Torus bundles over lens spaces

Abstract

Let p be an odd prime and let ρ : ℤ / p → GL n ⁡ ( ℤ ) {\rho:\mathbb{Z}/p\rightarrow\operatorname{{GL}}_{n}(\mathbb{Z})} be an action of ℤ / p {\mathbb{Z}/p} on a lattice and let Γ := ℤ n ⋊ ρ ℤ / p {\Gamma:=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/p} be the corresponding semidirect product. The torus bundle M := T ρ n × ℤ / p S ℓ {M:=T^{n}_{\rho}\times_{\mathbb{Z}/p}S^{\ell}} over the lens space S ℓ / ℤ / p {S^{\ell}/\mathbb{Z}/p} has fundamental group Γ. When ℤ / p {\mathbb{Z}/p} fixes only the origin of ℤ n {\mathbb{Z}^{n}} , Davis and Lück (2021) compute the L-groups L m 〈 j 〉 ⁢ ( ℤ ⁢ [ Γ ] ) {L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and the structure set 𝒮 geo , s ⁢ ( M ) {\mathcal{{S}}^{{\rm geo},s}(M)} . In this paper, we extend these computations to all actions of ℤ / p {\mathbb{Z}/p} on ℤ n {\mathbb{Z}^{n}} . In particular, we compute L m 〈 j 〉 ⁢ ( ℤ ⁢ [ Γ ] ) {L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and 𝒮 geo , s ⁢ ( M ) {\mathcal{{S}}^{{\rm geo},s}(M)} in a case where E ¯ ⁢ Γ {\underline{E}\Gamma} has a non-discrete singular set.