Groupoid homology and classifying-space homology are not isomorphic

Abstract

For an ample groupoid G, Matui-type groupoid homology H_•(G; Z) is built from the nerve G_• via the Moore complex of compactly supported locally constant chains C_c (G_n, 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 H₀ (G; Z) ≅ C(X, Z), a countable group, whereas H₀^sing (BG; Z) ≅ Σ_x∈X Z has cardinality 2^ℵ0. Cardinality alone separates the two groups, and it separates them in degree 0 already. In every positive degree they agree for this groupoid.