Topological spaces associated to higher-rank graphs

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 $C^{\ast }$-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.

Updated to agree with published version

References in corpus (3)

• On some fundamental results about higher-rank graphs and their C*-algebras
• Fundamental groupoids of k-graphs
• Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs

Cited by in corpus (2)

• Non-surjective pullbacks of graph C*-algebras from non-injective pushouts of graphs
• Branching Systems for Higher-Rank Graph C*-algebras