On the group cohomology of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911155764592640 |
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| author | Saldaña, Luis Jorge Sánchez Velásquez, Mario |
| author_facet | Saldaña, Luis Jorge Sánchez Velásquez, Mario |
| contents | We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ free of squares, by means of formulas that only depend on $n$, $m$ and the action of $\mathbb{Z}/m$ on $\mathbb{Z}^n$. We want to highlight the fact that we are not impossing any conditions on the $\mathbb{Z}/m$-action on $\mathbb{Z}^n$, and as far as we know our formulas are the first in the literature in this generality. This generalizes previous computations of Lück-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form $\mathbb{Z}^5\rtimes \mathbb{Z}/6$ where its cohomology groups are described in full detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14569 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the group cohomology of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free Saldaña, Luis Jorge Sánchez Velásquez, Mario Algebraic Topology Group Theory 20J06, 55T25, 20K25, 20H15 We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ free of squares, by means of formulas that only depend on $n$, $m$ and the action of $\mathbb{Z}/m$ on $\mathbb{Z}^n$. We want to highlight the fact that we are not impossing any conditions on the $\mathbb{Z}/m$-action on $\mathbb{Z}^n$, and as far as we know our formulas are the first in the literature in this generality. This generalizes previous computations of Lück-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form $\mathbb{Z}^5\rtimes \mathbb{Z}/6$ where its cohomology groups are described in full detail. |
| title | On the group cohomology of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free |
| topic | Algebraic Topology Group Theory 20J06, 55T25, 20K25, 20H15 |
| url | https://arxiv.org/abs/2403.14569 |