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

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