{Θ} = |I_{l}^{Θ}|

where Θ ∈ I_{l+k}, l + k < dim V.

Note that P_{0} can be identified with P (Def. IV.2.8). So our assumption of this chapter is that L_{0} is finite. The proof of the following lemma is straightforward.

Lemma 2.3 If L_{0} is finite then L_{l}^{Θ} is finite for all Θ and l. □

The proof of the following five theorems will be given in Section 5. The first one is the converse of Theorem IV.2.9.

Theorem 2.4 Suppose given data (V, Γ, W) as in (2.1) with L_{0} finite. Then H_{p}(Γ, CV_{C}) ⊗ ℚ has finite rank over the rational numbers ℚ.

We denote

D_{p} = rk H_{p}(Γ, CV_{C}) ⊗ ℚ.

Theorem 2.5 Suppose given data (V, Γ, W) as in (2.1) with L_{0} finite. Then H_{p}(Γ, CV_{C}) is free abelian. In particular it is uniquely determined by its rank which is D_{p}.

Note that Theorems 2.4 and 2.5 exclude the possibility that H(Γ, CV_{C}) would contain e.g. the dyadic numbers as a summand, a case which occurs frequently in cohomology groups of substitution tilings.

Recall that, if the normals of the hyperplanes W form an indecomposable set in the sense of IV.3.4 and L_{0} is finite then Theorem IV.6.7 implies that the rank of the stabilizer Γ^{Θ} depends only on the dimension of the plane it stabilizes, i.e.

rk Γ^{Θ} = ν dim Θ

where ν = \frac{rk Γ}{dim V} and dim Θ = l provided Θ ∈ I_{l}.

Recall Theorem III.3.1 applied to the present situation where we have data (V, Γ, W) with L_{0} finite and dim V = 1. Then H_{p}(Γ, CV_{C}) = ℤ^{D_{p}} with

D_{p} = \binom{ν}{p + 1}, p > 0,  (2.1)
D_{0} = (ν − 1) + L_{0}.  (2.2)

If {M_{i} : i ∈ I} is a family of submodules of some bigger module we denote by ⟨M_{i} : i ∈ I⟩ their span. For a finitely generated lattice G we let ΛG be the exterior ring (which is a ℤ-module) generated by it. In [FHK] we obtained the following theorem.

Theorem 2.6 Given data (V, Γ, W) as in (2.1) with dim V = 2. Suppose that L_{0} is finite and that the normals of the hyperplanes W form an indecomposable set in the sense of IV.3.4. Then

D_{p} = \binom{2ν}{p + 2} + L_{1} \binom{ν}{p + 1} − r_{p+1} − r_{p}, p > 0,

D_{0} = \binom{2ν}{2} − 2ν + 1 + L_{1}(ν − 1) + e − r_{1}

where

r_{p} = rk ⟨Λ_{p+1}Γ^{α} : α ∈ I_{1}⟩,

and the Euler characteristic is

e := ∑_{p}(−1)^{p}D_{p} = −L_{0} + ∑_{α∈I_{1}} L_{0}^{α}.

The main result of this chapter is an extension of this result to codimension 3.

Theorem 2.7 Given data (V, Γ, W) as in (2.1) with dim V = 3. Suppose that L_{0} is finite and that the normals of the hyperplanes W form an indecomposable set in the sense of IV.3.4. Then, for p > 0,

D_{p} = \binom{3ν}{p + 3} + L_{2} \binom{2ν}{p + 2} + L̃_{1} \binom{ν}{p + 1} − R_{p} − R_{p+1},

D_{0} = ∑_{j=0}^{3}(−1)^{j} \binom{3ν}{3 − j} + L_{2} ∑_{j=0}^{2}(−1)^{j} \binom{2ν}{2 − j}

+L̃_{1} ∑_{j=0}^{1}(−1)^{j} \binom{ν}{1 − j} + e − R_{1}

where L̃_{1} = −L_{1} + ∑_{α∈I_{2}} L_{1}^{α},

R_{p} = rk ⟨Λ_{p+2}Γ^{α}: α ∈ I_{2}⟩ − rk ⟨Λ_{p+1}Γ^{Θ}: Θ ∈ I_{1}⟩

+ ∑_{α∈I_{2}} rk ⟨Λ_{p+1}Γ^{Θ}: Θ ∈ I_{1}^{α}⟩ ,

and the Euler characteristic is

e := ∑_{p}(−1)^{p}D_{p} = L_{0} − ∑_{α∈I_{2}} L_{0}^{α} + ∑_{α∈I_{2}} ∑_{Θ∈I_{1}^{α}} L_{0}^{Θ} − ∑_{Θ∈I_{1}} L_{0}^{Θ}.

The last two theorems make combinatorial patterns for higher codimensional tilings apparent. We are able to present one such for the Euler characteristic of the general case. To set this up define a singular sequence to be a (finite) sequence c = Θ_{1}, Θ_{2}, ..., Θ_{k} of Γ-orbits of singular spaces strictly ascending in the sense that Θ_{j} ∈ I_{dim Θ_{j}}^{Θ_{j+1}}, dim Θ_{j} < dim Θ_{j+1}, and dim Θ_{1} = 0. An example of such a sequence is the Γ-orbit of a singular flag, but the dimension in the sequence can also jump by more than one. The length of the chain c is k, written |c| = k.

Theorem 2.8 Given data (V, Γ, W) with L_{0} finite. Then the Euler Characteristic equals

e := ∑_{p}(−1)^{p}D_{p} = ∑(−1)^{|c|+dim V}

where the sum is over all singular chains c.

3 Complexes defined by the singular spaces

Let C′ be an arbitrary countable collection of affine hyperplanes of V′, a linear space, and define C′-topes as in IV.3.2: compact polytopes which are the closure of their interior and whose boundary faces belong to hyperplanes from C′.

Definition 3.1 For n at most the dimension of V′ let C_{C′}^{n} be the ℤ-module generated by the n-dimensional faces of convex C′-topes satisfying the relations

[U_{1}] + [U_{2}] = [U_{1} ∪ U_{2}]

for any two n-dimensional faces U_{1}, U_{2}, for which U_{1} ∪ U_{2} is as well a face and U_{1} ∩ U_{2} has no interior (i.e. nonzero codimension in U_{1}). (The above relations then imply [U_{1}] + [U_{2}] = [U_{1} ∪ U_{2}] + [U_{1} ∩ U_{2}] if U_{1} ∩ U_{2} has interior.)

If we take C′ = C := {W + x : W ∈ W, x ∈ Γ}, our collection of singular planes, then C^{n} := C_{C}^{n} carries an obvious Γ-action, namely x · [U] = [U + x]. It is therefore an Γ-module. Recall Definition IV.3.2 in which we defined V_{C} by identifying CV_{C} with C^{dim V} as Γ-module. An isomorphism between C^{dim V} and CV_{C} is given by assigning to [U] the indicator function on the connected component containing U\S (which is a clopen set). Moreover, C^{0} is a free ℤ-module, its generators are in one to one correspondence to the elements of P. The following proposition is from [FHK].

Proposition 3.2 There exist Γ-equivariant module maps δ and ϵ such that

0 → C^{dim V} ⟶^{δ} C^{dim V −1} ⟶^{δ} ⋯ C^{0} ⟶^{ϵ} ℤ → 0

is an exact sequence of Γ-modules and ϵ[U] = 1 for all vertices U of C-topes.

Proof For a subset R of Γ let C_{R} := {W + r : W ∈ W, r ∈ R} and S_{R} = {x ∈ W : W ∈ C_{R}}. Let R be the set of subsets R ⊂ Γ such that all connected components of V \S_{R} are bounded and have interior. R is closed under union and hence forms an upper directed system under inclusion. Let V_{R} be the disjoint union of the closures of the connected components of V \S_{R}. R ⊂ R′ gives rise to a natural surjection V_{R′} → V_{R} which is the continuous extension of the inclusion V \S_{R′} ⊂ V \S_{R} and V_{C} is the projective limit of the V_{R}. For any R ∈ R, the C_{R}-topes define a regular polytopal CW-complex

0 → C_{C_{R}}^{dim V} ⟶^{δ_{R}} C_{C_{R}}^{dim V −1} ⟶^{δ_{R}} ⋯ C_{C_{R}}^{0} → 0,  (3.1)

with boundary operators δ_{R} depending on the choices of orientations for the n-cells (n > 0) [Mas]. Moreover, this complex is acyclic (V is contractible), i.e. upon replacing C_{C_{R}}^{0} → 0 by C_{C_{R}}^{0} ⟶^{ϵ_{R}} ℤ → 0 where ϵ_{R}[U] = 1, (3.1) becomes an exact sequence. Let us constrain the orientation of the n-cells in the following way: Each n-cell belongs to a unique singular n-space WΘ̂, Θ̂ ∈ P_{n}. We choose its orientation such that it depends only on the class of Θ̂ in I_{l} (i.e. we choose an orientation for all parallel WΘ̂ and then the cell inherits it as a subset). By the same principle, all cells of maximal dimensions are supposed to have the same orientation. Then the cochains and boundary operators δ_{R} share two crucial properties: first, if R ⊂ R′ for R, R′ ∈ R, then we may identify C_{C_{R}}^{n} with a submodule of C_{C_{R′}}^{n} and under this identification δ_{R}(x) = δ_{R′}(x) for all x ∈ C_{C_{R}}^{n}, and second, if U and U + x are C_{R}-topes then δ_{R}[U + x] = δ_{R}[U] + x. The first property implies that the directed system R gives rise to a directed system of acyclic cochain complexes, and hence its direct limit is an acyclic complex, and the second implies, together with the fact that for all x ∈ Γ and R ∈ R also R + x ∈ R, that this complex becomes a complex of Γ-modules. The statement now follows since C_{C}^{n} is the direct limit of the C_{C_{R}}^{n}, R ∈ R. □

Let C_{[x⃗,A]}^{l} be the restriction of C^{l} to polytopes which belong to W_{[x⃗,A]}. We can naturally identify C_{[x⃗,A]}^{l} with C_{[x′,A′]}^{l} if y · [x⃗, A] = [x′, A′] for some y ∈ Γ. Moreover, in this case the complexes obtained by restriction of δ,

0 → C_{[x⃗,A]}^{l} ⟶^{δ^{[x⃗,A]}} C_{[x⃗,A]}^{l−1} ⋯ C_{[x⃗,A]}^{0} → 0,

are isomorphic for all [x⃗, A] of the same orbit class in I_{l}. With this in mind we define ((C_{Θ}^{l})_{l∈ℤ}, δ^{Θ}) as a complex isomorphic to the above complex where Θ ∈ I_{l} is the class of [x⃗, A]. Thus every Θ ∈ I_{l} defines a new acyclic complex.

Lemma 3.3

C^{l} ≅ ⨁_{Θ∈I_{l}} C_{Θ}^{l} ⊗ ℤ[Γ/Γ^{Θ}]

and if Θ ∈ I_{l+k}, l + k < dim V, then

C_{Θ}^{l} ≅ ⨁_{Ψ∈I_{l}^{Θ}} C_{Ψ}^{l} ⊗ ℤ[Γ^{Θ}/Γ^{Ψ}].

Proof First observe that

C^{l} ≅ ⨁_{[x⃗,A]∈P_{l}} C_{[x⃗,A]}^{l},

because an l-face of a C-tope belongs a unique singular l-space. From the definition of C_{Θ} and the observation that y · [x⃗, A] = [x⃗, A] whenever y leaves W_{A} invariant the first statement of the lemma follows. The proof for the second is similar. □

4 Group homology

Recall that the homology of a discrete group Γ′ with coefficients in a Γ′-module M is defined as the homology of a complex which is obtained from a resolution of ℤ by (projective) Γ′-modules upon application of the functor ⊗_{Γ′}M. We use here a free and finite resolution of Γ′ = Γ ≅ ℤ^{N} by its exterior module so that the complex looks like

0 → Λ_{N}Γ ⊗ M ⟶^{∂} ⋯ Λ_{0}Γ ⊗ M → 0

with boundary operator ∂ given by

∂(e_{i_{1}} ⋯ e_{i_{k}} ⊗ m) = ∑_{j=1}^{k} e_{i_{1}} ⋯ ê_{i_{j}} ⋯ e_{i_{k}}(e_{i_{j}} · m − m)

denoting by ê_{i_{j}} that it has to be left out and by g· the action of g ∈ Γ on M. In particular, H_{k}(Γ, ℤΓ) is trivial for all k > 0 and isomorphic to ℤ for k = 0.

Suppose that we can split Γ = Γ′ ⊕ Γ′′ and let us compute H_{k}(Γ, ℤΓ′′) where ℤΓ′′ is the free ℤ-module generated by Γ′′ which becomes an Γ-module under the action of Γ given by (g ⊕h)·h′ = h+h′. Then we can identify

Λ_{k}Γ ⊗ ℤΓ′′ ≅ ⨁_{i+j=k} Λ_{i}Γ′ ⊗ Λ_{j}Γ′′ ⊗ ℤΓ′′  (4.1)

and under this identification ∂ ⊗ 1 becomes (−1)^{deg} ⊗ ∂′ where ∂′ is the boundary operator for the homology of Γ′′. It follows that

H_{k}(Γ, ℤΓ′′) ≅ ⨁_{i+j=k} Λ_{i}Γ′ ⊗ H_{j}(Γ′′, ℤΓ′′) ≅ Λ_{k}Γ′.

As a special case, H_{k}(Γ, ℤ) ≅ Λ_{k}Γ ≅ ℤ^{\binom{N}{k}}. Now let ϵ : ℤΓ′′ → ℤ be the sum of the coefficients, i.e. ϵ[h] = 1 for all h ∈ Γ′′. We shall later need the following lemma:

Lemma 4.1With the identifications H(Γ, ℤΓ′′) ≅ ΛΓ′ and H(Γ, ℤ) ≅ ΛΓ the induced map ϵ_{k} : H_{k}(Γ, ℤΓ′′) → H_{k}(Γ, ℤ) becomes the inclusion Λ_{k}Γ′ ↪ Λ_{k}Γ.

Proof Using the decomposition (4.1) it is not difficult to see that the induced map ϵ_{k} : ⨁_{i+j=k} Λ_{i}Γ′ ⊗ H_{j}(Γ′′, ℤΓ′′) → ⨁_{i+j=k} Λ_{i}Γ′ ⊗ H_{j}(Γ′′, ℤ) preserves the bidegree and must be the identity on the first factors of the tensor product. Since H_{k}(Γ′′, ℤΓ′′) is trivial whenever k ≠ 0 and one dimensional for k = 0, ϵ_{k} can be determined by evaluating ϵ_{0} on the generator of H_{0}(Γ′′, ℤΓ′′) and one readily checks that this gives a generator of H_{0}(Γ′′, ℤ) as well. □

Finally, we note two immediate corollaries of Lemma 3.3.

Corollary 4.2

H(Γ, C^{l}) ≅ ⨁_{Θ∈I_{l}} H(Γ^{Θ}, C_{Θ}^{l}).  □

Corollary 4.3 If Θ ∈ I_{l+k}, l + k < dim V, then

H(Γ^{Θ}, C_{Θ}^{l}) ≅ ⨁_{Ψ∈I_{l}^{Θ}} H(Γ^{Ψ}, C_{Ψ}^{l}).  □

5 The spectral sequences

In the last two sections we described two complexes. They can be combined to yield a double complex and then spectral sequence techniques can be applied to obtain the information we want. We do not explain these techniques in detail but only set up the notation, see, for example, [Br] for an introduction. Consider a double complex (E^{0}, ∂, δ) which is a bigraded module E^{0} = (E_{pq}^{0})_{p,q∈ℤ} with two commuting differential operators ∂ and δ of bidegree (−1, 0) and (0, −1), respectively. The associated total complex is T E^{0} = ((T E^{0})_{k})_{k∈ℤ} with (T E^{0})_{k} = ⨁_{p+q=k} E_{pq}^{0} and differential δ + ∂. Furthermore one can form two spectral sequences of bigraded modules [Br]. The first one, (E^{k})_{k∈ℕ_{0}}, is equipped with differential operators d^{k} of bidegree (k − 1, −k), i.e.

d_{pq}^{k} : E_{pq}^{k} ⟶ E_{p+k−1 q−k}^{k},  (5.1)

such that E^{k+1} = H_{d^{k}}(E^{k}). (Here and below we use also the notation Z_{d}(C^{k}) and H_{d}(C^{k}) for the degree k cycles and homology resp. of a complex (C, d).) E^{0} is the original module, d^{0} = ∂, d^{1} = δ_{∗} and the higher differentials become more and more subtle. The second sequence, (Ẽ^{k})_{k∈ℕ_{0}}, is obtained in the same way except that one interchanges the role of the two differentials, i.e. Ẽ^{0} = E^{0}, d̃^{0} = δ, d̃^{1} = ∂_{∗}. The two spectral sequences may look rather different but their ”limits” are related to the homology of the total complex. This relation may be quite subtle but in our case we need only the following. Suppose that E_{pq}^{0} is non-trivial only for finitely many p, q. Then the higher differentials vanish for both sequences from some k on so that the modules stabilize and we have well defined limit modules E^{∞}, Ẽ^{∞}. If the modules are moreover vector spaces then

⨁_{p+q=k} E_{pq}^{∞} ≅ ⨁_{p+q=k} Ẽ_{pq}^{∞} ≅ H_{∂+δ}((T E^{0})_{k}).

If the modules are only over some ring such as the integer numbers we may at least conclude that finitely generated E^{∞} implies that Ẽ^{∞} and H_{∂+δ}(T E^{0}) are finitely generated as well.

Inserting the complex from Proposition 3.1 for M in the last section we get the double complex (Λ_{p}Γ ⊗ C^{q}, ∂_{pq}, δ_{pq}) where ∂_{pq} = ∂_{p} ⊗ 1 and δ_{pq} = (−1)^{p} ⊗ δ_{q}.

Proposition 5.1 Consider the spectral sequence Ẽ^{k} derived from the double complex (E_{0}^{0} = Λ_{p}Γ⊗C^{q}, ∂_{pq}, δ_{pq}) when starting with homology in δ. It satisfies

⨁_{p+q=k} Ẽ_{pq}^{∞} ≅ Λ_{k}Γ.

In particular, rk (H_{δ+∂}(⨁_{p+q=k} Λ_{p}Γ ⊗ C^{q}) ⊗ ℚ) = \binom{rk Γ}{k}.

Proof The first page of that spectral sequence is

Ẽ_{pq}^{1} = H_{δ}(Λ_{p}Γ ⊗ C^{q}) ≅ \{ \begin{matrix}Λ_{p}Γ & for q = 0, \\ 0 & for q > 0. \end{matrix}

Under the isomorphism Ẽ_{p0}^{1} ≅ Λ_{p}Γ, d̃^{1} becomes trivial and all other differentials d̃^{k} vanish because they have bidegree (−k, k − 1). Hence Ẽ^{∞} = Ẽ^{1} from which the first statement follows. The second statement is then clear. □

The more difficult spectral sequence will occupy us for the rest of this chapter. Its first page is obtained when starting with homology in ∂, i.e.

E_{pq}^{1} = H_{∂}(E_{pq}^{0}) = H_{p}(Γ, C^{q}),

and the first differential is d^{1} = δ_{∗}. We realize that E_{p dim V}^{1} is what we want to compute. For that to carry out we have to determine the ranks of the higher differentials which is quite involved. On the other hand the proofs of Theorems 2.4, 2.5, and 2.8 involve only the general fact that the higher differentials d^{k} have bidegree (k − 1, −k) and begin with that. All three proofs are by induction on the dimension of V the one-dimensional case following from Eqs. (2.1),(2.2).

Proof of Theorem 2.4 Theorem 2.4 is a rational result. So we work with a rationalized version of the above. Corollary 4.2 yields

E_{pq}^{1} ≅ ⨁_{Θ∈I_{q}} H_{p}(Γ^{Θ}, C_{Θ}^{q}).

Hence by Lemma 2 and induction on the dimension of V we find that E_{pq}^{1} ⊗ ℚ is a finite dimensional vector space, provided q < dim V. This implies that E_{pq}^{k} ⊗ ℚ is finite dimensional, provided q < dim V, for arbitrary k, and is trivial for q > dim V or p > rk Γ. On the other hand by Proposition 5.1 E_{pq}^{∞} ⊗ ℚ has to be finite dimensional for all p, q. By construction E_{p dim V}^{∞} is obtained upon taking successively the kernels of the higher differentials. So if E_{p dim V}^{1} ⊗ ℚ was infinite, one of the ranks of the higher differentials would have to be infinite. But this would contradict the finite dimensionality of E_{pq}^{k} ⊗ ℚ for q < dim V. □

Proof of Theorem 2.5 It is a result of [FH] that H_{p}(Γ, CV_{C}) is torsion free. Therefore Theorem 2.5 follows if we can show that H_{p}(Γ, CV_{C}) is finitely generated. Now we use Lemma 2.2 and Corollary 4.2 to see inductively that E_{pq}^{1} is finitely generated, provided q < dim V. This implies that E_{pq}^{k} is finitely generated, provided q < dim V, for arbitrary k. By Eq. (5.1) we get sequences

E_{p dim V}^{k+1} ↪ E_{p dim V}^{k} ⟶^{d^{k}} E_{p+k−1 dim V −k}^{k}

which are exact at E_{p dim V}^{k} and whose right hand side modules are finitely generated if k ≥ 1. Hence E_{p dim V}^{k} is finitely generated if E_{p dim V}^{k+1} is finitely generated. But E_{p dim V}^{∞} is finitely generated by Proposition 5.1. Again inductively we conclude therefore that E_{p dim V}^{1} is finitely generated. □

Proof of Theorem 2.8 By Theorem 2.4 the dimensions of the rational vector spaces E_{pq}^{k} ⊗ ℚ are finite. From Eq. (5.1) it follows therefore that ∑_{p,q}(−1)^{p+q} dim E_{pq}^{k} ⊗ ℚ is independent of k. Choosing k = ∞ we see from Proposition 5.1 that this sum vanishes. With e_{q} = ∑_{p}(−1)^{p} dim H_{p}(Γ, C^{q}) ⊗ ℚ we thus get

e = e_{dim V} = − ∑_{q=0}^{dim V −1}(−1)^{q+dim V} e_{q}.

But dim H_{p}(Γ, C^{q}) ⊗ ℚ = ∑_{Θ∈I_{q}} dim H_{p}(Γ^{Θ}, C_{Θ}^{q}) ⊗ ℚ by Corollary 4.2 and we can use induction on dim V to see that e_{q} = ∑_{Θ∈I_{q}} ∑(−1)^{q+|c|−1} where the second sum is over all singular sequences c whose last element is Θ. Inserting this into the above formula for e gives directly the result. □

The strategy to prove the remaining Theorems 2.6 and 2.7 is as follows. Suppose that we know d_{pq} = dim E_{pq}^{1} ⊗ ℚ for q < dim V. Then we can determine d_{p dim V} from these data, the dimensions of the rationalized total homology groups (Proposition 5.1), and the ranks of the differentials. To carry that out we have to consider only modules over ℚ. We will therefore simplify our notation by suppressing ⊗ℚ and e.g. write E_{pq}^{k} instead of E_{pq}^{k} ⊗ ℚ. We consider first a more abstract situation.

Definition 5.2 Consider a double complex (E_{pq}^{0}, δ_{pq}, ∂_{pq}) of finitely generated ℚ-modules with its spectral sequence (E^{k})_{k∈ℕ_{0}}, i.e. E_{pq}^{1} := H_{∂}(E_{pq}^{0}) and d^{1} = δ_{∗}. We write d_{pq} = dim E_{pq}^{1}. Assume that E1 There exist finite M, N such that E_{pq}^{0} is non-trivial only for 0 ≤

q ≤ M and 0 ≤ p ≤ N, E2 d_{p0} = 0 for p ≥ 1. This is sufficient to ensure that the spectral sequence converges and we can define N_{k} := ∑_{p+q=k} dim E_{pq}^{∞}.

For M = 1, 2, 3, we now determine the ranks d_{pM} in terms of d_{pq}, q < M, and N_{k} and the ranks of the higher differentials d^{k}.

Lemma 5.3 If M = 1 then, with the notation of Definition 5.2,

d_{p1} = N_{p+1} + \{ \begin{matrix}d_{00} − N_{0} & for p = 0, \\ 0 & for p > 0. \end{matrix}

Proof If M = 1 then (5.1) implies that d^{k} = 0 for k ≥ 2 and hence E^{∞} = E^{2}. With the notation

a_{pq} := rk d_{p q+1}^{1} : E_{p q+1}^{1} → E_{pq}^{1}  (5.2)

we get

dim E_{pq}^{∞} = \{ \begin{matrix}d_{00} − a_{00} & for p = q = 0, \\ d_{01} − a_{00} & for p = 0, q = 1, \\ d_{p1} & for N ≥ p ≥ 1, q = 1, \\ 0 & otherwise, \end{matrix}

which when a_{00} is eliminated for N_{0} yields the statement. □

Let

b_{pq} := rk δ̃_{p q+1 ∗} : E_{p q+1}^{1} → H_{∂}(Z_{δ}(E_{pq}^{0})),  (5.3)

where δ̃_{p q+1 ∗} is the map induced from δ̃_{p q+1} which in turn is obtained from δ_{p q+1} by restricting its target space, namely it forms the exact

sequence 0 → E_{p q+2}^{0} ⟶^{δ_{p q+2}} E_{p q+1}^{0} ⟶^{δ̃_{p q+1}} Z_{δ}(E_{pq}^{0}) → 0.

Lemma 5.4 If M = 2 then, with the notation of Definition 5.2,

d_{p2} = N_{p+2} + d_{p1} − b_{p+1 0} − b_{p0}, p > 0
d_{02} = N_{2} + d_{01} − b_{10} − N_{1} + N_{0} − d_{00}.

Proof If M = 2 then (5.1) still implies that d^{k} = 0 for k ≥ 2, because of E2. But now, with the notation (5.2),

dim E_{pq}^{∞} = \{ \begin{matrix}d_{00} − a_{00} & for p = q = 0, \\ d_{01} − a_{00} − a_{01} & for p = 0, q = 1, \\ d_{p1} − a_{p1} & for N ≥ p ≥ 1, q = 1, \\ d_{p2} − a_{p1} & for N ≥ p ≥ 0, q = 2, \\ 0 & otherwise. \end{matrix}

Recall that d_{p q+1}^{1} : E_{p q+1}^{1} → E_{pq}^{1} is the map induced on the homology

groups from δ_{p q+1}. From the exact sequence 0 → E_{p2}^{0} ⟶^{δ_{p2}} E_{p1}^{0} ⟶^{δ̃_{p1}} Z_{δ}(E_{p0}^{0}) → 0 we conclude

a_{p1} = dim ker δ̃_{p1 ∗} = d_{p1} − b_{p0}

which now implies the statement. □

Lemma 5.5 If M = 3 then, with the notation of Definition 5.2 and (5.3)

d_{p3} = N_{p+3} + d_{p2} + d_{p+1 1} − b_{p1} − b_{p+1 1} − b_{p+1 0} − b_{p+2 0}, p > 0,

d_{03} = N_{3} − N_{2} + N_{1} − N_{0} + d_{02} + d_{11} − d_{01} + d_{00} − b_{11} − b_{20}.

Proof Now (5.1) implies that d^{k} = 0 only for k > 2 so that E^{3} = E^{∞}. With the notation

μ_{pq} := rk d_{p−1 q+2}^{2} : E_{p−1 q+2}^{2} → E_{pq}^{2}

and (5.2) we get

dim E_{pq}^{∞} = \{ \begin{matrix}d_{00} − a_{00} & for p = q = 0, \\ d_{01} − a_{00} − a_{01} & for p = 0, q = 1, \\ d_{p1} − a_{p1} − μ_{p1} & for N ≥ p ≥ 1, q = 1, \\ d_{p2} − a_{p1} − a_{p2} & for N ≥ p ≥ 0, q = 2, \\ d_{p3} − a_{p2} − μ_{p+1 1} & for N > p ≥ 0, q = 3, \\ d_{N3} − a_{N2} & for p = N, q = 3, \\ 0 & otherwise. \end{matrix}

Let us recall the definition of d^{2}. Look at the commuting diagram

\begin{aligned}0 & → & E_{p3}^{0} & ⟶^{δ_{p3}} & E_{p2}^{0} & ⟶^{δ_{p2}} & E_{p1}^{0} & ⟶^{δ_{p1}} & ⋯ \\ & & & & ↓δ̃_{p2} & ↗^{ı_{p0}} & ↓δ̃_{p1} & ↗^{ı_{p0}} \\ & & & & Z_{δ}(E_{p1}^{0}) & & Z_{δ}(E_{p0}^{0}) \end{aligned} .

Let θ : H_{∂}(Z_{δ}(E_{p1}^{0})) → H_{∂}(E_{p−1 3}^{0}) = E_{p−1 3}^{1} be the connecting map

of the exact sequence 0 → E_{p3}^{0} ⟶^{δ_{p3}} E_{p2}^{0} ⟶^{δ̃_{p2}} Z_{δ}(E_{p1}^{0}) → 0. Then d_{p−1 3}^{2} : E_{p−1 3}^{2} → E_{p1}^{2} is the map induced by ı_{p1 ∗} ∘ θ^{−1} : E_{p−1 3}^{1} → E_{p1}^{1}.

Since θ^{−1} identifies E_{p−1 3}^{2} = Z_{d^{1}}(E_{p−1 3}^{1}) ≅^{θ^{−1}} H_{∂}(Z_{δ}(E_{p1}^{0}))/Imδ̃_{p2 ∗} we

get

μ_{p1} = rk ı_{p1 ∗} − rk δ_{p2 ∗} = dim ker δ̃_{p1 ∗} − rk δ_{p2 ∗} = (d_{p1} − b_{p0}) − a_{p1}.

Furthermore, using a_{p2} = dim ker δ̃_{p2 ∗} = d_{p2} − b_{p1} one obtains the statement. □

Definition 5.6 We now specify to double complexes of the form E_{pq}^{0} = Λ_{p}Γ ⊗ C^{q}, ∂_{pq} = ∂_{p} ⊗ 1 and δ_{pq} = (−1)^{p} ⊗ δ_{q} as they arise in our application however tacitly changing the ring to be ℚ. Below, d_{pq} shall always denote the dimension of E_{pq}^{1} := H_{δ}(Λ_{p}Γ ⊗ C^{q}).

Clearly, E_{pq}^{0} is non-trivial only for 0 ≤ p ≤ rk Γ and 0 ≤ q ≤ dim V. Furthermore, E_{p0}^{1} = H_{δ}(Λ_{p}Γ ⊗ C^{0}) = H_{p}(Γ, C^{0}). Since C^{0} is always a free Γ-module one has H_{p}(Γ, C^{0}) = 0 for p > 0 and the dimension of H_{0}(Γ, C^{0}) is simply |I_{0}|, the number of orbits of in P:

d_{00} = L_{0}.

In particular E1 and E2 are satisfied. Furthermore, N_{k} = \binom{rk Γ}{k}.

The case dim V = 1 (2.1,2.2) can now be immediately obtained using Lemma 5.3 and Proposition 5.1.

Proof of Theorem 2.5 A proof can also be found in [FHK] but we repeat it here using the framework of spectral sequences. So let dim V = 2. We seek to apply Lemma 5.4. For that we need to determine d_{p1} and b_{p0}.

To determine d_{p1} we use Corollary 4.2. H(Γ^{α}, C_{α}^{1}) can be computed using the double complex (Λ_{p}Γ^{α} ⊗ C_{α}^{q}, ∂_{pq}^{α}, δ_{pq}^{α}). So let us consider the spectral sequence which arises when starting with homology in ∂^{α}. As for the case dim V = 1 one concludes that E_{p0}^{1} is non-trivial only for p = 0 and the dimension of E_{p0}^{1} equal to L_{0}^{α}, the number of orbits of points in P ∩ W_{α}. Since rk Γ^{α} = ν we get

d_{p1} = L_{1} \binom{ν}{p + 1} , p > 0,

d_{01} = L_{1}(ν − 1) + ∑_{α∈I_{1}} L_{0}^{α}.

It remains to compute b_{p0} = rk δ̃_{p1 ∗} : H_{p}(Γ, C^{1}) → H_{p}(Γ, Z_{δ}(C^{0})). For that look at the following commuting diagram of exact sequences

0 → C_{α}^{1} ⊗ ℤ[Γ/Γ^{α}] ⟶^{δ_{1}^{α}⊗1} C_{α}^{0} ⊗ ℤ[Γ/Γ^{α}] → ℤ[Γ/Γ^{α}] → 0
↓ δ̃_{1}^{α} ⊗ 1  ↓  ↓ ϵ^{α}
0 →  Z_{δ}(C^{0})  ↪  C^{0}  ⟶^{ϵ}  ℤ  → 0

where the middle verticle arrow is the inclusion, the right vertical arrow the sum of the coefficients, ϵ^{α}[γ] = 1, and the direct sum over all α ∈ I_{1} of the left vertical arrows is δ̃_{p1}. The above commutative diagram gives rise to two long exact sequences of homology groups together with vertical maps, all commuting, (δ̃_{p1}^{α} ⊗ 1)_{∗} : H_{p}(Γ, C_{α}^{1} ⊗ ℤ[Γ/Γ^{α}]) → H_{p}(Γ, Z_{δ}(C^{0})) being one of them. Now we use that for p > 0, H_{p}(Γ, C_{α}^{0} ⊗ℤ[Γ/Γ^{α}]) = 0 and H_{p}(Γ, C^{0}) = 0 which implies H_{p}(Γ, C_{α}^{1} ⊗ ℤ[Γ/Γ^{α}]) ≅ H_{p+1}(Γ, ℤ[Γ/Γ^{α}]) ≅ H_{p+1}(Γ^{α}, ℤ) and H_{p}(Γ, Z_{δ}(C^{0})) ≅ H_{p+1}(Γ, ℤ) and

(δ̃_{p1}^{α} ⊗ 1)_{∗} = ϵ_{p+1}^{α}.

By Lemma 4.1 the map ϵ_{p}^{α} can be identified with the inclusion Λ_{p}Γ^{α} ↪ Λ_{p}Γ. For p > 0 therefore,

b_{p0} = rk ⟨Im(δ̃_{p1}^{α} ⊗ 1)_{∗} : α ∈ I_{1}⟩ = rk ⟨Λ_{p+1}Γ^{α} : α ∈ I_{1}⟩.  (5.4)

Direct application of Lemma 5.4 yields now the formulas for the dimensions stated in Theorem 2.2. □

Proof of Theorem 2.6 Let dim V = 3. To apply Lemma 5.5 we need to determine d_{p1}, d_{p2}, b_{p0}, b_{p1}. To calculate d_{p1} we use Corollary 4.2 and Lemma 5.3 to obtain, for Θ ∈ I_{1},

dim H_{p}(Γ^{Θ}, C_{Θ}^{1}) = \binom{ν}{p + 1} + \{ \begin{matrix}L_{0}^{Θ} − 1 & for p = 0, \\ 0 & for p > 0. \end{matrix}

Hence

d_{p1} = L_{1} \binom{ν}{p + 1} + \{ \begin{matrix}−L_{1} + ∑_{Θ∈I_{1}} L_{0}^{Θ} & for p = 0, \\ 0 & for p > 0. \end{matrix}

To calculate d_{p2} we consider the complexes, α ∈ I_{2},

0 → C_{α}^{2} ⟶^{δ^{α}} C_{α}^{1} ⟶^{δ^{α}} C_{α}^{0} → 0.

The appropriate double complex to consider is therefore (Λ_{p}Γ^{α} ⊗ C_{α}^{q}, ∂_{pq}^{α}, δ_{pq}^{α}). Repeating the arguments of the proof of Theorem 2.2 (but now using Corollary 4.3 in place of 4.2) we get

dim H_{p}(Γ^{α}, C_{α}^{2}) = \binom{2ν}{p + 2} + L_{1}^{α} \binom{ν}{p + 1} − r_{p}^{α} − r_{p+1}^{α} p > 0,

dim H_{0}(Γ^{α}, C_{α}^{2}) = \binom{2ν}{2} − 2ν + 1 + L_{1}^{α}(ν − 1) − r_{1}^{α} − L_{0}^{α} + ∑_{Θ∈I_{1}^{α}} L_{0}^{Θ}

where

r_{p}^{α} = rk ⟨Λ_{p+1}Γ^{Θ} : Θ ∈ I_{1}^{α}⟩.

Writing F = ∑_{α∈I_{2}} L_{1}^{α} we thus get

d_{p2} = L_{2} \binom{2ν}{p + 2} + F \binom{ν}{p + 1} − ∑_{α∈I_{2}}(r_{p}^{α} + r_{p+1}^{α}), p > 0,

\begin{matrix}d_{02} = L_{2} (\binom{2ν}{p + 2} − 2ν + 1) + & F (ν − 1) \\ + ∑_{α∈I_{2}} ( ∑_{Θ∈I_{1}^{α}} L_{0}^{Θ} − L_{0}^{α} − r_{1}^{α}) . \end{matrix}

We now compute b_{p0} and b_{p1}. For the first we can again repeat the arguments which yield (5.4) to obtain

b_{p0} = rk ⟨Λ_{p+1}Γ^{Θ} : Θ ∈ I_{1}⟩.

b_{p1} is the rank of δ̃_{p2 ∗}, i.e. the rank of the map induced from δ̃_{2} : C^{2} → Z_{δ}(C^{1}). To determine it we consider the commutative diagram with exact rows

0 → C_{α}^{2} ⊗ ℤ[Γ/Γ^{α}] ⟶^{δ_{2}^{α}⊗1} C_{α}^{1} ⊗ ℤ[Γ/Γ^{α}] ⟶^{δ̃_{1}^{α}⊗1} Z_{δ}(C_{α}^{0}) ⊗ ℤ[Γ/Γ^{α}]→ 0
↓δ̃_{2}^{α} ⊗ 1  ↓  ↓
0 → Z_{δ}(C^{1})  ↪  C^{1}  ⟶^{δ_{1}}  Z_{δ}(C^{0})  → 0.

The middle and right vertical maps are the obvious inclusions. By Corollary 4.2 the direct sum of the left vertical arrows of the diagrams which arise if α ∈ I_{2} is δ̃_{2}. The diagram gives rise to a commutative diagram of which the following is one degree

H_{p}(Γ, C_{α}^{2} ⊗ ℤ[Γ/Γ^{α}]) → H_{p}(Γ, C_{α}^{1} ⊗ ℤ[Γ/Γ^{α}]) ⟶
↓(δ̃_{p2}^{α} ⊗ 1)_{∗}  ↓
H_{p}(Γ, Z_{δ}(C^{1}))  →  H_{p}(Γ, C^{1})  ⟶
⟶^{(δ̃_{p1}^{α}⊗1)_{∗}}H_{p}(Γ, Z_{δ}(C_{α}^{0}) ⊗ ℤ[Γ/Γ^{α}])
↓
⟶^{δ_{p1 ∗}}  H_{p}(Γ, Z_{δ}(C^{0}))

(5.5)

also with exact sequences. In particular,

b_{p1} = rk ⟨Im(δ̃_{p2}^{α} ⊗ 1)_{∗} : α ∈ I_{2}⟩.

Now we can follow the same analysis as in the proof of Theorem 2.2 to see that, for p > 0, (5.5) is isomorphic to

→ H_{p}(Γ, C_{α}^{2} ⊗ ℤ[Γ/Γ^{α}]) → ⨁_{Θ∈I_{1}^{α}} Λ_{p+1}Γ^{Θ} ⟶^{β_{p}^{α}} Λ_{p+1}Γ^{α} →
↓(δ̃_{p2}^{α} ⊗ 1)_{∗}  ↓j_{p}^{α}  ↓ı_{p}^{α}
→ H_{p}(Γ, Z_{δ}(C^{1})) → ⨁_{Θ∈I_{1}} Λ_{p+1}Γ^{Θ} ⟶^{γ_{p}} Λ_{p+1}Γ →
(5.6)

where β_{p}^{α} is the direct sum over Θ ∈ I_{1}^{α} of the inclusions Λ_{p+1}Γ^{Θ} ↪ Λ_{p+1}Γ^{α}, γ_{p} is the direct sum over Θ ∈ I_{1} of the inclusions Λ_{p+1}Γ^{Θ} ↪ Λ_{p+1}Γ, and j_{p}^{α} and ı_{p}^{α} are the obvious inclusions. As we are working with finitely generated ℚ-modules we obtain from (5.6)

Im(δ̃_{p2}^{α} ⊗ 1)_{∗} ≅ Imı_{p+1}^{α}/(Imı_{p+1}^{α} ∩ Imγ_{p+1}) ⊕ j_{p}^{α}(ker β_{p}^{α}).

Since I_{1} = ⋃_{α_{∈}I_{2}} I_{1}^{α} the direct sum over α ∈ I_{1}^{α} of j_{p}^{α} is surjective. Hence Imγ_{p} ⊂ ⟨Imı_{p}^{α} : α ∈ I_{2}⟩ and therefore

⟨Imı_{p}^{α}/(Imı_{p}^{α} ∩ Imγ_{p}) : α ∈ I_{2}⟩ = ⟨Imı_{p}^{α} : α ∈ I_{2}⟩/Imγ_{p}.

Furthermore, using also that ı_{p}^{α} is injective, we get j_{p}^{α}(ker β_{p}^{α}) = Imj_{p}^{α} ∩ ker γ_{p} and

⟨Imj_{p}^{α} ∩ ker γ_{p} : α ∈ I_{2}⟩ = ker γ_{p}.

Thus we have

⟨Im(δ̃_{p2}^{α} ⊗ 1)_{∗} : α ∈ I_{2}⟩ ≅ ⟨Imı_{p+1}^{α} : α ∈ I_{2}⟩/Imγ_{p+1} ⊕ ker γ_{p}

which, since rk γ_{p} = b_{p0}, implies

b_{p1} = rk ⟨Λ_{p+2}Γ^{α} : α ∈ I_{2}⟩ − b_{p+1 0} + L_{1} \binom{ν}{p + 1} − b_{p0}.

With these results we obtain from Lemma 5.5 the ranks stated in Theorem 2.3. □

6 Example: Ammann-Kramer tilings

In Chapter IV we showed that generically, canonical projection tilings have infinitely generated cohomology. Nevertheless, all tilings known to us which are used to describe quasicrystals have finitely generated cohomology. In [GK] a list of results for the known 2-dimensional tilings used by quasicrystallographers was presented. Here we present the first result for a 3-dimensional projection tiling with finitely generated cohomology. It is the Ammann-Kramer tiling. The tiling was invented before the discovery of quasicrystals [KrNe] and rediscovered in [DK] [Els]. It has been used to describe icosahedral quasicrystals in [EH] [HE]. It is sometimes also called 3-dimensional Penrose tiling because it generalizes in a way Penrose’s 2-dimensional tilings. A short description of this tiling and further 3-dimensional tilings related icosahedral symmetry is given in [KrPa]. The tiling is obtained by the canonical projection method from the data (ℤ^{6}, E), ℤ^{6} being the lattice generated by an orthonormal basis of ℝ^{6}, and E being obtained from symmetry considerations involving the representation theory of the icosahedral group. Projecting ℤ^{6} orthogonally onto the orthocomplement V of E one obtaines the (dense) lattice Γ generated by the six vectors

( \begin{matrix}τ \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}−τ \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ τ \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ τ \\ 1 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ −τ \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ τ \end{matrix} ) .

with respect to an orthonormal basis of V. Here τ = \frac{\sqrt{5}−1}{2} and we have the usual relation τ^{2} + τ − 1 = 0. The acceptance domain is the orthogonal projection of the 6-dimensional unit cube into V and forms the triacontrahedron whose vertices are the linear combinations with coefficients 0, 1 of the above vectors. It has 30 faces which are all triangles and the affine hyperplanes W are the planes spanned by these triangles. Although we have an action of the icosahedral group (which even permutes the faces) the geometry of their intersections is quite complex, but can be summarized as follows. Singular subspaces in V: The two triangles which are in opposite position of the triacontrahedron belong to the same orbit (under the action of Γ) and representatives of the 15 orbits of singular planes are given by the spans of all possible pairs of vectors from our list of 6 vectors above. Thus I_{2} has L_{2} = 15 elements. The 15 planes can be gathered into five orthogonal triples. Pairs of these planes intersect in singular lines and triples intersect in singular points. Singular lines occur in three classes: Intersections of 5 planes simultaneously at a line (class I - 6 possible orientations): Intersections of 3 planes only (class II - 15 possible orientations): Intersections of 2 planes only (class III - 10 possible orientations). Each of the class I and III orientations is a single orbit, but each of the class II orientations splits into two orbits. Thus if we write I_{1} = I_{1}^{I} ∪ I_{1}^{II} ∪ I_{1}^{III}, a disjoint union according to the class, we have |I_{1}^{I}| = 6, |I_{1}^{II}| = 2 × 15, and |I_{1}^{III}| = 10. In particular, we have L_{1} = 46 orbits of singular lines, arranged in 31 possible orientations. A similar calculation finds L_{0} = 32 orbits of singular points formed at the intersection of three or more singular planes: Briefly, every singular point lies at the intersection of some orthogonal triple of planes. Given a particular othogonal triple, those points which are at the intersection of three planes in this orientation split into 8 orbits (because intersections of orthogonal planes give class II lines). However, two of those orbits are found at intersections of planes parallel to every other orientation (e.g. the orbit of the origin), and the remaining 6 are unique to the particular orthogonal triple chosen. This adds up to 6 × 5 + 2 = 32 orbits therefore. Singular subspaces in W_{i}, i ∈ P_{2}: Since all W_{i} lie in one orbit of the semidirect product of π^{⊥}(ℤ^{6}) with the icosahedral group it doesn’t matter which i to take. Within a singular plane W_{i} there are 2 directions of each class of lines, but each of the class II lines is in two orbits. Thus if we denote the class of i in I_{2} by α and decompose I_{1}^{α} = I_{1}^{α I}∪I_{1}^{α II}∪I_{1}^{α III} disjointly according to the classes of its singular lines we have |I_{1}^{α I}| = 2, |I_{1}^{α II}| = 2 × 2, and |I_{1}^{α III}| = 2, giving a total of L_{1}^{α} = 8 orbits of singular lines in 6 possible directions. In particular L̃_{1} = 15 × 8 − 46 = 74. A careful calculation finds the singular points arranged into L_{0}^{α} = 8 orbits. Singular subspaces in WΘ̂, Θ̂ ∈ P_{1}: In each class I and class III line we have exactly two orbits of singular points. In each class II line we have exactly 4 orbits. Hence L_{0}^{Θ} = 2 if Θ, the class of Θ̂ in I_{1}, is a class I or class III line whereas otherwise L_{0}^{Θ} = 4. This gives ∑_{Θ∈I_{1}} L_{0}^{Θ} = 2 × 6 + 4 × 30 + 2 × 10 = 152 and ∑_{Θ∈I_{1}^{α}} L_{0}^{Θ} = 2 × 2 + 4 × 4 + 2 × 2 = 24. Altogether e = 120.

To find the ranks R_{1}, R_{2}, a more detailed analysis of the generators of the stabilizers and their inner products must be made. Since this involves finding the rank of matrices of up to 15 × 15 in size, it is best checked by computer: R_{1} = 69 and R_{2} = 9.

Specializing Theorem 2.3 to dim V = 3 and rk Γ = 6 we obtain as the only nonzero ranks

D_{3} = 1
D_{2} = 6 + L_{2} − R_{2}
D_{1} = 15 + 4L_{2} + L̃_{1} − R_{1} − R_{2}
D_{0} = 10 + 3L_{2} + L̃_{1} + e − R_{1}.

This yields the following numbers for the ranks of the homology groups:

D_{0} = 180, D_{1} = 71, D_{2} = 12, D_{3} = 1.

So to conclude we have:

K_{0}(C^{∗}(GT )) ≅ ℤ^{192} K_{1}(C^{∗}(GT )) ≅ ℤ^{72}

for the Ammann-Kramer tiling T . We are grateful to Franz Gähler for assistence in completing the last step on the computer and for verifying these results generally.

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