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arXiv · 1310.6100

Topological spaces associated to higher-rank graphs

Alex Kumjian·David Pask·Aidan Sims·Michael F. Whittaker
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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.

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Alex Kumjian, David Pask, Aidan Sims, Michael F. Whittaker. 2013-10-23. Topological spaces associated to higher-rank graphs. https://doi.org/10.1016/j.jcta.2016.04.005

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