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.Comment: Updated to agree with published versio
