Let \(X\) be obtained from a lens space of dimension \(2n+1\) by deleting a point. Compute \(\pi_{2n}(X)\) as a module over \(\mathbb{Z}[\pi_1(X)]\).
Solution
Let the lens space be \[ L^{2n+1}=S^{2n+1}/G, \qquad G\cong\mathbb Z_m, \] and let \(X=L-\{x\}\). The inverse image of \(x\) in the universal cover consists of one free \(G\)-orbit of \(m\) points, so the universal cover of \(X\) is \[ \widetilde X=S^{2n+1}-G\widetilde x. \] A sphere with \(m\) points deleted is homotopy equivalent to a wedge of \(m-1\) copies of \(S^{2n}\). Therefore \(\widetilde X\) is \((2n-1)\)-connected and \[ \pi_{2n}(X) \cong\pi_{2n}(\widetilde X) \cong H_{2n}(\widetilde X) \cong\mathbb Z^{m-1} \] by Hurewicz.
We now keep track of the \(G\)-action. Equivariant Alexander duality identifies \[ H_{2n}(S^{2n+1}-G\widetilde x) \cong \widetilde H^0(G\widetilde x). \] The latter is the reduced permutation module on the regular \(G\)-set. If \[ \mathbb Z[G]\cong\mathbb Z[t]/(t^m-1), \qquad N=1+t+\cdots+t^{m-1}, \] then \[ \widetilde H^0(G\widetilde x) \cong \mathbb Z[G]/\mathbb Z N. \] Multiplication by \(t-1\) gives an exact sequence \[ 0\longrightarrow\mathbb ZN \longrightarrow\mathbb Z[G] \xrightarrow{\ t-1\ } I_G\longrightarrow0, \] where \[ I_G=\ker(\varepsilon:\mathbb Z[G]\to\mathbb Z) \] is the augmentation ideal. Hence \[ \mathbb Z[G]/\mathbb ZN\cong I_G \] as \(\mathbb Z[G]\)-modules.
Consequently \[ \boxed{\pi_{2n}(X)\cong I_G=(t-1)\mathbb Z[G]} \] as a module over \(\mathbb Z[\pi_1(X)]\cong\mathbb Z[G]\).