2011/10/07 by Alex Kumjian, Kumjian, Alex, David Pask +3 · 2 citations
Computer Science · Mathematics · #55N10 #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Primary 46L05 #Secondary 18G60 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1110.1433
openalex publication_date 2011/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set Q(Λ) from a k-graph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of Q(Λ) as defined by Grandis.