Torus bundles over lens spaces

Abstract

Let $p$ be an odd prime and let $\rho :\mathbb{Z}/p\to {\mathrm{GL}}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $\Gamma :={\mathbb{Z}}^n{⋊}_{\rho }\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T_{\rho }^n{\times }_{\mathbb{Z}/p}{S}^{\ell }$ over the lens space $S^{\ell }/\mathbb{Z}/p$ has fundamental group $\Gamma$. When $\mathbb{Z}/p$ fixes only the origin of ${\mathbb{Z}}^n$, Davis and Lück \cite{DavisLuckTorusBundles} compute the $L$-groups $L_m^{⟨j⟩}(\mathbb{Z}[\Gamma ])$ 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 $L_m^{⟨j⟩}(\mathbb{Z}[\Gamma ])$ and ${\mathcal{S}}^{geo,s}(M)$ in a case where $\underline{E}\Gamma$ has a non-discrete singular set.

arXiv comment39 pages