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Topological spaces associated to higher-rank graphs

Journal Article


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Abstract


  • We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-k graphs gives rise to pullbacks of the associated C*-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a rank-2 graph. We also show how to construct k-spheres and wedges of k-spheres as topological realisations of rank-k graphs.

Publication Date


  • 2016

Citation


  • Kumjian, A., Pask, D., Sims, A. & Whittaker, M. F. (2016). Topological spaces associated to higher-rank graphs. Journal of Combinatorial Theory, Series A, 143 19-41.

Scopus Eid


  • 2-s2.0-84971671418

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6746&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5718

Number Of Pages


  • 22

Start Page


  • 19

End Page


  • 41

Volume


  • 143

Place Of Publication


  • United States

Abstract


  • We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-k graphs gives rise to pullbacks of the associated C*-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a rank-2 graph. We also show how to construct k-spheres and wedges of k-spheres as topological realisations of rank-k graphs.

Publication Date


  • 2016

Citation


  • Kumjian, A., Pask, D., Sims, A. & Whittaker, M. F. (2016). Topological spaces associated to higher-rank graphs. Journal of Combinatorial Theory, Series A, 143 19-41.

Scopus Eid


  • 2-s2.0-84971671418

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6746&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5718

Number Of Pages


  • 22

Start Page


  • 19

End Page


  • 41

Volume


  • 143

Place Of Publication


  • United States