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 $H^{\text{sing}}_0(B\mathcal{G};\mathbb{Z})\cong\bigoplus_{x\in X}\mathbb{Z}$ 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.