On the LS-category of group homomorphisms (Q6063911)

Let \(\phi: \Gamma \to \pi\) be a homomorpism of groups. Mark Grant [mathoverflow.net/questions/89178/Cohomological dimension of a homomorphism] defined the cohomological dimension \(\operatorname{cd}(\phi)\) as the maximum of \(k\) such that there is a \(\pi\)-module \(M\) with the nonzero induced homomorphism \(\phi^*:H^k(\pi;M)\to H^k(\Gamma;M)\). Let \(\operatorname{cat}\) denote Lusternik-Schnirelmann category. Motivated by the well-known equality \(\operatorname{cd}(G)=\operatorname{cat}(G)\) for discrete groups (Eilenberg-Ganea), the authors conjecture that \(\operatorname{cd}(\phi)=\operatorname{cat}(\phi)\) for all homomorphism \(\phi\). They prove the conjecture for homomorphisms \(\phi: \Gamma \to \pi\) of any torsion free finitely generated nilpotent group \(\Gamma\). Goodwillie constructed a \(\phi\) that serves as a counterexample to the conjecture with non-finitely generated \(\Gamma\). The authors present a counterexample with geometrically finite groups.