# Topological spaces associated to higher-rank graphs

@article{Kumjian2013TopologicalSA, title={Topological spaces associated to higher-rank graphs}, author={Alex Kumjian and David Pask and Aidan Sims and Michael F. Whittaker}, journal={J. Comb. Theory, Ser. A}, year={2013}, volume={143}, pages={19-41} }

## 5 Citations

### BRANCHING SYSTEMS FOR HIGHER-RANK GRAPH C*-ALGEBRAS

- MathematicsGlasgow Mathematical Journal
- 2018

Abstract We define branching systems for finitely aligned higher-rank graphs. From these, we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a…

### Relative graphs and pullbacks of relative Toeplitz graph algebras

- Mathematics
- 2022

. In this note we generalize a result from a recent paper of Hajac, Reznikoﬀ and Tobolski ([5]). In that paper they give conditions they call admissibility on a pushout diagram in the category of…

### Non‐surjective pullbacks of graph C*‐algebras from non‐injective pushouts of graphs

- MathematicsBulletin of the London Mathematical Society
- 2020

We find a substantial class of pairs of ∗ ‐homomorphisms between graph C*‐algebras of the form C∗(E)↪C∗(G)↞C∗(F) whose pullback C*‐algebra is an AF graph C*‐algebra. Our result can be interpreted as…

### The contravariant functoriality of graph algebras

- Mathematics
- 2022

A BSTRACT . We study pushouts in the category of directed graphs. Our ﬁrst result is that the subcategory given by strongly admissible monomorphisms is closed with respect to pushouts. Next, we…

### Pullbacks of graph C*-algebras from admissible pushouts of graphs

- MathematicsBanach Center Publications
- 2020

We define an admissible decomposition of a graph $E$ into subgraphs $F_1$ and $F_2$, and consider the intersection graph $F_1\cap F_2$ as a subgraph of both $F_1$ and $F_2$. We prove that, if the…

## References

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Abstract We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization.…

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Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of…

### Remarks on some fundamental results about higher-rank graphs and their C*-algebras

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2013

Abstract Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the…

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k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz- Krieger type. Here we develop a theory of the…

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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…

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We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the C∗-algebra of the graph. We show that the crossed product by this action is stably…

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- 2005

This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of…