Groupoid Homology and Classifying-Space Homology Are Not Isomorphic

Abstract

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 the alternating sum of pushforwards along the face maps. For a discrete group the theory agrees with the singular homology of the classifying space. For a totally disconnected locally compact Hausdorff space, viewed as a groupoid of units, it computes compactly supported cohomology instead, which is not a homotopy invariant. We make the resulting discrepancy explicit: 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 $H_0^{\text{sing}}(B\mathcal{G};\mathbb{Z})\cong {⨁}_{x\in X}\mathbb{Z}$ has cardinality $2^{{\aleph }_0}$. Cardinality alone separates the two groups, and it separates them in degree $0$ already. In every positive degree they agree for this groupoid.