Torus bundles over lens spaces

TL;DR

This paper extends the computation of algebraic and geometric invariants of torus bundles over lens spaces to all actions of a prime order group on lattices, including cases with non-discrete singular sets.

Contribution

It generalizes previous results by computing $L$-groups and structure sets for all $ ho:bZ/p o ext{GL}_n(bZ)$ actions, even with non-trivial fixed points.

Findings

Computed $L$-groups $L^{raket{j}}_m(bZ[bZ^n times_ ho bZ/p])$ for all actions.

Determined structure sets $bS^{geo,s}(M)$ in cases with non-discrete singular sets.

Extended previous work to more general group actions on lattices.

Abstract

Let $p$ be an odd prime and let $\rho:\mathbb{Z}/p\rightarrow\operatorname{GL}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $\Gamma:=\mathbb{Z}^n\rtimes_{\rho}\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T^n_{\rho}\times_{\mathbb{Z}/p}S^{\ell}$ 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 L\"uck \cite{DavisLuckTorusBundles} compute the $L$-groups $L^{\langle j\rangle}_m(\mathbb{Z}[\Gamma])$ 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 $L^{\langle j\rangle}_m(\mathbb{Z}[\Gamma])$ and $\mathcal{S}^{geo,s}(M)$ in a case where $\underline{E}\Gamma$ has a non-discrete singular set.