4 A. FORREST, J. HUNTON AND J. KELLENDONK commutative and non-commutative invariants, to demonstrate their applicability as providing obstructions to a tiling arising as a substi- tution, and finally to provide a practical method for computing them this we illustrate with a number of examples, including that of the Ammann-Kramer (43 dimensional Penrose') tiling. Broadly speaking, one of the perspectives of this memoir is that non-commutative in- variants 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 ZN which is generated by an orthonormal base of R^. Reserving detail and elaborations for later, we call a projection method pattern T on E = Rd a pattern of points (or a finite decoration of it) given by the orthogonal projection onto E of points in a strip (K x E)DZN c RN, where E is a subspace of RN and K x E is the so-called acceptance strip, a fattening of E in M.N defined by some suitably chosen region K in the orthogonal complement E1- of E in R^. The pattern T thus depends on the dimension iV, the positioning of E in WN and the shape of the accept- ance domain K. When this construction was first made [dBl] [KD] the domain K was taken to be the projected image onto E1- of the unit cube in B,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 E1- 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 ZN repositioned by some vector u G R^, i.e., ZN replaced by ZN + u. Completing certain subsets of positioning vectors u with respect to an appropri- ate pseudo-metric gives us the continuous hull MT. This analysis shows in particular that MT contains another transversal XT which gives rise to d independant commuting Z actions and hence to a genu- ine discrete dynamical system (Xj-,2^) whose mapping torus is also MT. This is a key point in relating the K-theory of AT with the cohomology of T in the process, the latter is also identified with the Cech cohomology of MT and with the group cohomology of Zd with coefficients in the continuous integer valued functions over XT- The space XT arises in another way. Let V be a connected com- ponent of the euclidian closure of 7rJ-(ZiV), where TT1- denotes the
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