Groupoid Homology and Classifying-Space Homology Are Not Isomorphic: The Cantor Unit Groupoid

For an ample groupoid $\mathcal{G}$, Matui-type groupoid homology $H_{\bullet }(\mathcal{G};\mathbb{Z})$ is built from the nerve ${\mathcal{G}}_{\bullet }$ via the Moore complex of compactly supported, locally constant chains $C_c(\mathcal{G}n,\mathbb{Z})$, with differential given by the alternating sum of pushforwards along the face maps. As both this complex and the singular complex of the classifying space $B\mathcal{G}$ originate from the same nerve, one might expect the two homologies to coincide. We show that they need not, and that the failure is sharp: for the unit groupoid on the Cantor set $X$, we compute $H_0(\mathcal{G};\mathbb{Z})\cong C(X,\mathbb{Z})$, a countable group, whereas $H_0^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(B\mathcal{G};\mathbb{Z})\cong {⨁}_{x\in X}\mathbb{Z}$ has cardinality $2^{{\mathrm{\aleph }}_0}$. Thus the two groups are non-isomorphic for cardinality reasons alone, exhibiting a discrepancy already in degree $0$.