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

By Alex Kumjian, David Pask, Aidan Sims. 2011

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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 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/18925a626964918728c6e56e4b58f1a6506c41b1