Torus bundles over lens spaces (Q6587567)

Let \(p\) be an odd prime and \(\rho\colon{\mathbb Z}_p\to \mathrm{GL}_n({\mathbb Z})\) be a homomorphism. Let \(T_\rho^n\) be the torus with the \({\mathbb Z}_p\)-action determined by \(\rho\). If \(S^\ell\) is a sphere with a prescribed \({\mathbb Z}_p\)-action, then the manifold \(M=T_\rho^n\times_{{\mathbb Z}_p}S^\ell\) is a torus bundle over a Lens space with fundamental group \(\Gamma={\mathbb Z}^n\rtimes {\mathbb Z}_p\). The goal is the paper is to make various computations culminating at the structure sets \(S^s(M)\) and \(S^h(M)\).\N\NUsing the Farrell-Jones conjecture, \textit{J. F. Davis} and \textit{W. Lück} [J. Noncommut. Geom. 7, No. 2, 373--431 (2013; Zbl 1279.19003), Commun. Pure Appl. Math. 74, No. 11, 2348--2397 (2021; Zbl 1487.57037)] previously studied the case when \(\rho\) has no nonzero fixed points. This paper extends the discussion as there are nonzero fixed points in the action of \({\mathbb Z}_p\) on \(T_\rho^n\).\N\NAs a \({\mathbb Z}[{\mathbb Z}_p]\)-module, the \(K\)-cohomology \(K^m(T_\rho^n)\) is a sum \(\bigoplus_\ell H^{m+2\ell}(T^n)\) of even or odd cohomology groups of the standard torus depending on the parity of \(m\). This isomorphism is proved in the paper by the collapse of the Atiyah-Hirzebuch spectral sequence for tori. This result then identifies the major calculation of the paper, i.e.~the \(K\)-cohomology of the classifying space \(B\Gamma\), which can be expressed in term of the equivariant \(K\)-theory of \(T_\rho^n\) and the \(p\)-adic integers \(\hat{{\mathbb Z}}_p\). Additionally, after inverting 2, one can see that \(KO_m(B\Gamma)\) is the sum of a finitely generated free \({\mathbb Z}[1/2]\)-module and a \(p\)-torsion group.\N\NAfter computing the \(L\)-groups for \({\mathbb Z}[\Gamma]\) and the Whitehead groups \(\mathrm{Wh}_m(\Gamma)\), the paper assembles these calculations to compute the structure set for \(M\). For \(j\le 2\) there is an isomorphism of the geometric structure set \N\[\NS^{\langle j\rangle}_{n+\ell+1}(M)\cong H_n(T^n_\rho; L({\mathbb Z})\langle 1\rangle)^{{\mathbb Z}_p}\oplus \bigoplus_P L^{\langle j\rangle}_{n+\ell+1} ({\mathbb Z}[N_\Gamma P])/L^{\langle j\rangle}_{n+\ell+1}({\mathbb Z}[W_\Gamma P]).\N\]\NHere \(P\) ranges over the set of conjugacy classes of nontrivial finite subgroups of \(\Gamma\). Also, the group \(N_\Gamma P\) denotes the normalizer of \(P\) in \(\Gamma\) and \(W_\Gamma P=N_\Gamma P/P\) denotes the Weyl group. A similar isomorphism can be written for the periodic structure set. The \(L\)-groups appearing in these computations are computable. Since the normalizers and the Weyl groups are infinite abelian, the theory of Shaneson splittings allows us describe these groups in terms of the \(L\)-groups of \({\mathbb Z}\) and \({\mathbb Z}[{\mathbb Z}_p]\).