Euler characteristics and cohomology for quasiperiodic projection patterns

This thesis investigates quasiperiodic patterns and, in particular, polytopal projection patterns, which are produced using the projection method by choosing the acceptance domain to be a polytope. Cohomology theories applicable in this setting are defined, together with the Euler characteristic.;Formulae for the Cech cohomology Hˇ* ( M P ) and Euler characteristic eP are determined for polytopal projection patterns of codimension 2 and calculations are carried out for several examples. The Euler characteristic is shown to be undefined for certain codimension 3 polytopal projection patterns. The Euler characteristic eP is proved to be always defined for a particular class of codimension n polytopal projection patterns P and a formula for eP for such patterns is given. The finiteness or otherwise of the rank of Hˇm(M P ) ⊗ Q for m ≥ 0 is also discussed for various classes of polytopal projection patterns. Lastly, a model for M P is considered which leads to an alternative method for computing the rank of Hˇm(M P ) ⊗ Q for P a d-dimensional codimension n polytopal projection pattern with d > n..