Torus bundles over lens spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909212392554496 |
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| author | Wang, Oliver H. |
| author_facet | Wang, Oliver H. |
| contents | Let $p$ be an odd prime and let $ρ:\mathbb{Z}/p\rightarrow\operatorname{GL}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $Γ:=\mathbb{Z}^n\rtimes_ρ\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T^n_ρ\times_{\mathbb{Z}/p}S^{\ell}$ over the lens space $S^{\ell}/\mathbb{Z}/p$ has fundamental group $Γ$. When $\mathbb{Z}/p$ fixes only the origin of $\mathbb{Z}^n$, Davis and Lück \cite{DavisLuckTorusBundles} compute the $L$-groups $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and the structure set $\mathcal{S}^{geo,s}(M)$. In this paper, we extend these computations to all actions of $\mathbb{Z}/p$ on $\mathbb{Z}^n$. In particular, we compute $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and $\mathcal{S}^{geo,s}(M)$ in a case where $\underline{E}Γ$ has a non-discrete singular set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_02566 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Torus bundles over lens spaces Wang, Oliver H. Geometric Topology Let $p$ be an odd prime and let $ρ:\mathbb{Z}/p\rightarrow\operatorname{GL}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $Γ:=\mathbb{Z}^n\rtimes_ρ\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T^n_ρ\times_{\mathbb{Z}/p}S^{\ell}$ over the lens space $S^{\ell}/\mathbb{Z}/p$ has fundamental group $Γ$. When $\mathbb{Z}/p$ fixes only the origin of $\mathbb{Z}^n$, Davis and Lück \cite{DavisLuckTorusBundles} compute the $L$-groups $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and the structure set $\mathcal{S}^{geo,s}(M)$. In this paper, we extend these computations to all actions of $\mathbb{Z}/p$ on $\mathbb{Z}^n$. In particular, we compute $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and $\mathcal{S}^{geo,s}(M)$ in a case where $\underline{E}Γ$ has a non-discrete singular set. |
| title | Torus bundles over lens spaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2203.02566 |