INTRODUCTION 7 the projection of slices of Z d + 1 to Wd for more or less arbitrary accept- ance domains. In Chapter IV we give descriptions of the invariants for generic projection patterns arising from arbitrary projections ZN to WLd 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 gen- erated 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 formulae for the cohomology and i^-theory of projection patterns of codimension 1, 2 and 3: in principle the procedure can be iterated to higher codimensions indef- initely, though in practice the formulae 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 de- scribed 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 res- ults now in Chapters IV and V, aiming to found a calculus for projec- tion method tiling cohomology. These papers are currently available as a preprint-series Projection Quasicrystals I-III [FHKI-III] cover- ing 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 acknow- ledge 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 dynam- ics 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
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