Cohomology of tiling spaces: beyond primitive substitutions

This thesis explores the combinatorial and topological properties of tiling spaces associated to 1-dimensional symbolic systems of aperiodic type and their associated algebraic invariants. We develop a framework for studying systems which are more general than primitive substitutions, naturally partitioned into two instances: in the first instance we allow systems associated to sequences of substitutions of primitive type from a finite family of substitutions (called mixed substitutions); in the second instance we concentrate on systems associated to a single substitution, but where we entirely remove the condition of primitivity. We generalise the notion of a Barge-Diamond complex, in the one-dimensional case, to any mixed system of symbolic substitutions. This gives a way of describing the associated tiling space as an inverse limit of complexes. We give an effective method for calculating the Cech cohomology of the tiling space via an exact sequence relating the associated sequence of substitution matrices and certain subcomplexes appearing in the approximants. As an application, we show that there exists a system of substitutions on two letters which exhibit an uncountable collection of minimal tiling spaces with distinct isomorphism classes of Cech cohomology. In considering non-primitive substitutions, we naturally divide this set of substitutions into two cases: the minimal substitutions and the non-minimal substitutions. We provide a detailed method for replacing any non-primitive but minimal substitution with a topologically conjugate primitive substitution, and a more simple method for replacing the substitution with a primitive substitution whose tiling space is orbit equivalent. We show that an Anderson-Putnam complex with a collaring of some appropriately large radius suffices to provide a model of the tiling space as an inverse limit with a single map. We apply these methods to effectively calculate the Cech cohomology of any substitution which does not admit a periodic point in its subshift. Using its set of closed invariant subspaces, we provide a pair of invariants which are each strictly finer than the usual Cech cohomology for a substitution tiling space.