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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