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

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

2011-10-07

arxiv.org/abs/1110.1433v1

Mathematics

Operator Algebras

33 pages, 9 pictures and one diagram prepared in TiKZ

Scientific paper

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.

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