Ample groupoids: equivalence, homology, and Matui's HK conjecture

Citation

• Farsi, C., Kumjian, A., Pask, D., & Sims, A. (2018). Ample groupoids: equivalence, homology, and Matui's HK conjecture. Retrieved from http://arxiv.org/abs/1808.07807v1

Web Of Science Accession Number

Abstract

• We investigate the homology of ample Hausdorff groupoids. We establish that a

  number of notions of equivalence of groupoids appearing in the literature

  coincide for ample Hausdorff groupoids, and deduce that they all preserve

  groupoid homology. We compute the homology of a Deaconu{Renault groupoid

  associated to k pairwisecommuting local homeomorphisms of a zero-dimensional

  space, and show that Matui's HK conjecture holds for such a groupoid when k is

  one or two. We specialise to k-graph groupoids, and show that their homology

  can be computed in terms of the adjacency matrices, using a chain complex

  developed by Evans. We show that Matui's HK conjecture holds for the groupoids

  of single vertex k-graphs which satisfy a mild joint-coprimality condition. We

  also prove that there is a natural homomorphism from the categorical homology

  of a k-graph to the homology of its groupoid.

UOW Authors

•   Pask, David
•   Sims, Aidan

Publication Date

• 2018

Citation

• Farsi, C., Kumjian, A., Pask, D., & Sims, A. (2018). Ample groupoids: equivalence, homology, and Matui's HK conjecture. Retrieved from http://arxiv.org/abs/1808.07807v1

Web Of Science Accession Number