We review the definition and elementary properties of the Crainic-Moedijk homology in the setting of ample groupoids. We show that there is a natural homomorphism from the homology of a higher-rank graph to that of its path groupoid. Given an ample groupoid G with compact unit space which is minimal and effective, Matui conjectured that the K-theory of the reduced C*-algebra of G is isomorphic to the homology, and verified his conjecture in a number of important cases. In joint work my coauthors, Carla Farsi, David Pask and Aidan Sims, and I have shown that the isomorphism in the conjecture holds for the path groupoids of higher-rank graphs of ranks one or two even when the unit space is no longer compact.
The homology of ample groupoids and the Matui conjecture (Joint with the Ulam Seminar) Sponsored by the Meyer Fund
We review the definition and elementary properties of the Crainic-Moerdijk homology in the setting of ample groupoids. We show that there is a natural homomorphism from the homology of a higher rank graph to that of its path groupoid. Given an ample groupoid G with compact unit space which is minimal and effective, Matui conjectured that the K-theory of the reduced C*-algebra of G is isomorphic to the homology and verified his conjecture in a number of important cases. In joint work my coauthors, Carla Farsi, David Pask and Aidan Sims, and I have shown that the isomorphism in the conjecture holds for the path groupoids of higher rank graphs of ranks one or two even when the unit space is no longer compact.