Homology for higher-rank graphs and twisted C*-algebras

TL;DR

This paper develops a homology theory for higher-rank graphs, linking algebraic topology with operator algebras, and demonstrates its applications to twisted C*-algebras including noncommutative tori.

Contribution

It introduces a new homology framework for k-graphs, connecting it with topological realizations and extending to twisted C*-algebras with novel combinatorial constructions.

Findings

Homology of k-graphs matches their topological realizations.

Compatible combinatorial and topological constructions are established.

Examples include all noncommutative tori.

Abstract

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 \tilde{Q}(\Lambda) from a k-graph {\Lambda} and demonstrate that the homology and topological realisation of {\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.