A split exact sequence of equivariant \(K\)-groups of virtually nilpotent groups (Q1360324)

Let \(\Gamma\) be a finite extension of a finitely presented torsionfree nilpotent group \(N\). Then the finite group \(G= \Gamma/N\) acts on a nil-manifold \(M_\Gamma\). There is the exact sequence of Steinberger and West involving equivariant PL-\(K\)-theory groups and their controlled analogues \[ \widetilde K^{\text{PL}}_{i,G} (M_\Gamma)_c @>f>> \widetilde K_{i, G}^{\text{PL}} (M_\Gamma) \to\widetilde K^{\text{Top}}_{i,G} (M_\Gamma) \to \widetilde K_{i-1, G}^{\text{PL}} (M_\Gamma)_c @>f>> \widetilde K_{i-1,G}^{\text{PL}} (M_\Gamma). \] In the paper the forget control maps \(f\) are studied. The main theorem says for \(G\) of odd order that \[ \widetilde K_{i,G}^{\text{PL}} (M_\Gamma)_c @>f>> \widetilde K_{i,G}^{\text{PL}} (M_\Gamma) \] is a split monomorphism for \(i\leq 1\). The result is put into context with the equivariant rigidity conjecture.