# Topological realizations and fundamental groups of higher-rank graphs

@article{Kaliszewski2012TopologicalRA, title={Topological realizations and fundamental groups of higher-rank graphs}, author={S. Kaliszewski and Alex Kumjian and John Quigg and Aidan Sims}, journal={Proceedings of the Edinburgh Mathematical Society}, year={2012}, volume={59}, pages={143 - 168} }

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. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological…

## 6 Citations

### Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

- MathematicsTransactions of the American Mathematical Society, Series B
- 2021

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph
Λ
\Lambda
, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in…

### Deformations of Fell bundles and twisted graph algebras

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract We consider Fell bundles over discrete groups, and the C*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles…

### Computing the fundamental group of a higher-rank graph

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2021

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental…

### The suspension of a graph, and associated C⁎-algebras

- MathematicsJournal of Functional Analysis
- 2019

### Textile systems, coloured graphs and their applications to higher-rank graphs

- Mathematics
- 2013

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We show that textile systems may be…

## References

SHOWING 1-10 OF 21 REFERENCES

### HOMOLOGY FOR HIGHER-RANK GRAPHS AND TWISTED

- Mathematics
- 2011

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homol- ogy of a k-graph coincides with the…

### Groupoid models for the C*-algebras of topological higher-rank graphs

- Mathematics
- 2006

We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical…

### Fundamental groupoids of k-graphs

- Mathematics
- 2004

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…

### Higher Rank Graph C-Algebras

- Mathematics
- 2000

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…

### Coverings of Directed Graphs and Crossed Products of C*-Algebras by Coactions of Homogeneous Spaces

- Mathematics
- 2002

We show that if p:F→E is a covering of directed graphs, then the Cuntz–Krieger algebra C*(F) of F can be viewed as a crossed product of C*(E) by a coaction of a homogeneous space for the fundamental…

### Crossed products of k-graph C*-algebras by Zl

- Mathematics
- 2007

An action of Zl by automorphisms of a k-graph induces an action of Zl by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k + l)-graph whose C*-algebra coincides…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

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…