On the homology of partial group representations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917013910192128 |
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| author | Jerez, Emmanuel |
| author_facet | Jerez, Emmanuel |
| contents | We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the \textit{universal globalization} $Λ(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$ is naturally isomorphic to the classical group homology $H_{\bullet}(G, Λ(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(Λ(K_{par}G), M))$, where $K_{par}G$ stands for the partial group algebra of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the homology of partial group representations Jerez, Emmanuel Algebraic Topology Group Theory K-Theory and Homology Primary 20J05, 20J06, Secondary 16E30 We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the \textit{universal globalization} $Λ(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$ is naturally isomorphic to the classical group homology $H_{\bullet}(G, Λ(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(Λ(K_{par}G), M))$, where $K_{par}G$ stands for the partial group algebra of $G$. |
| title | On the homology of partial group representations |
| topic | Algebraic Topology Group Theory K-Theory and Homology Primary 20J05, 20J06, Secondary 16E30 |
| url | https://arxiv.org/abs/2404.14650 |