INTRODUCTION

5

orthoprojection onto E1-. We first disconnect V along the boundaries

of all 7rJ-(ZiV)-translates of K (we speak loosely here, but make the

idea precise in Chapter I). Then X? 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 bound-

aries 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 Xj-

appear as continuous chains in the limit. This makes the group co-

homology of the dynamical action of

Zd

on Xj- 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

AT 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 AT as well and thus

define elements of

KQ(AT)-

Therefore, the ordered ifo-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] [Kl]. One

of the main results of this memoir is the determination in Chapter

V of closed formulae for the ranks of the if-groups corresponding

to canonical projection method patterns with small codimension (as

one calls the dimension of V). These formulae 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 KQ or the image on a

tracial state.

Further important results of this memoir concern the structure

of a if-group of a canonical projection method pattern. We find that

its ifo-group is generically infinitely generated. But when the rank

of its rationalization is finite then it has to be free abelian. We ob-

serve in Chapters III and IV that both properties are obstruction to

some kinds of self-similarity. More precisely, infinitely generated ra-

tionalized cohomology rules out that the tiling is a substitution tiling.

On the other hand, if we know already that the tiling is substitu-

tional then its if-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