Groupoid homology and classifying-space homology are not isomorphic
DOI:
https://doi.org/10.17398/Palabras clave:
ample groupoid, étale groupoid, groupoid homology, classifying space, singular homology, Cantor setResumen
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 Cc (Gn, 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 H0sing (BG; Z) ≅ Σx∈X Z has cardinality 2ℵ0. Cardinality alone separates the two groups, and it separates them in degree 0 already. In every positive degree they agree for this groupoid.
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