$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid

TL;DR

This paper demonstrates that for the Cantor unit groupoid, the homology computed via Matui's method differs from the singular homology of its classifying space, highlighting a fundamental discrepancy in degree zero.

Contribution

The paper provides an explicit example showing the divergence between Matui type homology and singular homology for a specific ample groupoid.

Findings

Homology via Moore chains can differ from singular homology.

Discrepancy appears in degree 0 for the Cantor unit groupoid.

Highlights limitations of existing homology computations for groupoids.

Abstract

For an ample groupoid $\mathcal{G}$, Matui type groupoid homology is computed from the nerve $\mathcal{G}_\bullet$ via Moore chains $C_c(\mathcal{G}_n,\mathbb{Z})$ and the alternating sum of pushforwards along the face maps. We give an explicit example showing that this homology need not agree with singular homology of the classifying space $B\mathcal{G}$. The discrepancy occurs already in degree $0$ for the unit groupoid on the Cantor set.