$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid
Algebraic Topology
2026-02-17 v1 Operator Algebras
Abstract
For an ample groupoid , Matui type groupoid homology is computed from the nerve via Moore chains and the alternating sum of pushforwards along the face maps. We give an explicit example showing that this homology need not agree with singular homology of the classifying space . The discrepancy occurs already in degree for the unit groupoid on the Cantor set.
Cite
@article{arxiv.2602.13375,
title = {$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid},
author = {Luciano Melodia},
journal= {arXiv preprint arXiv:2602.13375},
year = {2026}
}