2013/10/31 by Alex Kumjian, David Pask, Aidan Sims +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Cograph #Combinatorics #Discrete mathematics #Disjoint sets #Graph #Graph product #Homotopy and Cohomology in Algebraic Topology #Line graph #Mathematics #Pathwidth #Quotient #Rank (graph theory) #Topology (electrical circuits) #math.AT #math.OA
paper · pdf · doi:10.1016/j.jcta.2016.04.005
published as J. Comb. Th., Series A, 143 (2016), 19-41 · Updated to agree with published version
openalex publication_date 2016/05/16 · arxiv created 2016/06/08 · arxiv updated 2016/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.