$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912904253538304 |
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| author | Melodia, Luciano |
| author_facet | Melodia, Luciano |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid Melodia, Luciano Algebraic Topology Operator Algebras 22A22, 55R35, 55N35, 55U10, 54H11 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. |
| title | $H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid |
| topic | Algebraic Topology Operator Algebras 22A22, 55R35, 55N35, 55U10, 54H11 |
| url | https://arxiv.org/abs/2602.13375 |