Abstract. We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs, and show that the C ∗-algebras defined are coherent with the existing theory. 1.

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C ∗-algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C ∗ (Λ). The K-groups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances and we consider the case k =3. Weprovethat for arbitrary k, the torsion-free rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and for k = 2 we determine the position of

Abstract. We describe a class of rank-2 graphs whose C ∗-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C ∗-algebra. We identify rank-2 Bratteli diagrams whose C ∗-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C ∗-algebras contain as full corners the irrational rotation algebras and the Bunce-Deddens algebras. 1.

Abstract. A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C ∗-algebras. We show how to build a (k + 1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra Pn whose K-theory is opposite to that of On, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras. 1.

Abstract. We define the notion of a Λ-system of C ∗-correspondences associated to a higher-rank graph Λ. Roughly speaking, such a system assigns to each vertex of Λ a C ∗-algebra, and to each path in Λ a C ∗-correspondence in a way which carries compositions of paths to balanced tensor products of C ∗-correspondences. Under some simplifying assumptions, we use Fowler’s technology of Cuntz-Pimsner algebras for product systems of C ∗-correspondences to associate a C ∗-algebra to each Λ-system. We then construct a Fell bundle over the path groupoid GΛ and show that the C ∗-algebra of the Λ-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature. 1.

A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C∗-algebras. We show how to build a (k + 1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra Pn whose K-theory is opposite to that of On, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.

Abstract. We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C ∗-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set ˜ Q(Λ) from a k-graph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of ˜ Q(Λ) as defined by Grandis. 1.